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«THE MICRORHEOLOGY OF LIPID BILAYERS by TRISTAN HORMEL A DISSERTATION Presented to the Department of Physics and the Graduate School of the University ...»

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filling.png FIGURE 1. Space filling models for DPPC (top) and DOPC (bottom). Tail bonds are in the trans conformation, with the exception of the cis double bond at the ∆9 position in DOPC.

Images from Avanti Polar Lipids.

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TABLE 1. Lipid components of membranes in rat liver cells.

Taken from ref. [1] molar concentrations of up to 45%[12]. The enzymatic pathway that produces cholesterol is long and energy intensive, requiring over three dozen steps, including 18 steps to obtain cholesterol from a similar sterol, lanosterol[8]. In mammalian cell membranes, cholesterol seems to serve a number of purposes, including direct modulation of the activity of some proteins[13, 14].

Cholesterol also has a profound effect on membrane fluidity, and especially the phase behavior, of lipid bilayers.

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Because they are amphipathic, lipids in aqueous environments will form structures that simultaneously shelter hydrophobic regions from and expose hydrophilic regions to the surrounding solvent. The bilayer, oriented such that the acyl chains are sandwiched between two layers of polar headgroups, obviously has this property. But the bilayer is only one of several structures lipids form in aqueous solution[15]. Additionally, the bilayer itself is not organized in just a single way- there are in fact several bilayer phases. Collectively, these are termed lamellar phases, while any non-bilayer phase is nonlamellar4. What phases can form depend on the chemical identity of the lipid— even within bilayer forming lipids, like phospholipids, there is variation[16]. DPPC, for example, can form any lamellar phase, but some phospholipids with short chains in particular will not form some bilayer phases and will instead create micelles, a nonlamellar phase in which lipids pack into spheres with hydrophobic acyl chains pointing inward.

4 Another distinction often encountered is between unilamellar, as in giant unilamellar vesicle, and multilamellar phases. Multilamellar phases are usually considered nonlamellar.

FIGURE 3. Bond line (top) and space filling (bottom) models of cholesterol.

Images from wikipedia.

In general, both acyl chain length and polar headgroup size are important determinants of lipid phase behavior[17]. However, headgroup interactions are hard to modify[1], while liipids with different chain lengths are readily available. This means that in practice, it is often easier to study the effect of chain length and conformation on bilayer properties, an approach that we adopted in our experiments.

Bilayers transition between different lamellar phases due to changes in temperature, pressure, composition, tension and hydration. Of these, temperature is especially convenient to examine experimentally. Differential Scanning Calorimetry (DCM) can be used to determine the temperature of different bilayer phase transitions when other variables are held constant[18]; DCM studies can be combined with Nuclear Magnetic Resonance (NMR) or Fourier Transform Infrared spectroscopy (FTIR) to determine the conformation of the bilayer in the different phases[19].

The simplest lamellar phase behavior is seen in bilayers containing just a single component.

Such systems are therefore important models for investigating lipid phase behavior. Single component membranes exhibit one fluid, or liquid crystalline, and several sub-fluid gel and crystalline phases. Bilayers formed from DPPC, the most studied single component system5, will form all four phases[8].

In all of the gel and crystalline phases, saturated acyl chains are fully extended (i.e., the hydrocarbon chains contain only trans conformers; see Fig. 4). The lowest temperature state is the pseudocrystalline Lc phase. Here lipid motion is severely restricted. Lipids pack in a formation that is similar to their crystalline dehydrate. With increasing temperature, the bilayer enters the gel phase. This state has several polymorphisms, in each of which lipids slowly rotate.

One of the polymorphisms, confusingly, is termed the gel phase Lβ. Here,the lipids pack in to orthorhombic latice. In the tilted gel phase, Pβ, lipids tilt with respect to the bilayer normal.

As temperature is increased further, the bilayer will enter the Ripple Pβ phase. Here, the lipids exhibit faster rotation and are packed in a hexagonal latice. Lipids are displaced vertically, so that the bilayer no longer exists in a plane, but is rippled.[8] The main transition, called the chain melting transition, occurs when the highly ordered acyl chains of the gel phases melt; this creates the fluid Lα phase, also called the liquid crystalline or liquid disordered (LD ) phase. In contrast to gel phases, where acyl chains are extended, in the LD phase chains have both cis and trans bonds. For DPPC, with its 16 carbon chains, on average

3.9 bonds will be in the cis configuration[10]. The bonds most often form “‘kinks” by adopting a cis-trans-cis configuration[20]. These kinks, and the cis bond angles in general, have the effect of frustrating the hexagonal packing of the gel phases. As a result, lipids occupy a greater area, and are more free to rotate. All of this has the result of markedly increasing the fluidity of the membrane: the translational diffusion coefficient for lipids in a liquid crystalline bilayer is ≈ 1 µm/s2 [21], orders of magnitude faster than the equivalent value in a gel phase bilayer (≈ 1 × 10−2

- 1 × 10−3 µm/s2 )[22, 23].

Understanding the phase behavior of a single component bilayer will obviously be an important step in attaining a similar understanding for living membranes. Nonetheless, it is worth emphasizing that if a single component phospholipid bilayer membrane is to serve as a model of a biological membrane, it is an extremely reductionist one. At the very least, living membranes are replete with sometimes hundreds of different varieties of proteins[4] (which, in total, can outweigh the lipid components of living membranes[3]), and carbohydrates. Any of these might 5 DPPC is particularly well studied primarily for two reasons: (1) it readily forms bilayers, and so is easy to work with, and (2) it is of a headgroup type and chain length that are widely utilized in living cells.

FIGURE 4. Schematic.

Different bilayer phases, top to bottom: Psuedocrystalline, gel, tilted gel, ripple gel, and liquid disordered.

reasonably be expected to significantly affect the phase behavior of the constituent lipid bilayer.

Furthermore, cells employ not just a single species of lipid when building their membranes, but thousands[24]. It is therefore also important that we understand the phase behavior of lipid mixtures. In particular, two obvious questions are (1) do multicomponent bilayers form the same phases as single component membranes; and (2) how are different lipid species distributed in the bilayer?

These questions are actually related: multicomponent bilayers can, in the right conditions, phase separate into domains rich in a certain lipid species and depleted of another, creating heterogeneous bilayers with gel and fluid regions. An important case of this behavior is cholesterol-mediated, and involves a cholesterol dependent phase. This phase, termed liquid ordered (LO ), is in many ways intermediate between gel phases and the liquid disordered state.

It occurs at temperatures below the chain melting temperature (although it should be noted that cholesterol affects this temperature in the phospholipids in complex ways that depend on both headgroup structure and acyl chain length[17, 25]), and so acyl chains are extended. On the other hand lateral lipid diffusion more closely resembles the LD phase, though it is still slowed (6.9 µm2 /s in the LD phase[21] vs. 3.3 µm2 /s in LO phase[26] for DMPC at 43◦ C, for example). Cholesterol also affects acyl chain order and lipid packing above the melting transition temperature; lipid tails appear more ordered in the presence of cholesterol[27]. Finally, even when phospholipids will phase separate in the presence of cholesterol, the literature is somewhat unclear with respect to the location of cholesterol itself. There is broad agreement that living membranes contain LO domains rich in cholesterol[28], but at least in two PC component bilayers with cholesterol the cholesterol doesn’t seem to have a preference for LO or LD domains[22].

Multicomponent, phase separated bilayers are an especially important model systems for experimentalists. They have an instructive appeal, as they represent a midway point between gross complexity of cell membranes, from which it is difficult to draw organizing principles, and single-component or homogeneous bilayers, from which biological inference is questionable.

More importantly, they possess an immense experimental utility. Fluorescently labled lipids can themselves partition in to a particular lipid phase, and when included in small quantities in model membranes they provide a convenient means through which phase separation can be viewed[29]. A number of experiments utilize these visibly phase separated domains[30–33]. Particularly relevant is that, since the domains undergo Brownian diffusion, they can be used to measure membrane viscosity[34, 35]; see chapter 4.

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Now I will take a larger view, and consider the bilayer on a scale at which the motions and conformations of individual molecules are smeared out. That is, I want to investigate a description of bilayer fluidity that relies not on the organization of lipid molecules, but instead is parametrized by fluid properties such as viscosity. I will focus mostly on diffusion of objects embedded in membranes. There are two reasons that diffusion is an especially important in this context: (1) living organisms make extensive use of diffusion, and so the phenomenon is of intrinsic biological interest, and (2) measurements of diffusion can be used determine bilayer fluid properties. This second point is the motivation for microrheological experiments, and the logic is the reason that we are able to engage in such measurements in the first place.

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We begin by considering a suspension of particles of mass m in a homogeneous fluid6. The system is in equilibrium, and acted on by a gravitational force in the z direction Fz = mg. We’ll begin by writing the continuity equation

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where ρ (r, t) is the particle density at r at time t, and j (r, t) is the particle flux. In a steady ∂ρ state, = 0, so we only need concern ourselves with the particle flux. This term can be broken ∂t into parts: a deterministic portion due to particle sedimentation, and a stochastic portion due to Brownian motion. To obtain the former, we can balance forces to get the sedimentation speed

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6 This derivation follows my notes from one of John Toner’s classes.

where b is the drag on the particles. We’ll calculate the drag on particles in membranes in the next section, but for now we can treat it as a constant7. Then the deterministic flux is just

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7 In principle, the drag b will depend on properties of the fluid and the suspended particles, but for this derivation none of these quantities will vary.

FIGURE 5. Setup for calculating the Stokes drag on a circle embedded in a 2D fluid.

The fluid moves with velocity U far from the inclusion. At the edge of the inclusion, the fluid is stationary.

(2.9) is called the Einstein relation. The essential feature that makes (2.9) so useful is that it provides a connection between fluctuations (of which Brownian motion is an example) and energy dissipation (drag). In this respect it is just one of several instances of the fluctuationdissipation theorem, a subject that has received considerable theoretical attention[36].

Experimentally, it suggests a method of relating an observable phenomenon (the random motion of a particle) to underlying features of the material in which it is embedded. This connection forms the basis of passive microrheological techniques. However, if we are to use (2.9) to measure fluid properties, we still must connect the drag to these fluid properties. In principle, this is simple, and the drag on a sphere in three dimensions, for instance, takes a satisfyingly simple form. For an object embedded in a two-dimensional membrane, however, the calculation is problematic.

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The drag we wish to calculate is the translational drag8 on a cylinder of radius a embedded in a two dimensional surface, as shown in Fig 5.

8 A similar derivation for the rotational drag on a cylinder in a 2D membrane does not encounter the difficulties that will become evident below.

The incompressible fluid in which the cylinder is embedded moves with velocity u in one direction at a steady rate. To find the drag, we can begin with the equations of motion for the

system, which are given by the Navier-Stokes equations for an incompressible fluid:

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where ρ is the fluid density, η is the fluid viscosity, and g is any body accelerations acting on the fluid (for instance from gravity). (2.10) is notoriously difficult to work with in this general form, but we can get considerable mileage by making simplifications appropriate to the problem at hand. Specifically, we won’t consider forces acting on the fluid, so we can ignore the final term in (2.10).

A less trivial simplification involves rewriting (2.10) in a dimensionless form. The motivation for such an action is that we can sometimes construct dimensionless quantities that are small out of the lengthscales inherent to the problem. For instance, the Reynolds number

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where L is a characteristic length, represents the ratio of inertial to viscous forces. For cells and their associated machinery, Re is always small9. So if we are trying to derive drag on a cylinder in an cell membrane from (2.10), we can gain some traction by defining dimensionless versions of the

variables appropriate to the given scales:

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9 It almost goes with out saying that, since we are talking about biology, there are exceptions to this “universal” rule.

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