Structure and Interactions of Lipid Bilayers(cid:2)

The cell is the fundamental unit in biology. Each cell is spatially defined by its cytoplasmic membrane. The structural basis for each membrane is lipid in bilayer form. Following this reductionist point of view, it is therefore not surprising that lipid bilayers have been much studied using a great variety of techniques.


Introduction
The cell is the fundamental unit in biology. E a c h cell is spatially de ned by its cytoplasmic membrane. The structural basis for each m e m brane is lipid in bilayer form. Following this reductionist point of view, it is therefore not surprising that lipid bilayers have b e e n m uch studied using a great variety o f techniques.
What is surprising is the large uncertainty for simple structural quantities that has been generated by the di erent studies. Let us consider the most studied of all bilayers, the one composed of the lipid DPPC in the fully hydrated, biologically relevant phase (L = uid(F)=liquid-crystalline) at T = 5 0 o C. V arious di raction and NMR studies have g i v en values for the interfacial area A per DPPC lipid that range from 56 A 2 to 72 A 2 1,2]. A most distinguished neutron di raction study suggested A = 5 8 A 2 3] while a m uch used x-ray method obtained A = 7 1 A 2 4]. The real uncertainty i n these numbers is even larger than the nominal 24% obtained by dividing one result by the other because A for DPPC in the low temperature gel (G) phase is A G D PPC = 4 8 A 2 1,5,6]. Therefore, the e ect of uidizing the DPPC bilayer (i.e., making it biologically relevant) should be de ned to be A F ;A G . Using the above di erences for the uid phase area of DPPC yields an enormous uncertainty i n A F D PPC ; A G D PPC -from 8 A 2 to 24 A 2 . E v en though one does expect to achieve as good precision in biophysics as in the physical sciences, this 100% level of uncertainty is ridiculous! Uncertainty i n A is directly related to uncertainty in the bilayer thickness. A common de nition of bilayer thickness is D B = 2 V L =A, where V L is the volume of a lipid molecule in the bilayer V L has been measured accurately (0:2%) by a n umber of groups 7]. The thickness of lipid bilayers (vide infra for discussion of various de nitions of thickness) is an important structural quantity for discussing the incorporation of intrinsic membrane proteins.
Molecular dynamics simulations give m uch i n s i g h t i n to lipid bilayer structure at a level of detail not available experimentally, but uncertainty i n A negatively impacts such s i m ulations. Some simulations are done with lipids and water in a simulation box of xed size, in which c a s e A is xed in the simulation. Results obtained from such s i m ulations performed at the wrong A will be misleading and could even lead the unwary simulator to vary interaction parameters in order to t other data, such as the NMR order parameters. Many simulations are now done with constant lateral pressure 8{10]. It is then in principle possible for the simulation to nd the correct value of A, but the computer time necessary to equilibrate can be large if the starting A is far from the equilibrated value. Furthermore, even if the simulation can be equilibrated, experimental uncertainty i n A reduces the ability t o t e s t t h e interaction parameters used in the simulation.
This chapter will review some of the e orts to obtain structural results for L phase lipid bilayers with an emphasis on recent w ork from our lab. To obtain some of these structural results we found that it was necessary to deal with the e ects of uctuations. This in turn led us to the issue of interactions between bilayers, which is the second topic that will be discussed in this chapter. Underlying both these e orts is the central role of uctuations.
Fluctuations are important in biology. The fact that the biologically relevant uid phase of lipid bilayers is the one with the largest uctuations supports this paradigm. Bilayers with greater uidity can seal leaks and tears more readily. Local uctuations in the lipid molecules a ect passive permeability of solutes through the membrane and can facilitate the function of intrinsic membrane proteins by transiently reducing activation energy barriers 11]. In addition to uctuations at the molecular length scale, there are also longer length scale uctuations that can be adaptive for cell shape changes. Longer wavelength uctuations give rise to an additional force between membranes, and these are the uctuations that degrade di raction data necessary to obtain structure. This chapter will focus upon these longer length scale uctuations. We will nish this general introduction with Fig. 1 which s h o ws one key piece of structural information from our lab 12,13], the volume per lipid V L . The temperature dependence in Fig. 1 indicates the various thermodynamic phases of DPPC. This chapter will focus on the most biologically relevant phase, identi ed in di raction studies as the L phase. There will be some use made of results for the gel phase, which w e believe is the best characterized of all the phases 5]. The chapter of Katsaras and Raghunathan 14] complements this chapter by focussing on the lower temperature phases, especially the subgel and the ripple phases.

Levels of Description of Structure
It is important to appreciate that it makes no sense to contemplate an atomic level structure at the sub{ A level for lipid bilayers. This is not because of poor di raction technique or sample preparation. Lipid bilayers have biologically vital uctuations. This means that atoms are not inherently localized. The proper description for the positions of atoms in the lipid molecule is that of broad statistical distribution functions. Fig. 2a shows simulations for distribution functions for several of the component groups of the lipid molecule along the direction of the bilayer normal 15]. The widths in this direction are of order 5 A. I n c o n trast, in the`in-plane' direction the distribution functions for the L phase are just constants because the lipid molecules are in a twodimensional uid phase. (Of course, one can still consider pair correlation functions, which are important for di use wide angle scattering, but this is a little explored area.) In contrast, for the lower temperature phases there is interesting and valuable in-plane structure 14,5].
Fluctuations in biologically relevant fully hydrated uid phase bilayers mean that x-ray di raction data can only yield electron density pro les like the one shown in Fig. 2b. The peaks in such electron density pro les are associated with the electron dense phosphate group and the lower electron density in the center is associated with the hydrocarbon region and especially with the low electron density of terminal methyl groups of the fatty acids. Therefore, electron density pro les con rm the usual picture of bilayer structure and they give another measure of the bilayer thickness, namely, the head{ head thickness, D HH . H o wever, electron density pro les do not yield the z coordinate of molecular groups along the bilayer normal. Such information has been obtained using neutron di raction, either with selective deuteration of various component groups (DPPC at 98% RH 3]), or combined with x-ray di raction (DOPC at 67% RH 16]).
The transverse description of the bilayer as a set of distribution functions along the z axis is valuable, but it does not include other important information, such a s A in the lateral direction, or volumes. Therefore, a complementary description of bilayer structure, shown in Fig. 2c Fig. 2c divides the volume V L of the lipid into two regions. The tail region is essentially a hydrophobic hydrocarbon chain region by de nition, it includes only the methylenes and terminal methyls on the fatty a c i d c hains. The head region is essentially a hydrophilic region, which includes the remainder of the lipid molecule (carbonyls, glycerol, phosphate and choline). An average structure is depicted by d r a wing two sharp boundaries, one between chains and heads and one between heads and water, as shown on the left side of Fig. 2c. In view of the uctuations shown in Fig. 2a, such sharp boundaries with all the chains on one side and the heads on the other are clearly arti cial, but it is still a valid representation in the sense that the sharp lines can be justi ed as Gibbs dividing surfaces 17]. Nevertheless, in the case of the interface between the headgroups and the water, it is useful to consider a re nement to the simple description on the left side of Fig. 2c. This re nement, shown on the right side of Fig. 2c, explicitly mixes the heads and water in the polar, interfacial region. This gives better correspondence with the distribution function description in Fig. 2a in particular, it gives a better representation of the steric thickness, de ned to be D 0 B .

Problem with the Gravimetric Method
A popular method for obtaining structural information 4,6] is most easily explained from the description shown on the left side of Fig. 2c. The total volume V L of one lipid molecule and its associated n w water molecules is AD=2, where D is the repeat distance that is easily and accurately measured by di raction on stacks of lipid bilayers. (Using synchrotron x-rays and a high resolution setup, we h a ve measured D with accuracy of 0:01 A, though reproducibility with nominally identical samples is usually not so good.) Therefore, AD = 2 ( V L + n w V w ) (1) where V L is the measured lipid volume 12], V W is the volume of water and n w is the number of water molecules/per lipid. The gravimetric method simply weighs the amount o f w ater and the amount of lipid to obtain n w . Then, A is obtained as a function of n w from Eq. 1. The procedure is then to vary n w and measure D. In principle, as n w increases towards full hydration, D increases until an excess water phase forms at the fully hydrated value of n w . Increasing n w further just adds to the excess water phase and D should remain constant.
While the concept of the gravimetric method is simple and elegant, it has been criticized and a number of studies have obtained di erent r e s u l t s 1,18{21]. So let us elucidate the aw. The gravimetric method assumes that all the water added to the system goes between the lipid bilayers that are neatly stacked in regular one-dimensional arrays. In fact, gravimetric experiments are performed on lipid dispersions consisting of multilamellar vesicles (MLVs). Such samples have m a n y defect regions. For example, it is customary to visualize MLVs as consisting of spheres of about 10 m diameter composed of stacks of nearly a thousand bilayers. As is well known, packing of spheres leaves defect volumes between the spheres that amount to about 26% of the total volume (for a nice schematic see Fig. 3 in 21]). Such defect volumes, which m ust be lled with water, escape detection by di raction, which focusses on the more ordered structure. Therefore, the value of n w that should be used in Eq. 1 should be smaller than the gravimetric value of n w because the total weighed water includes defect water that is invisible to di raction.
This artifact suggests that the gravimetric method will tend to overestimate A. Direct veri cation of this tendency for the gravimetric method to overestimate n w and A was given for the gel phase of DPPC, for which i nplane chain-packing and tilt angle were measured directly from wide angle di raction. This gave A G DPPC = 4 8 A 2 and n w = 12 1,5]. The results of the most recent g r a vimetric studies 4,22] gave n w in the range 17:5 ; 19 which would require A G D PPC to be in the range 52 ; 54 A 2 1]. The only exception we k n o w to this tendency of the gravimetric method to overestimate is for EPC where 23] obtained A F EPC = 6 4 A 2 using the gravimetric method which is smaller than our best value of A F EPC = 6 9 :4 A 2 24].
The gravimetric method also indicated that A increases strongly as the limit of full hydration is approached 6,21]. Indeed, A should increase in this limit. Recall that less than full hydration is equivalent to exerting osmotic pressure P on the water. The major e ect of osmotic pressure is to decrease the water space D w and thereby t h e D space. However, osmotic pressure also decreases A because this too extracts water from stacks of bilayers. The appropriate formula to describe this second e ect is 25] where A 0 is the fully hydrated area when P = 0 and K A is the phenomenological area modulus. However, while A should increase as full hydration is approached, Rand and Parsegian 25] realized that the changes in A obtained from the unadulterated gravimetric method became much too large near full hydration for the measured values of K A 26]. They then used gravimetric values of A obtained under osmotic pressure at 10 atmospheres and they used Eq. 2 to extrapolate to fully hydrated P = 0. This reduced the estimate of A F D PPC from 71 A 2 obtained from the unadulterated gravimetric method to 68:1 A 2 25]. However, this is still larger than the value obtained by a n alternative method that we n o w proceed to discuss.

Electron Density Pro le Method
The electron density pro le (z) for symmetric bilayers with a lamellar repeat spacing D is where for the di erent orders h > 0, h is the phase factor which can only assume va l u e s o f + 1 o r ;1. F h is the bilayer form factor which is routinely obtained from the intensity I h = F 2 h =C h under the di raction peak. C h is the Lorentz polarization correction factor for low angle scattering C h is nearly proportional to h 2 for unoriented MLV samples and to h for oriented samples. The zeroth order form factor F(0) is given by 2 7 ] AF (0) = 2(n L ; W V L ) = 2 ( L ; W )V L (4) where A is the area per lipid, n L is the number of electrons in the lipid molecule, V L is the lipid molecular volume and L n L =V L is the average electron density of the lipid molecule. The form factors F h involve a n u nknown scale factor, so only the absolute ratio r h = jF h =F 1 j of form factors are measured directly and this means that only relative electron density proles are routinely reported. Obtaining absolute electron density pro les will be discussed in Sec. 2.7.
The most reliable quantitative information that can be obtained from the electron density pro le is the headgroup spacing D HH , de ned to be the distance between the two peaks in the electron density pro le. In practice, four orders (h max = 4) su ce to give a reasonably accurate estimate of D HH . Recently, w e h a ve found that, even with four orders, the measured D HH should be corrected due to the limited number of Fourier terms 24,28,29]. This realization came by examining reasonable model electron density proles. The model we prefer 30], and that adequately represents the results of several simulations 31], employs a Gaussian function for the headgroup region and a Gaussian function for the terminal methyls on the chains as well as a constant for the methylene region. When we F ourier analyze this hybrid model and compare D HH obtained from the 4th order Fourier reconstruction with the model D HH , w e nd that there is a small error that systematically varies with D HH =D 5]. In our current use of this method (Yufeng Liu, unpublished), the parameters used to construct the hybrid model are obtained from the experimental form factors and so the hybrid model used for corrections to D HH is tuned to the particular lipid being studied.
McIntosh and Simon 32] introduced a method to use D HH to obtain A for the L phase. The idea is to compare the more poorly determined L phase with the much better determined gel phase and to use di erences to extrapolate from the gel phase structure to the L structure. Then, the L phase area A F is obtained in terms of the di erence in bilayer thickness D HH = D F HH ; D G HH , the measured lipid volume V F L and gel phase values for the hydrocarbon thickness D G C and headgroup volume V G H , This method was rst applied to DLPE 32] which w as a favorable rst choice for two reasons. The chains in DLPE are perpendicular to the bilayer in the gel phase, so gel phase quantities are easier to obtain than for PCs where the chains are tilted. However, we h a ve been able to obtain a structural determination of gel phase DPPC in the sense of Fig. 2c 1,5]. The second reason DLPE was more favorable than the PCs is that there were four orders of di raction for fully hydrated L phase DLPE, but not for DPPC, and we now turn to this major hurdle.

Why are there so few orders of di raction?
The immediate shortcoming of the electron density pro le approach i s t h a t fully hydrated samples of many lipids, such as unoriented DPPC dispersions in the L phase, have o n l y t wo robust orders of di raction. Electron density pro les using two orders of di raction are not quantitative, even for D HH . The generic explanation for so few orders is that uctuations and disorder reduce higher order intensities. However, to make sense of di raction data, it is necessary to understand that there are two quite di erent pieces to this explanation.
Most of the analyses of electron density and neutron scattering length pro les implicitly make the assumption that stacks of bilayers are one dimensional crystals with regular D spacing. Disorder and local molecular uctuations within each bilayer give rise to the broad distribution functions in Fig. 2a. Broad electron distribution functions, in turn, require that higher order terms in the Fourier expansion be small, so the higher order peak intensities are small. This point, which has been made forcefully by Wiener and White 33], is, however, only the rst part of the explanation for the absence of higher order peaks.
The second reason for the absence of higher order peaks is that stacks of lipid bilayers are not one dimensional crystals, but smectic liquid crystals. Smectic liquid crystals have large scale (long wavelength) uctuations that destroy crystalline long range order and replace it with quasi-long-range-order (QLRO) in which pair correlation functions diverge logarithmically instead of remaining bounded as in crystals. Because long range order is destroyed, Debye-Waller theory of scattering from crystals with uctuations is not appropriate (see appendix to 31]). Instead, QLRO c hanges the scattering peak shape from an intrinsic delta function by removing intensity from the central scattering peak and spreading it into tails of di use scattering centered on the original peaks. The magnitude of this shifting of intensity increases with increasing di raction order. For high enough order, the scattering peaks are completely converted to di use scattering even if the Fourier component for the local lipid bilayer is large. In-plane x-distance (Å) The preceding distinction between short range and long range uctuations can be summarized as follows. Short range uctuations are intrinsic to the single lipid bilayer. These are the uctuations that one sees in MD simulations. They correspond to disorder within a unit cell in a crystalline stack of repeat units. In contrast, long range uctuations are uctuations in the relative positions of the unit cells which m a y be thought of as the centers of the bilayer. These longer range uctuations do not change the distribution functions of molecular components relative to the bilayer center, so they do not a ect the structure of the single lipid bilayer.
Both kinds of uctuations reduce the intensity of the higher orders. The rst kind of uctuations are local and their reduction in higher orders faithfully re ects the true bilayer structure. In contrast, the reduction in intensity due to the second kind of uctuations at large length scale is an artifact that should be removed in order to obtain bilayer structure. This removal requires taking data with high instrumental resolution and then analyzing it using liquid crystal theory.
A v ery appropriate name for this method is \liquid crystallography". This name, however, should not be confused with the same name that has been used by Wiener and White 16] in a series of papers that introduced a di erent major innovation, namely, t h e j o i n t re nement method for combined x-ray and neutron di raction data. Wiener and White properly emphasized that the rst kind of molecular uctuations within each unit cell are intrinsic to liquid crystals. However, this rst kind of short range disorder is also present in highly disordered solids and no particular properties of liquid crystals appear in the Wiener and White analysis. It is the second kind of long range uctuations that requires an analysis speci cally tailored to liquid crystals and that we suggest should be called \liquid crystallography".

Liquid Crystallography
The beginning of liquid crystallography w as a remarkably succinct three page paper by Caill e 35] and communicated to the French Academy of Sciences by Guinier. That paper predicted power law tails for smectic liquid crystals (including stacks of bilayers) and it related the powers (exponents) to bulk phenomenological material properties, the bending modulus K c and the bulk compression modulus B the latter represents the interactions between adjacent bilayers in a stack. These predictions of the theory were later veri ed by highly precise experiments on general smectics 36] and later on lipid bilayers 37].
Readers of Guinier's ne book on di raction 38] will recall that, before Caill e's paper, Guinier had discussed the important distinction between disorder of the rst and second kind, where disorder of the second kind destroys crystalline long range order. Applied to a one-dimensional stack of bilayers, Guinier's theory is the same as the paracrystalline theory of Hosemann 39]. The Caill e theory also treats uctuations of the second kind, but it is considerably di erent from the earlier theories 38,39]. The earlier theories assumed that any disorder in the unit cell dimension propagated uniformly in the in-plane direction. This is clearly arti cial because bilayers can also undulate so the local water spacing can vary with in-plane coordinates (x,y). Another major distinction between the theories is that Caill e's is based on a realistic Hamiltonian model rather than the purely stochastic approach o f paracrystalline theory. H o wever, the Caill e theory is considerably more difcult to apply, and paracrystalline theory has been used in biophysics, so it was necessary to test whether Caill e theory really represents a signi cant improvement. Our group has documented the de nite superiority of Caill e theory for L phase DPPC bilayers 40]. On the other hand, for low temperature phases with smaller undulation uctuations, we h a ve found that the scattering peaks are broader and appear not to follow the Caill e form, but perhaps are dominated by frozen-in defects.
There are two main e ects of liquid crystallography. The rst is that the proportion of di use scattering to total scattering increases with order h.
Indeed, for high enough h the scattering is entirely di use and no central peak can be seen. The second is that the proportion of the scattering that is di use increases for all orders as the lipids become more fully hydrated.
Before the e ect of uctuations was understood, it was well known that more orders of di raction could be obtained by drying lipid bilayer stacks. For example, Wiener and White 16] obtained h=8 orders of di raction for DOPC at 67% relative h umidity. H o wever, drying the sample raises the spectre that the bilayer parameters one wishes to measure are changed. Indeed, they found A F67 DOPC = 5 9 :4 A 2 for DOPC 16] whereas we n d A F100 DOPC = 7 2 :2 A 2 for fully hydrated DOPC at 100%RH 29]. As mentioned in Sec. 2.2, the unadulterated gravimetric method generally gives quite large increases in A near full hydration. The method of electron density pro les would seem to agree that there were large structural changes if one interprets the data from a purely crystallographic viewpoint. The higher orders of di raction disappear and even the second order of di raction for DPPC systematically falls o the continuous transform obtained at 98% RH as the humidity is increased to full hydration 31]. However, liquid crystallography predicts these very same e ects, at least qualitatively.
To v erify that liquid crystallography predictions are quantitative requires considerably more e ort. The rst e ort, skillfully carried out by R. Zhang in our lab, was to improve the Caill e theory to give quantitative predictions, not just for the power laws, but also for the amplitudes of the scattering 41]. The ensuing modi ed Caill e theory enables us to predict the shapes of the scattering peaks for all orders using only a few parameters, primarily the average domain size L, which a ects the width of the central peak, and the Caill e 1 parameter 35], 1 is proportional to the mean square uctuations 2 in the water space and 1 governs the size of the scattering tails as well as the power law decay.
The second e ort was to obtain the peak shapes experimentally. W e use a silicon analyser crystal with instrumental resolution q = 0 :0001 A ;1 40]. However, with such high resolution, most scattered x-rays do not get to the detector, so a synchrotron source is helpful and we use the CHESS facility a t Cornell. We are able to measure su ciently far into the power law tails (before signal/noise becomes too small) so that we can obtain the 1 parameter. It might be noted that the classic way of obtaining power law exponents such as 1 is to use log{log plots 36,37]. This is di cult because the number of decades in q over which straight line behavior on a log{log plot can be seen is rather small. The small q range in the central peak is limited by t h e crossover to a regime dominated by the sample domain/correlation size L and the large q range is limited by signal/noise and is further degraded by continuous changes in the form factor F(q). In contrast, our method relies, not only on the power law behavior, but also on the larger amplitudes in the tails when 1 is larger. Once we h a ve obtained the parameters in the model, we can extrapolate the di use scattering that is in the tails of the structure factor S(q). Even though this extrapolated di use scattering intensity i s s o small that it can't be separated from background, the total amount o f i t i s large because it extends all the way b e t ween scattering peaks. Fig. 4 indicates the amount o f i n tegrated intensity that is recovered using this extrapolation. When this missing intensity is added, the result is that liquid crystallography does indeed predict the e ects in the preceding paragraph quantitatively, and the use of it enables more accurate form factors F h to be obtained that are true to the bilayer structure. Our former student, Horia Petrache, has made available a program to perform liquid crystallography data analysis. (Send e-mail to nagle@andrew.cmu.edu to obtain access.)

Structural Results
The method we h a ve been using to obtain structural results rst obtains uctuation corrected form factors for unoriented samples using liquid crystallography. Electron density pro les are drawn for those samples that have four orders of di raction. Such samples are typically under osmotic stress of 20-60 atmospheres, corresponding to relative h umidities of 95 ; 98%. To exert osmotic pressure we use the now classic method of Rand and Parsegian 25] with the polymer PVP or dextran. There is never a problem of choosing the correct phases for F h either plots of F(q) a t m a n y osmotic pressures or tting hybrid electron density models to the intensities always give u n a mbiguous phases, which are often (; ; +;). For PC lipids, we use Eq. 5 to obtain A as a function of osmotic pressure P . The reference phase that we have used in Eq. 5 is the gel phase of DPPC, for which headgroup volume V G H and hydrocarbon thickness D G C are accurately known from gel phase studies 1,5]. The volume V F L in the L phase is accurately ( 0:2%) measured 12]. The value of D HH used in Eq. 5 is obtained from the electron density pro les. Inspired by Eq. 2, we plot the ensuing values of A against AD w P, where the slope is ;1=K A and the intercept is the full hydration value A o .
To d o t h i s w e also need the Luzzati water thickness D w , w h i c h is obtained from the partitioning indicated in Eq. 1, namely, AD w = n w V w , w h e r e n w is now obtained directly from Eq. 1 and the value of A. (Note that Eq. 1 is valid, even if the gravimetric method of using it is not.) Fig. 5  In the case of DMPC, K A for giant unilamellar vesicles is 145 dyn/cm 26]. One concern is that this kind of measurement o f K A is in the expansion mode whereas our use of K A is in the compression mode. This issue was recently addressed using NMR as a function of osmotic pressure with nearly the same result K A = 141 dyn/cm 21]. This K A was combined with a g r a vimetric x-ray determination of A limited to osmotic pressures greater than 20 atmospheres (98%RH) to obtain A o = 5 9 :5 A 2 21]. In comparison, our unconstrained result K A = 108 dyn/cm is again somewhat smaller than the measured values, but this makes little di erence in our value of A o . C o nstraining the slope K A gives excellent agreement of our A o = 5 9 :4 A 2 with 21].
In for a non-in nite K A . If they had, our best A would probably increase to about 64 A 2 from our quoted value A = 6 2 :9 1:3 A 2 31].

Absolute Electron Density P r o l e s
Obtaining absolute electron density pro les requires information in addition to low angle scattering. Wide angle x-ray studies of the gel phase and volumetric studies as a function of temperature give the electron density of the lipid molecule V L and some of its component groups, especially the methylenes V CH2 and the terminal methyls V CH3 in the chains 7,12]. This kind of information is better used with the hybrid electron density model 30] than with the Fourier representation. Also, the hybrid model has the additional advantage over the Fourier representation in that data for many samples, including di erent osmotic pressures and D spacings if there is little change in structure, can be used simultaneously to obtain the best t.
If one ts any model to measured relative form factors, the model must contain an unknown scale factor K. O n e w ay to constrain K in the hybrid model is to require that the model have the correct value for the electron density in the methylene plateau. Another way is to require that the methyl trough be the correct size to account for the known de cit of electron density in the terminal chain methyls. Yet another way to constrain K is to require that the model has the value of F(0) that is obtained from V L and A using Eq. 4. (This latter way cannot work when F(0) is e ectively zero, which i s the case for many L phases, but even then F(0) still acts as a constraint o n the overall hybrid model parameters.) Although any one of these constraints should su ce in principle, in practice, when only one or two are applied, the others are then not satis ed. It is therefore best to use all three constraints simultaneously 31]. This is not surprising or disturbing because the low a n g l e di raction information is con ned to small q, corresponding to h = 4 , s o l o w angle x-ray information should be supplemented as much as possible by other information.
The preceding, somewhat strenuous, method of constructing electron density pro les has only been applied to the DPPC L phase 31]. It has also been applied to the L 0 phase, but with data only at full hydration 30]. Derivation of absolute electron density pro les for other PC lipids is based on a simple argument. Since the headgroups are the same, the integrated electron density under the headgroup peak in excess of the level due to water on one side, and hydrocarbon on the other, should scale inversely with the area A, and the prefactor can be determined from V H and the number of electrons in the headgroup 24].
It might also be noted that one could contemplate scaling the electron density pro les from simulations. However, di erent s i m ulations give rather di erent scaling factors (see Fig. 7 in 31]), so a more immediate use of absolute electron density pro les is to test simulations.

Interactions between Bilayers
From the preceding section, it is clear that long range uctuations of the second kind are really a nuisance for obtaining average structure of lipid bilayers in the highly uctuating, fully hydrated, biologically relevant L phase. From a structural point of view these uctuations have n o i n trinsic value. We n o w turn to a topic where these uctuations do have i n trinsic importance that is directly addressed by liquid crystallography.

Hard versus Soft Con nement Regimes
As was originally shown by Helfrich 43], long range uctuations are the cause of an e ective i n teraction between lipid bilayers that is called the uctuation interaction. The conceptual basis for this interaction is that two bilayers close to one another cannot uctuate as much a s t wo bilayers far from each other. Mutual suppression of independent uctuations leads to a decrease in entropy and an increase in free energy as the average water separation distance a is decreased, so this interaction is repulsive and entropic. It is an entropic energy (-TS) that is absent at absolute zero temperature, rather than a bare energetic interaction (E).
Helfrich showed that, when the only bare energetic interaction between bilayers is steric (excluded volume interaction), the form of the e ective u ctuation free energy is 43] f U = 0 :42 (k B T) 2 K c a 2 : This result has been con rmed experimentally in systems where the bare interaction V B (a) can be closely approximated as zero over most of the relevant range in water spacing a 44]. This regime is called the hard con nement regime because the bare potential can be thought of as con nement of each bilayer between hard walls formed by neighboring bilayers. Of course, the neighboring bilayers can also move, so the hard wall potential also uctuates, but this does not change the form of the uctuation interaction. Everything about the hard con nement regime is quite well established, except for the magnitude of the prefactor 45].
In general there are additional bare interactions besides the obvious steric interaction. If these interactions have ranges that are comparable to the average water spacing a, then the approximation of the bare interaction V B (a) by a hard box-like potential is obviously de cient. It is then appropriate to consider a soft con nement regime 46,47].
One important b a r e i n teraction is the strong repulsive h ydration force which, even though not so well understood, has been well documented experimentally 25,42] to have the form V hyd (z) = P h e ;z= (8) with parameters (decay length) and prefactor P h . Another important b a r e interaction is the van der Waals attractive i n teraction, V vdW (z) = ; H 12 1 z 2 ; 2 (z + D B ) 2 + 1 (z + 2 D B ) 2 (9) where D B is the bilayer thickness and H is the Hamaker parameter. This is the interaction assumed to be responsible for limiting the swelling in bilayers composed of lipids with no net charge. We will de ne a o to be the limiting water space for fully hydrated bilayers with osmotic pressure P = 0. Because a o is only 10 ; 30 A, a graph of bare potential V B (z) v ersus z on this length scale shows considerable variation, especially as z approaches 0. For charged lipids in low salt, one should also consider an electrostatic interaction, but this is absent for the neutral lipids. An additional very short range repulsion has been measured and attributed to protrusions 48]. We do not include it since it only plays a role for lipids under high osmotic pressure and small water space a. I t d o e s p l a y the formal role of suppressing the singularity i n the van der Waals potential at z = 0 .
It has been proposed for the soft con nement regime that the uctuation interaction free energy in Eq. (7) should be modi ed 47,46] and a formula involving an exponential with decay length fl f U2 = ( k B T= 16)(P h =K c ) 1=2 exp(;a= fl ) (10) has been o ered 26,47]. This exponential functional form is quite di erent from the power law form in Eq. 7 established for the hard con nement regime. Furthermore, the decay length fl was predicted to be twice the decay length 2 of the hydration force.
For lipid bilayers the now traditional way 2 5 ] t o i n vestigate interbilayer forces experimentally is to measure the average water space a as osmotic pressure P is varied such data are usually plotted as logP(a) as in Fig.  6. The data from many groups clearly show an exponential increase for P greater than 10 atmospheres and this is the experimental basis for the force that is named the hydration force. However, there are at least three energies involved with four parameters ( , P h , H and K c ). There are also uncertainties in how one de nes and obtains water space a (gravimetric D W versus steric D 0 W -see Fig. 2c) and bilayer thickness (D B vs D 0 B ) 7]. While it has been encouraging that ts to the P(a) data make sense with reasonable values for the parameters 49], there are too few data to provide ts that uniquely separate P(a) i n to its constituent forces. As noted by P arsegian and Rand 17], \... dissection of the measured pressure P into its physically distinct components is a problem almost as di cult as the theoretical explanation of these components themselves". In particular, the functional form of the uctuation pressure is an important assumption in carrying out such ts. Since the derivation of the soft con nement f o r m ula for uctuation pressure was non{trivial (involving some close self{consistency arguments that are di cult to improve upon and di cult to validate by analytic theory), it therefore seemed appropriate to test Eq. 10.

Experimental Window on the Fluctuation Force
We realized that our experimental study of the uctuation correction for structural studies also provided an experimental window on uctuational forces that could help reduce the ambiguity inherent in only using P (a) data this was a major part of Horia Petrache's research in our group. The most direct connection is that the uctuational free energy F fl is related to the Caill e 1 parameter 50] by Since the bending modulus K c is a property of the single, isolated bilayer, the functional form of F fl (a) can be obtained from 1 and D. A plot of 1= 1 D 2 therefore shows the functional form of F fl . Data for EPC are shown in Fig. 7. Data for DPPC, DMPC, EPC 50] and DOPC 29] are all inconsistent with the hard con nement functional form in Eq. 7, proving that a theory of soft con nement is necessary. The data are consistent with the prediction of the soft con nement theory that the uctuation free energy has an exponential decay with a. H o wever, the e ective decay length of the uctuation free energy, w h i c h is de ned to be fl , is consistently larger than the theoretical prediction fl = 2 as indicated in Fig. 7 for EPC. For the four lipids that we h a ve studied, the ratio fl = is in the range 2:5 ; 3.

Simulations
We undertook simulations to address several questions regarding interactions. The rst one, from the preceding subsection, concerns our experimental result that fl = is consistently greater than 2. This result could have b e e n d u e t o several reasons, including: (i) the soft con nement theory (Eq. 10) may n o t b e correct, (ii) the bare interactions may be incorrectly described or (iii) there may be experimental discrepancies. By doing a simulation with the same form of the interactions as in Eq. 8 and Eq. 9, we b ypassed (ii) and (iii) and tested (i) directly. The result of the simulation is that fl = is about 2. 4 34].
Although this is a bit smaller than the experimental ratio, it clearly agrees with the experimental conclusion that the ratio is larger than the value of 2 given by Eq. 10.
The simulations were at the nano-scale in which the bilayer was treated as an elastic continuum with a bending modulus K c and the interactions between bilayers consisted of the phenomenological van der Waals and hydration forces. This is the appropriate length scale for the long wavelength uctuation forces. In particular, atomic level molecular dynamics simulations are not able to probe this longer length scale because the system size and number of atoms becomes prohibitively large to equilibrate in reasonable computer times. Since motional dynamics are unclear at the nano-scale, the appropriate kind of simulation is Monte Carlo rather than molecular dynamics. Monte Carlo also has the advantage that sampling phase space can be done more e ciently than following equations of dynamical motion. Nikolai Gouliaev in our lab implemented a particularly e cient algorithm that employed changes in the Fourier coe cients of the bilayer positions so that rather large, multiple moves in real space could be made while changing the bending energy by small amounts. This method, which w e c a l l F ourier Monte Carlo, decreases the equilibration time by a factor of at least 30 when compared to the usual method of moving one local point on the bilayer at each step 51]. This allowed Gouliaev to extrapolate thermodynamic properties to stacks consisting of many larger bilayers by using sequences consisting of as many a s M = 3 2 b i l a yers in a stack, N 2 = 1 0 2 4 F ourier modes (equivalent to lattice sites) and lateral extents L up to 2800 A 34,51]. Fig. 3 shows a snapshot of a slice through a simulation of M=8 bilayers.
The thermodynamic quantities of greatest interest are the osmotic pressure P (a) and the root mean square uctuation (a) i n w ater spacing, both as a function of mean interbilayer spacing a. is simply related to the measured Caill e 1 parameter 50] by 2 = 1 D 2 = 2 : (12) The simulation results compare favorably with the analytic theory 47] for small a and when there are no van der Waals interactions, but the discrepancy grows as a approaches full hydration when P = 0 34,51]. These discrepancies are too large to ignore when trying to t data.
The simulations also allowed us to address another issue 34]. Our experimental analysis that determines the Caill e 1 parameter is based on liquid crystal theory which assumes that the interaction between bilayers is harmonic. However, the phenomenological potential consisting of the sum of van der Waals and hydration interactions is not harmonic. Furthermore, the analytic theory approximates the true potential self-consistently with a harmonic potential. Since the analytic theory shows discrepancies with the results of the simulations, the obvious question was whether the uctuational data analysis is also awed. However, the di raction line shapes depend upon the functional form of the pair correlation functions. The simulations show t h a t the functional form for the true potentials agrees well with that obtained from harmonic theory, thereby v alidating the data analysis. The problem with the analytic theory is that the self-consistency relation gives a di erent v alue for the harmonic modulus than the one that best describes the results of the simulations 34].

Determination of Interaction Parameters
An important goal now is to determine the values of the interaction parameters. The basic experimental approach 50] determined the decay length fl of the uctuation force and its magnitude up to a factor of the bending modulus K c . G i v en a value of K c , ts to the bare pressure P bare (a) = P (a) ; P fl gave w ell determined values for H Hamaker , and P h . H o wever, ts with different v alues of K c over the range spanned by literature values gave equally good ts, essentially because variations in H compensated for variations in K c whereas values of (about 2 A) a n d P h were robustly determined 50].
The basic experimental approach of the preceding paragraph used the uctuation data 1 (a) only to eliminate the e ective modulus B for interbilayer interactions and this throws away information about the absolute size of the uctuations. Simulations, however, give both P(a) and 1 (a). Requiring both to agree with the data is a stronger constraint o n t h e i n teraction parameters. Detailed ts of simulations and data have not yet been carried out for all the lipids our group has worked on. However, for DMPC at 30 o C the following parameter set ts both P (a) a n d 1 (a) fairly well 52]: H = 7 :13 10 ;14 erg, K c = 0 :5 10 ;12 erg, = 1 :91 A and P h = 1 :32 10 9 erg/cm 2 and it is clear that larger values of K c provide inferior ts. This value of K c agrees well with 53] and is smaller than the value given by 5 4 ] . T h e v alue of H is somewhat larger than preferred by 55]. However, more lipid systems should be studied before drawing de nitive conclusions for these parameter values.