## Abstract

A variety of observations—sometimes controversial—have been made in recent decades when attempting to elucidate the roles of interfacial slip on tracer diffusion in phospholipid membranes. Evans–Sackmann theory (1988) has furnished membrane viscosities and lubrication-film thicknesses for supported membranes from experimentally measured lateral diffusion coefficients. Similar to the Saffman and Delbrück model, which is the well-known counterpart for freely supported membranes, the bilayer is modelled as a single two-dimensional fluid. However, the Evans–Sackman model cannot interpret the mobilities of monotopic tracers, such as individual lipids or rigidly bound lipid assemblies; neither does it account for tracer–leaflet and inter-leaflet slip. To address these limitations, we solve the model of Wang and Hill, in which two leaflets of a bilayer membrane, a circular tracer and supports are coupled by interfacial friction, using phenomenological friction/slip coefficients. This furnishes an exact solution that can be readily adopted to interpret the mobilities of a variety of mosaic elements—including lipids, integral monotopic and polytopic proteins, and lipid rafts—in supported bilayer membranes.

## 1. Introduction

Singer & Nicolson [1] proposed the fluid-mosaic model of cell membranes as comprising a two-dimensional fluid embedded with proteins and functionalized lipids. Lateral diffusion of these mosaic elements facilitates biological signalling and controls transport into and out of the cell. Many studies have therefore sought to interpret the lateral diffusivity [2–5]. The first theoretical model was proposed by Saffman & Delbrück [6], and this remains the cornerstone of continuum hydrodynamic membrane transport models. It considers a flat membrane supported on both sides by solvent half-spaces. Treating the membrane as a two-dimensional Newtonian fluid and the tracer as a cylinder that spans the membrane thickness, the dimensionless ratio of the tracer velocity *U* to the force *F*, otherwise termed the dimensionless mobility 4*πη*_{m}*h*|*U*/*F*|, depends on a single dimensionless parameter *Λ*=*ηa*/(*η*_{m}*h*). Here, *η* is the solvent viscosity, *η*_{m} is the membrane viscosity, *a* is the tracer radius and *h* is the membrane wall thickness. Saffman & Delbrück [6] derived an analytical solution that is valid when *Λ*≪1. The key conceptual result was that the drag force can never become independent of the solvent viscosity, no matter how small *Λ* might be.

The Saffman–Delbrück theory has not been without controversy. For example, although experiments and simulations of Peters & Cherry [7] and Weiss *et al.* [8], respectively, have supported the theory, Gambin *et al.* [9] have demonstrated the diffusion coefficients of transmembrane proteins to scale with the reciprocal (rather than logarithmic) radius, suggesting—among other possibilities—non-continuum influences due to the small protein size. Nevertheless, the Saffman–Delbrück theory has also motivated computational and asymptotic methods to solve the full model for all values of *Λ* [10]. Shortcomings were revealed by experiments in which membranes are in close proximity to supporting substrates. This prompted Evans & Sackmann [11] to develop a model in which the membrane experiences a drag force from frictional coupling to a proximal wall, furnishing a formula for the dimensionless tracer mobility that depends on a single dimensionless parameter *b* is the phenomenological membrane-wall friction coefficient, which yields a shear traction exerted by the support on the membrane ** t**=−

*b*

*u*_{m}, where

*u*_{m}is the membrane velocity. For a thin solvent film sandwiched between the membrane and a solid support,

*b*=

*η*/

*δ*with

*δ*being the (apparent) lubrication-film thickness.

Interestingly, Evans & Sackmann showed that their model could also mimic the Saffman–Delbrück model (for any *Λ*) with a suitable choice of the membrane-support friction coefficient; this was achieved for *ϵ*^{2}-term) furnished close correspondence with the calculations of Hughes *et al.* [10]. Figure 1 compares these approximations (lines) with numerically exact solutions (circles) of the full membrane model; note that while the Evans–Sackmann model does not agree exactly with the full model at any value of *Λ*, it furnishes a convenient approximation with an error that is small compared with typical experimental uncertainty.

Stone & Ajdari [13] undertook numerical calculations to address the degree to which the Evans–Sackmann model, based on its phenomenological friction coefficient, can mimic supporting fluid layers that have a finite thickness *H*. They showed that the Evans–Sackmann equation is valid when _{1} and ℓ_{2}. These calculations identified regions in the parameter space where the Evans–Sackmann equation can be applied to model hydrogel-supported membranes. Close correspondence was established when the friction lengths *δ*_{1} and *δ*_{2} in the Evans–Sackmann model are set equal to the respective Brinkman screening lengths. As demonstrated by the comparison in figure 1, setting *ϵ*^{2}=*Λ*(*a*/*δ*_{1}+*a*/*δ*_{2}) in the Evans–Sackmann equation furnishes an excellent approximation to computations [12] for the small values of *Λ* where it also approximates the full Saffman–Delbrück model.

Motivated by recent interest in the interface between cell membranes and soft supports, such as gels and extra-cellular matrices [14], Wang & Hill [12] proposed a dual-leaflet model by which friction between the leaflets and between the tracer and the distal leaflet are accounted for using phenomenological friction coefficients. Such a model would help to distill the many factors [9,15] that control the lateral mobility of integral monotopic membrane tracers and nanoparticulates that bind to bilayer membranes. Lipids are in this category, although the degree to which a continuum-membrane approximation might hold for such small tracers is uncertain, in part because no such continuum model has been available to address the dual-leaflet characteristics and the frictional coupling of the tracer, leaflet and support elements [9]. Nevertheless, hydrodynamic friction coefficients are widely used—often with great practical utility—for modelling electromigration, and ion and solute diffusion. As a simple example, we note that the self-diffusion coefficient of a water molecule, assigned a radius *a*≈0.1 *nm* and bulk shear viscosity *η*≈0.001 Pa s, is *D*=*k*_{B}*T*/(6*πηa*)≈2×10^{3} μm^{2} s^{−1}, which is (perhaps fortuitously here) within 3% of the most precise measurements, e.g. *D*≈2299±5 μm^{2} s^{−1} [16].

The role of inter-leaflet friction membrane diffusion was first studied by Merkel *et al.* [2]. For flat membranes supported on glass, their FRAP experiments did not reveal differences between the mobilities of lipid tracers in the proximal and distal leaflets. However, using NMR, Hetzer *et al.* [17] found the diffusion coefficients of lipids in the proximal leaflet of bilayer-coated silica spheres to be ≈50% lower than in the distal leaflet. They attributed the difference with respect to planar bilayers on glass as being due to the higher curvature and stronger interaction with the support. Zhang & Granick [18] introduced fluorescence correlation spectroscopy [19] to study diffusion in supported bilayer membranes, distinguishing the proximal and distal leaflets by quenching fluorescence in the distal leaflet with iodide. These experiments furnished diffusion coefficients in proximal leaflets ≈10% lower than in the distal leaflet, suggesting that leaflet–leaflet coupling is stronger than leaflet–substrate coupling. In another study, Zhang & Granick [20] adsorbed polymer onto the distal leaflet, finding that the lipid mobilities in the proximal and distal leaflets adopted fast and slow modes equally. Recently, Scomparin *et al.* [21] have further elaborated the significant roles that membrane preparation (Langmuir–Schaefer, Langmuir–Blodget and vesicle-fusion deposition) and substrate (glass and mica) can have on the degree of leaflet coupling. The foregoing literature also acknowledges the complicating influence of flip-flop, i.e. the transfer of lipids between leaflets.

Attempts to measure the inter-leaflet friction coefficient *b*_{12} have found it to vary substantially with the lipid and temperature. Based on lateral diffusion coefficients of tracers NBD-DMPE and NBD-DOPE in DMPC and DOPC bilayers (one leaflet was covalently or electrostatically bonded to glass), Merkel *et al.* [2] showed that leaflet friction increases with the degree of hydrocarbon chain inter-digitation and decreases with temperature. Their pioneering experiments furnished *b*_{12} in the range 10^{4}−10^{9} Pa s m^{−1} with the membrane-support/wall friction coefficient *b*_{mw} in the range 10^{4}−10^{6} Pa s m^{−1}. Letting *b*_{mw}=*η*/*δ*_{mw} with *η*=0.001 Pa s furnishes an effective lubrication-film thicknesses *δ*_{mw} in the range 1−100 nm. From molecular dynamics simulations, den Otter & Shkulipa [22] found the inter-leaflet friction coefficient to be much less sensitive to the tail length, reporting *b*_{12} in the range 10^{6}−10^{7} Pa s m^{−1}. Finally, from pressure-driven flow in microfluidic channels, Jönsson *et al.* [23] ascertained

Recently, Camley & Brown [24] applied the method of regularized Stokeslets to compute the translational and rotational diffusion coefficients of complex objects in free and supported lipid bilayer membranes. They focused on the roles of shape anisotropy and leaflet structure, also demonstrating agreement with the Evans–Sackman theory for circular objects when the sub-phase is sufficiently thin. In the regularized Stokeslet approach, tracers are represented by a collection of many hundreds to discrete blobs. While the continuum model developed in this paper is limited to circular domains with a thin supporting phase, it has an advantage of furnishing an exact analytical solution that also accounts for *all* possibilities of slip among the tracer, support, and the proximal and distal leaflets.

To improve the quantitative utility of continuum hydrodynamic models at small (nanometric) scales, tangential interfacial slip is often modelled using the Navier-slip condition, which carries a slip-length parameter *δ*_{N}. Thus, when the slip length is much larger than the tracer radius, perfect slip results, whereas a vanishing slip length recovers the familiar no-slip condition. Navier-slip conditions have been adopted in many fields of research (see [25–27] for recent examples), including membrane diffusion [28], and generally has a minor impact on the low-frequency hydrodynamic drag. In this paper, we focus on inter-leaflet, leaflet–substrate, and tracer–substrate slip. However, because tracers can represent large lipid assemblies that are rigidly bound by associations with monotopic integral membrane proteins, polymers, cholesterol inclusions, peripheral proteins and membrane-anchored nanoparticles, we also address friction between a monotopic tracer and its distal leaflet.

Accordingly, we derive an analytical solution of the dual-leaflet model proposed by Wang & Hill [12]. The tracer may span one leaflet, with frictional coupling between (i) the tracer and its distal leaflet, (ii) between the two leaflets, and (iii) between each leaflet and a rigid support. We show that this model also furnishes a dual-leaflet model in which the tracer spans both leaflets. In this paper, we focus mainly on the lateral mobility of monotopic tracers/lipid assemblies that adopt a proximal or distal configuration, also considering (in the concluding discussion) the effect of slip at the tracer–leaflet peripheral interface and the role of inter-leaflet slip on transmembrane tracer mobility.

## 2. Theory

The principal components of the dual-leaflet membrane model are illustrated in figure 2. Figure 2*a* identifies tracers in either the top or bottom leaflet of a flat bilayer that is sandwiched between flat supports. Between the bilayer leaflets and their respective supporting wall are depicted fluid layers, the thickness of which ultimately specifies the respective leaflet–wall friction coefficients (*γ*_{1} and *γ*_{2}). We will apply the model to physical situations in which the leaflet–wall coupling is much weaker for one of the two leaflets, as illustrated in figure 2*b*(i),(ii). In figure 2*b*(i), the tracer is in what we will refer to as the proximal configuration, whereas in figure 2*b*(ii) the tracer occupies the distal configuration. Similarly, leaflets will be referred to using the terms proximal and distal to identify the leaflet with the largest or smallest, respectively, leaflet–wall friction coefficient. By considering each leaflet as a two-dimensional viscous fluid, we calculate the proportionality between the force and translational velocity of the cylindrical tracer. The reciprocal of this friction coefficient is termed the mobility, which is directly proportional to the experimentally measurable lateral diffusion coefficient.

Without loss of generality, we consider a tracer embedded in the leaflet denoted number 2 of a bilayer membrane, as depicted in figure 2. For the leaflet denoted 1 where *r*>*a*, the leaflet momentum and mass conservations equations (** u** and

*p*denote the velocity and pressure) are

*r*>

*a*

*r*<

*a*will be addressed separately below. Note that, for notational convenience,

*η*

_{m}is the (three-dimensional) membrane shear viscosity, and the friction coefficients

*γ*

_{1},

*γ*

_{2}and

*γ*

_{12}have been scaled to have dimensions of squared reciprocal length. In this manner, each term in the equations above has units of force per unit volume. For these reasons, some care is required when integrating the tractions to obtain the force.

Friction arising from protruding tracer element is accounted for by the tracer–wall friction coefficient *γ*_{tw}, since rigorously accounting for such protrusions clearly presents significant mathematical challenges. Nevertheless, the friction coefficients *γ*_{1} and *γ*_{2} can be rigorously linked to a lubrication flow—and thus to the film thickness and fluid viscosity—between the respective leaflets and their solid support [11]. Replacing a solid support by a hydrogel with Brinkman permeability ℓ^{2} (square of the Brinkman screening length) permits the lubrication film thickness to be replaced by ℓ [12].

Denoting the (two-dimensional) membrane viscosity of Evans and Sackmann by *b*_{i}≡*η*_{m}*hγ*_{i} and *b*_{i}=*η*/*δ*_{i}, where *η* is the solvent shear viscosity, and *δ*_{i} is a solvent film thickness parameter, then *γ*_{i}*a*^{2}=*Λ*(*a*/*δ*_{i}) or *Λ*=(*η*/*η*_{m})(*a*/*h*) is the well-known dimensionless parameter that governs the mobility in the Saffman–Delbrück (*Λ*≪1) and Hughes *et al.* (arbitrary *Λ*) membrane diffusion models. The foregoing equalities are summarized in table 1 for convenient reference.

The boundary conditions are as follows. For a tracer in leaflet 2, the Navier-slip condition is *δ*_{N} is the Navier-slip length, *t*_{2}(*r*=*a*) is the leaflet traction (*t*_{i}=*T*_{i}⋅*e*_{r} with *T*_{i}=−*p*_{i}** I**+

*η*

_{m}[

**∇**

*u*_{i}+(

**∇**

*u*_{i})

^{T}] the leaflet stress) and

**−**

*I*

*e*_{r}

*e*_{r}is the tangential operator with

*e*_{r}the radial unit vector. In the far-field,

**′ the membrane velocity in leaflet 1 where**

*u**r*<

*a*,

**′(**

*u**r*=0) must be finite, and the velocity and traction at

*r*=

*a*must be continuous, i.e.

**′(**

*u**r*=

*a*)=

*u*_{1}(

*r*=

*a*) and

**′(**

*t**r*=

*a*)=

*t*_{1}(

*r*=

*a*). Note that this dual-leaflet model for a tracer spanning just one leaflet reduces to a dual-leaflet model for a cylindrical tracer that spans both leaflets. In this case, the velocity in each leaflet needs to satisfy only the Navier-slip condition at

*r*=

*a*, i.e.

### (a) Leaflet velocity disturbances and force

Taking the curl of equations (2.1) and (2.2) furnishes
*ω*_{1}(*r*,*θ*) and *ω*_{2}(*r*,*θ*) are the scalar magnitudes of the vorticity [*ω*_{1}=*ω*_{1}(*r*,*θ*)*e*_{z} and *ω*_{2}=*ω*_{2}(*r*,*θ*)*e*_{z}] in leaflets 1 and 2, respectively. These equations can be decoupled by arranging as
_{1} and λ_{2} of the symmetric matrix ** S** are given by the characteristic equation

**is symmetric) when the scaling factors**

*S**e*

_{1}and

*e*

_{2}are selected to ensure |

*e*_{1}|=|

*e*_{2}|=1. Now writing

**=**

*ω**ϕ*

_{1}

*e*_{1}+

*ϕ*

_{2}

*e*_{2}gives

*e*_{1}and

*e*_{2}are orthogonal,

For leaflet 1 sandwiched between the tracer and the distal wall, the equations of motion where *r*<*a* are
*γ*_{tm} is the friction coefficient acting between the tracer and membrane. Again, taking the curl gives

Since
*ξ*_{i} can be written [29]
*r*>*a*,

Similarly, for leaflet 1 where *r*<*a*, the velocity
*h*_{1i}, *h*_{2i}, *c*_{1i} and *c*_{2i}, etc.) of the general solution are easily calculated to satisfy the boundary conditions (see the electronic supplementary material).

Finally, the frictional force on the tracer from the proximal wall is
*T*_{2} is the leaflet stress with *h* d*l* the differential surface area.

Evaluating the foregoing integrals from velocity and pressure fields in the form ** u**=

*A*(

*r*)

**+**

*U**B*(

*r*)

**⋅**

*U*

*e*_{r}

*e*_{r}and

*p*=(

*Dr*+

*Cr*

^{−1})

**⋅**

*U*

*e*_{r}furnishes

*A*=

*A*

_{1}(

*κ*

_{1},

*r*)

*e*

_{21}+

*A*

_{2}(

*κ*

_{2},

*r*)

*e*

_{22}and

*B*=

*B*

_{1}(

*κ*

_{1},

*r*)

*e*

_{21}+

*B*

_{2}(

*κ*

_{2},

*r*)

*e*

_{22}, and

*C*

_{2}=−

*η*

_{m}{

*h*

_{21}[

*γ*

_{12}

*e*

_{21}+

*γ*

_{2}

*e*

_{21}−

*γ*

_{12}

*e*

_{11}]+

*h*

_{22}[

*γ*

_{12}

*e*

_{22}+

*γ*

_{2}

*e*

_{22}−

*γ*

_{12}

*e*

_{12}]}, where, recall,

*e*

_{21}=(

*γ*

_{1}+λ

_{1})/

*γ*

_{12}and

*e*

_{22}=(

*γ*

_{1}+λ

_{2})/

*γ*

_{12}.

The total force acting on the tracer from friction and the membrane traction is
*γ*=|*F*/*U*| is termed the friction coefficient (*γ*^{−1} is the mobility). According to the Stokes–Einstein relation, the translational diffusion coefficient is *D*=*k*_{B}*T*/*γ*. Following Evans & Sackmann, we will examine the scaled mobility 4*πη*_{m}*h*/*γ*=4*πη*_{m}*hD*/(*k*_{B}*T*).

### (b) Parametric analysis

Dimensionless analysis of the foregoing mathematical model (taking *δ*_{N}=0) identifies the following independent dimensional parameters that affect the friction coefficient: *γ*_{1}, *γ*_{2}, *γ*_{12}, *γ*_{tm}, *γ*_{tw}, *η*_{m}, *h* and *a*. These eight parameters, which involve three dimensions, yield the following six independent dimensionless parameters: *h*/*a*, *f* is independent of *h*/*a*. Further reductions in the parameter space come when applying the model to bilayers that are separated far from one of the walls by a low-viscosity fluid. Thus, for tracers embedded in the leaflet that is proximal to the solid support, *ϵ*_{tm}≈*ϵ*_{12} and *ϵ*_{tw}≈*ϵ*_{2}, in which case we have the low-dimensional models

It is expedient to compare the dual-leaflet model with the theory of Evans & Sackmann [11]
*h* (also contained in *Λ*) denotes the *tracer* and *membrane* thickness. Note that we apply this formula to monotopic tracers taking *h* to be the *tracer* and *leaflet* thicknesses. Moreover, we set the tracer–wall friction coefficient in equation (2.6) *b*_{w}/*b*_{s}=(*ϵ*_{tw}/*ϵ*_{2})^{2}=*δ*_{2}/*δ*_{tw} for proximal tracers, and *b*_{w}/*b*_{s}=0 for distal tracers. Thus, for very weakly coupled leaflets (*πη*_{m}*h*/*γ* for a proximal monotopic tracer (*ϵ*≈*ϵ*_{2} because *δ*_{2}≪*δ*_{1}) should take the Evans–Sackmann value, because the tracer and its leaflet (each with thickness *h*) interact principally with the solid support. For very tightly coupled leaflets (*ϵ*≈*ϵ*_{1} because *δ*_{1}≪*δ*_{2}) should approach half the Evans–Sackmann value, because now the tracer interacts with the solid support via a membrane that has an effective thickness 2*h*. In §4, we present a formula that generalizes equation (2.6) to account for slip between the tracer and membrane.

## 3. Results

The following results focus on the roles of inter-leaflet and substrate friction, so the influence of slip between the tracer and its leaflet is deferred to §4, where we present dimensional self-diffusion coefficients. First, we assess the order-of-magnitude of the parameters that can be achieved in experiments. Camley & Brown [24] identify an inter-leaflet friction coefficient *b*_{12}≈2×10^{8} Pa s m^{−1} with (two-dimensional) membrane viscosity *hη*_{m}≈2×10^{−10} Pa s m. Thus, with a nominal bilayer thickness 2*h*≈5 nm [30–32], we find *η*_{m}≈0.08 Pa s and *a* measured in nanometres. Note that a characteristic slip length in a solvent with the viscosity of bulk water (*η*≈0.001 Pa s) *δ*_{12}=(*η*/*η*_{m})/(*γ*_{12}*h*)∼0.07 nm. Thus, the inter-leaflet friction coefficient can be considered much greater than for aqueous supporting films with nanometre thickness. Pan *et al.* [33] measured a DOPC bilayer thickness 2*h*≈3.7 nm, which is somewhat smaller than the value we have adopted to analyse the characteristic scales. The magnitude of the tracer–leaflet coefficient is presently unknown; we will consider, for simplicity, that it has the same magnitude as the inter-leaflet friction coefficient, but we will also explore the consequences of it being asymptotically large and small.

Next, considering the friction coefficient acting between a leaflet and a solid support, when the gap is mediated by a solvent film with thickness *δ*_{i} and bulk viscosity *η*, we have *h*≈5 nm and *η*_{m}≈0.08 Pa s, we find *a* and *δ*_{i} are measured in nanometres. For lipids with *a*≈0.5 nm in membranes supported on a film with *δ*_{i}≈1 nm, for example, it follows that *ϵ*_{i}≈0.04. On the other hand, for large (i.e. micrometre-sized) raft domains, *ϵ*_{i}≈70.

Finally, as highlighted by Evans & Sackmann [11], the order of magnitude of the friction coefficient appropriate for leaflets that contact a solvent half-space with viscosity *η* is obtained by setting *γ*_{i}*a*^{2}=*Λa*/*δ*_{i} where *h*≈5 nm, we find *ϵ*_{i}≈0.005*a* when *a* is measured in nanometres.

For convenient reference, the foregoing results are summarized in table 2 with several of their ratios that are independent of the tracer radius. Note that the parameters can span three orders of magnitude, and that *ϵ*_{i} for supporting films can span as many as five orders of magnitude, depending on the tracer size and film thickness.

### (a) Reduced proximal and distal problems: *f*_{p}(*ϵ*_{2},*ϵ*_{12};*ϵ*_{1}≪1,*ϵ*_{tm}=*ϵ*_{12},*ϵ*_{tw}=*ϵ*_{2}) and *f*_{d}(*ϵ*_{1},*ϵ*_{12};*ϵ*_{2}≪1,*ϵ*_{tm}=*ϵ*_{12},*ϵ*_{tw}=*ϵ*_{2}).

Here, we consider the minimal parameter space in which *ϵ*_{tm}=*ϵ*_{12} with *ϵ*_{tw}=*ϵ*_{2}. The scaled mobilities are presented as functions of the scaled inter-leaflet friction coefficient *ϵ*_{12} for several fixed values of the respective leaflet–wall friction coefficient, *ϵ*_{2} (proximal) or *ϵ*_{1} (distal) in figure 3. Here, the particle radius can be considered fixed, so, as expected, mobilities decrease with increasing inter-leaflet friction. While both the proximal and distal configurations plateau to the Evans–Sackmann mobility when the inter-leaflet friction coefficient *ϵ*_{12} is large or small enough, the proximal and distal configurations otherwise furnish qualitatively different mobilities when the coupling is weak. In the proximal configuration (figure 3*a*), the Evans–Sackmann limit is achieved irrespective of the inter-leaflet friction, as long as the scaled proximal leaflet–wall friction coefficient *b*), however, weak coupling furnishes high mobilities, even when the proximal leaflet is tightly bound to the support.

The results in a form where the friction coefficients can be considered as having fixed ratios while varying the particle size are shown in figure 4. Accordingly, the scaled mobility is plotted versus the scaled particle size, *ϵ*_{2} (proximal) or *ϵ*_{1} (distal), for several fixed values, respectively, of *ϵ*_{12}/*ϵ*_{2} or *ϵ*_{12}/*ϵ*_{1}. Again, the proximal and distal configurations highlight the distinct asymmetry. For example, in the proximal configuration (figure 4*a*), large tracers (*ϵ*_{2}>1) experience the Evans–Sackmann drag force, irrespective of the inter-leaflet coupling. In the distal configuration (figure 4*b*), however, weak coupling (*ϵ*_{12}/*ϵ*_{1}<1) produces much higher mobilities than in the Evans–Sackmann limit where the mobilities again plateau to half the Evans–Sackmann value.

### (b) Strong tracer–leaflet friction: *f*_{p}(*ϵ*_{2},*ϵ*_{12}/*ϵ*_{2};*ϵ*_{1}/*ϵ*_{2}≪1,*ϵ*_{tm}=10,*ϵ*_{tw}/*ϵ*_{2}=1) and *f*_{d}(*ϵ*_{1},*ϵ*_{12}/*ϵ*_{1};*ϵ*_{2}/*ϵ*_{1}≪1,*ϵ*_{tm}=10,*ϵ*_{tw}/*ϵ*_{2}=1)

The result of increasing the tracer–leaflet friction coefficient to a fixed value *ϵ*_{tm}=10, again with *ϵ*_{tw}/*ϵ*_{2}=1, is demonstrated in figure 5. Here, the strong coupling of the tracer to its respective distal leaflet furnishes low mobilities that are much less sensitive to the respective membrane-wall friction. In the proximal configuration (figure 5*a*), mobilities are less than the Evans–Sackmann mobility for all values of *ϵ*_{2} and *ϵ*_{12}/*ϵ*_{2}. With strong leaflet–wall coupling (*ϵ*_{2}≫1), the mobility asymptotes to the Evans–Sackmann mobility. In the distal configuration (figure 5*b*), the mobility asymptotes to half the Evans–Sackmann mobility when the inter-leaflet coupling is strong (*ϵ*_{12}/*ϵ*_{1}≫1); otherwise, the mobility can be significantly higher (when *ϵ*_{12}/*ϵ*_{1}≪1), but still small because of the strong direct coupling to the proximal leaflet.

### (c) Tracer–leaflet coupling: *f*_{p}(*ϵ*_{tm}/*ϵ*_{12},*ϵ*_{2};*ϵ*_{1}/*ϵ*_{2}≪1,*ϵ*_{12}/*ϵ*_{2}=1,*ϵ*_{tw}/*ϵ*_{2}=1) and *f*_{d}(*ϵ*_{tm}/*ϵ*_{12},*ϵ*_{1};*ϵ*_{2}/*ϵ*_{1}≪1,*ϵ*_{12}/*ϵ*_{1}=1,*ϵ*_{tw}/*ϵ*_{1}=1)

The influence of the tracer–leaflet friction coefficient on the proximal (*a*) and distal (*b*) mobilities is demonstrated in figure 6. Here, the scaled mobility versus the ratio of the tracer–leaflet to inter-leaflet friction coefficients, *ϵ*_{tm}/*ϵ*_{12}, is shown for various values of the scaled particle size *ϵ*_{2} (proximal configuration) or *ϵ*_{1} (distal configuration). In this manner, each curve can be considered to have a fixed tracer radius. In these examples, *ϵ*_{12}/*ϵ*_{2}=1 (proximal) and *ϵ*_{12}/*ϵ*_{1}=1 (distal). The scaled mobility decreases monotopically with increasing tracer–leaflet coupling, from weak- to strong-coupling plateaus that depend on the scaled particle size and proximal/distal configuration. For large tracers in the proximal configuration (*ϵ*_{2}≫1, figure 6*a*), the mobility is well approximated by the Evans–Sackmann mobility for any degree of tracer–leaflet coupling. This is because the drag is dominated by the direct interactions between the tracer and the proximal leaflet. Note that small tracers tend to have a much higher mobility than the Evans–Sackmann theory because of the weaker contribution of tracer–leaflet coupling to the overall drag. In the distal configuration, the mobility asymptotes to half the Evans–Sackmann mobility only for large tracers (*ϵ*_{1}≫1) when the tracer–leaflet coupling is strong (*ϵ*_{tm}/*ϵ*_{12}≫1).

### (d) Membrane velocity disturbances

When observed on the tracer length scale, the leaflet streamlines can vary greatly, depending not only on the leaflet–wall coupling parameters, *ϵ*_{1} and *ϵ*_{2}, but also on the tracer–leaflet and inter-leaflet coupling parameters, *ϵ*_{tm} and *ϵ*_{12}. Figure 7 illustrates the general situation for the proximal configuration that prevails when all the primary dimensionless parameters are order 1. Under these conditions, there exists recirculating flow on the tracer length scale in both leaflets. Note also that qualitatively similar streamlines prevail in the distal configuration (not shown).

The degree to which the tracer–leaflet coupling (*ϵ*_{tm}=10) can indirectly influence the apparent leaflet–leaflet coupling, even when the direct inter-leaflet coupling is weak (*ϵ*_{12}=0.01), is demonstrated in figure 8. In the proximal configuration (figure 8*a*), the distal leaflet dynamics are driven by the direct coupling to the tracer. Because the leaflet–leaflet coupling is weak, the distal disturbance decays on the dimensionless length scale *b*), the tracer drives an order 1 proximal-leaflet velocity disturbance. Now the disturbance in the distal leaflet is long-ranged, while the proximal disturbance decays on the dimensionless length scale

## 4. Discussion

By accounting for solvent in the half-spaces on each side of a membrane supported on a soft substrate, such as an hydrogel, Wang & Hill [12] showed that the Evans and Sackmann model furnishes an accurate approximation when the Brinkman screening length *et al.* [10] models when the thickness of the supporting film *a*≈0.5 nm, this restricts the dual-leaflet model to situations where the gap between the proximal leaflet and its respective wall

The influence of several key model parameters on the diffusion coefficient of lipid tracers (*a*=0.5 nm) is investigated in figure 9. Note that the upper pair of solid lines are calculated with zero tangential friction (‘perfect slip’) between the tracer and its proximal leaflet, whereas the lower pair have the no-slip boundary condition. The influence of the slip condition is attenuated when increasing the tracer–leaflet or inter-leaflet friction, because the coupling of the tracer to both leaflets is strengthened by the tracer–leaflet interaction, as measured by *ϵ*_{tm}. The relatively weak influence of this slip on the diffusion coefficient justifies our focus above on the leaflet–leaflet and leaflet-support friction coefficients. Note, however, that we have generalized the Evans–Sackman theory (see the electronic supplementary material for details) to accommodate a Navier-slip boundary condition (slip length *δ*_{N}), furnishing a force
*δ*_{N}/*a*≪1 and *κ*_{2}*δ*_{N}≪1 (

In figure 9*a*, the diffusion coefficient is plotted versus the tracer–leaflet coupling parameter *ϵ*_{tm} scaled with the inter-leaflet coupling parameter *b*_{12}. Note that we have adopted a leaflet viscosity *η*_{m}≈0.08 Pa s and inter-leaflet friction coefficient *b*_{12}≈2×10^{8} Pa s m^{−1} based on the values used by Camley & Brown [24] for mixtures of DPPC (1,2-dipalmitoyl-sn-glycero-3-phosphocholine) and DPhPC (1,2-diphytanoyl-sn-glycero-3-phosphocholine) lipids. The distance between the proximal leaflet and the support *δ*=2*a*=1 nm, furnishing *b*=*η*/*δ*=1×10^{6} Pa s m^{−1}. Accordingly, because the leaflet–wall coupling is weak compared with the leaflet–leaflet coupling, the proximal (red) distal (blue) diffusion coefficients are practically the same (solid lines). Note that increasing the coupling of the tracer to the distal leaflet decreases the diffusion coefficient from about 4 μm^{2} s^{−1} to about 3 μm^{2} s^{−1} when *ϵ*_{tm}>*ϵ*_{12}. Figure 9*b* shows how the diffusion coefficients vary with *b*_{12} and *b*_{tm}=*b*_{12}. Note that *δ*=1 nm, so when *c* shows how the diffusion coefficients vary with *δ* when *b*_{tm}=*b*_{12}≈2×10^{8} Pa s m^{−1}. As expected, the diffusion coefficients differ only when the proximal-leaflet friction coefficient is greater than the inter-leaflet friction coefficient, i.e. when *η*=0.001 Pa s).

Finally, using the foregoing membrane viscosity and inter-leaflet friction coefficient, the diffusivities of larger membrane tracers can be assessed. Note that integral membrane proteins have radii in the range 0.5–4 nm [7,34]. Here, we apply the model to tracers that span one or both leaflets. Interestingly, even for integral membrane proteins that span both leaflets, the model exhibits dual-leaflet characteristics, i.e. a sensitivity to the leaflet-coupling parameter and asymmetry with respect to the top and bottom friction coefficients. For tracers with a radius *a*=3.5 nm, the proximal and distal diffusion coefficients with an effective lubrication-film thickness *δ*=4.0 pm (corresponding to a glass or dense polymeric support) are *D*_{p}≈0.15 μm^{2} s^{−1} and *D*_{d}≈0.41 μm^{2} s^{−1}. For the same tracers on a support with *δ*=1.4 nm (corresponding to a 0.5% agarose-gel support), the model predicts *D*_{p}≈2.92 μm^{2} s^{−1} and *D*_{d}≈2.94 μm^{2} s^{−1}; and with *δ*=11.1 nm (corresponding to a 6% polyacrylamide-gel support), *D*_{p}≈5.86 μm^{2} s^{−1} and *D*_{d}≈5.86 μm^{2} s^{−1}. We ascertained the foregoing lubrication film-thicknesses from lipid-tracer diffusivities in bilayers supported on agarose and polyacrylamide gels (to be reported elsewhere). Next, for a tracer having the same (*a*=3.5 nm) radius, but spanning both leaflets, the dual-leaflet model predicts *D*≈0.14 μm^{2} s^{−1} with *δ*=4.0 pm, *D*≈2.79 μm^{2} s^{−1} with *δ*=1.4 nm, and *D*≈5.72 μm^{2} s^{−1} with *δ*=11.1 nm. Thus, variations in the diffusivity with respect to proximal and distal partitioning can be significant when the leaflet–wall coupling is strong and can be attenuated by supporting the bilayers on very soft, hydrophilic supports. We hope that the quantitative insights provided by this model will assist future efforts to control and interpret the lateral mobility of mosaic elements in biological and technological membrane processes.

## Funding statement

Supported by the Natural Sciences and Engineering Research Council of Canada (NSERC), the Canada Research Chairs program, the Centre for Self-Assembled Chemical Structures (CSACS) and a McGill Engineering Doctoral Award (MEDA) to C.-Y.W.

- Received December 19, 2013.
- Accepted April 9, 2014.

- © 2014 The Author(s) Published by the Royal Society. All rights reserved.