Thermal fluctuations in shape, thickness, and molecular ... · I. INTRODUCTION Surface tension...

Thermal fluctuations in shape, thickness, and molecular orientation in lipidbilayers. II. Finite surface tensionsMax C. Watson, Alex Morriss-Andrews, Paul M. Welch, and Frank L. H. Brown Citation: J. Chem. Phys. 139, 084706 (2013); doi: 10.1063/1.4818530 View online: http://dx.doi.org/10.1063/1.4818530 View Table of Contents: http://jcp.aip.org/resource/1/JCPSA6/v139/i8 Published by the AIP Publishing LLC. Additional information on J. Chem. Phys.Journal Homepage: http://jcp.aip.org/ Journal Information: http://jcp.aip.org/about/about_the_journal Top downloads: http://jcp.aip.org/features/most_downloaded Information for Authors: http://jcp.aip.org/authors

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THE JOURNAL OF CHEMICAL PHYSICS 139, 084706 (2013)

Thermal fluctuations in shape, thickness, and molecular orientation in lipidbilayers. II. Finite surface tensions

Max C. Watson,1,2,3 Alex Morriss-Andrews,2 Paul M. Welch,3 and Frank L. H. Brown2,4

1NIST Center for Neutron Research, National Institute of Standards and Technology, Gaithersburg,Maryland 20899, USA2Department of Physics, University of California, Santa Barbara, California 93106, USA3Theoretical Division, Los Alamos National Laboratory, Los Alamos, New Mexico 87545, USA4Department of Chemistry and Biochemistry, University of California, Santa Barbara, California 93106, USA

(Received 24 May 2013; accepted 30 July 2013; published online 29 August 2013)

We investigate the role of lipid chemical potential on the shape, thickness, and molecular orienta-tion (lipid tilting relative to the monolayer surface normal) of lipid bilayers via a continuum-levelmodel. We predict that decreasing the chemical potential at constant temperature, which is associ-ated with an increase in surface tension via the Gibbs-Duhem relation, leads both to the well knownreduction in thermal membrane undulations and also to increasing fluctuation amplitudes for bilayerthickness and molecular orientation. These trends are shown to be in good agreement with molecu-lar simulations, however it is impossible to achieve full quantitative agreement between theory andsimulation within the confines of the present model. We suggest that the assumption of lipid volumeincompressibility, common to our theoretical treatment and other continuum models in the litera-ture, may be partially responsible for the quantitative discrepancies between theory and simulation.© 2013 AIP Publishing LLC. [http://dx.doi.org/10.1063/1.4818530]

I. INTRODUCTION

Surface tension regulates shape of eukaryotic cells ina variety of situations including endocytosis,1 exocytosis,2

membrane repair,3 cell spreading,4 and cell motility.5 Osmot-ically induced tension may play a decisive role during shapechanges such as the fusion and fission of membranes.6, 7 Theactivity of mechanosensitive channels is also very sensitive tothe surface tension within the surrounding bilayer.8, 9

Molecular dynamics simulations provide a means tostudy the detailed effects of membrane surface tension onshort length scales. Numerous investigations10–14 have mea-sured how the area per molecule, thickness, diffusion coeffi-cient, order parameter, and pressure profile within the mem-brane change as a function of applied tension.

Though height (undulation) fluctuations have been mea-sured in many simulations of membranes in the tensionlessstate,14–38 only a small number of groups14, 30–33, 35, 39 have an-alyzed fluctuations under tension. Presumably, this is due tothe fact that lipid bilayers exhibit a vanishingly small sur-face tension under common experimental conditions41 and thebending rigidity of membranes is most easily determined viasimulations by analyzing height fluctuations in the tensionlessstate.

Within the standard continuum theory ofmembranes,40–42 the effect of surface tension is captured ina simple way. The membranes are modeled as homogeneousand structureless two dimensional elastic sheets, and theirfree energy is given by

F = γ

∫dA + κ

2

∫(J − c0)2 dA, (1)

where γ is the surface tension and the integration takes placeover the surface A of the membrane. The first term opposesthe creation of new surface area. The second term is the en-ergy associated with bending the membrane, where κ is thebending rigidity, J is twice the mean curvature, and c0 isthe total spontaneous curvature. Here, we consider a closedsurface that does not change its topology, so that the con-tribution due to Gaussian curvature is constant43 and maybe neglected. Assuming no overhangs, any point R on thesurface may be described within the Monge representation:R = (x, y, h (x, y)). Within this representation, the surface el-ement is given by dA =

√1 + (∇h)2 dx dy and J = ∇2h for

small curvatures. Expanding the square root, keeping termsup to second order and neglecting additive constants, the freeenergy may be written as

F = 1

2

∫[γ (∇h)2 + κ(∇2h)2] dx dy. (2)

Thermal fluctuations in membrane shape as a function ofwave number q are then given by

〈|hq|2〉 = kBT

κq4 + γ q2. (3)

The implications of this expression have been very successfulfor interpreting aspiration experiments on lipid vesicles.44–47

However, this picture only describes overall membrane shapefluctuations and has nothing to say about the many phys-ical observables which are readily measured at the shortwavelengths studied in molecular simulations, including fluc-tuations in lipid number density, bilayer thickness, andlipid orientation. Furthermore, it should be mentioned thatthe quantity γ appearing in Eq. (3) is subject to multipleinterpretations and significant controversy, especially when

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084706-2 Watson et al. J. Chem. Phys. 139, 084706 (2013)

applied to the small membranes studied in computersimulations.48–51

In this work, we will always define the surface tension γ

to be the thermodynamic variable conjugate to the lateral areaof a rectangular simulation box, which the membrane spans(see Sec. II D). This definition differs from the implicationof Eq. (1), where γ appears conjugate to the total membranearea. (Since lipid bilayer membranes can exhibit out-of-planemotions, the membrane area is, in general, greater than theprojected area of the simulation box.) However, to the accu-racy available within the quadratic Monge model (Eq. (2)), itis impossible to distinguish between the various possible sur-face tension definitions.49 We refer the interested reader to arecent paper by Diamant51 for an illuminating discussion re-lated to the thermodynamics of membrane surface tension.

Since the area per lipid within the plane of the mem-brane increases with surface tension, one would expect the ac-companying changes in molecular packing to affect the mem-brane’s resistance to bending (and other elastic properties).Indeed this effect has been observed in molecular simulationsby analyzing the height fluctuations of monolayers.27–29, 52

Though Eq. (3) with a constant value of κ was found to bereasonably accurate for bilayers over a range of tensions inRefs. 30 and 31, these studies involve very aggressive coarse-graining (each lipid was represented by a single point particle)and it is unclear that these results would hold for more realis-tic lipid architectures.

Through the studies of the zero tension case, it hasbeen long established15, 19, 25, 34, 36, 37 that Eq. (3) is only validon length scales considerably longer than the bilayer thick-ness. On smaller length scales, the elastic energy in Eq. (2)must be generalized to include the effects of microscopicprotrusions14, 19, 25, 53, 54 and lipid tilting.34, 36, 37, 55

In contrast to height fluctuations, we are aware of onlyone investigation (Neder et al.14) in which the fluctuations inbilayer thickness were studied as a function surface tension.In that study, a theory of the zero tension case19 was usedto fit data for finite tensions by rescaling the values of theelastic parameters. The magnitude of the rescaling was deter-mined by fitting the data and did not have a physical basis.A recent publication56 has discussed a theoretical model for

bilayer thickness energetics as a function of chemical poten-tial/surface tension. This model generalizes the approach ofRefs. 19 and 57 to include non-vanishing tension, but does notconsider lipid orientation and focuses on the physical prob-lem of membrane deformations in the vicinity of protein in-clusions as opposed to thermal fluctuation spectra of homo-geneous membranes.

In this paper, we present a continuum model for mem-brane deformations which includes the effects of surface ten-sion, membrane shape, thickness, molecular orientation, andmicroscopic noise within a unified framework. The treatmentpresented here is an extension of recent work by Watson,Penev, Welch, and Brown,34 which we will refer to as ‘Pa-per I” throughout this work. We modify the zero-tension treat-ment of Paper I34 to include arbitrary tensions. Assumingsmall deformations, we calculate analytical predictions for thethermal fluctuation spectra of a membrane at various tensions.The analytic expressions are found to be in good, but imper-fect, agreement with molecular simulations.

II. THEORETICAL MODEL

Many aspects of the necessary theory have been de-scribed previously.34 The purpose of the present section is toilluminate those aspects of the theoretical description that arecritical to understanding the effects of applied surface tensionand which were not discussed in prior work.

A. Geometric description

We adopt the same notation, methods, and geometricsetup as in Paper I,34 and the reader is referred there forfurther detail. Here, we briefly define the variables appear-ing in our later equations. The cross sectional area per lipidat the hydrocarbon-water interface that minimizes the mono-layer free energy per molecule in a flat geometry at zero sur-face tension is denoted by �0. b0 = v/�0 is the hydrocarbonchain length under the same conditions, where v is the lipidvolume. The corresponding quantities for arbitrary values ofthe surface tension are denoted by �∗ and b∗ = v/�∗, respec-tively (Fig. 1); lipid volume is assumed constant regardless

h(1)

h(2)

z(1)

z(2)

z(m) z+z(1)

z(2)

h(1)

h(2)z(m)z+

FIG. 1. For each molecule, dark grey circles mark the portion of the lipid separating the polar head from the hydrocarbon tails. On a macroscopic level, thepolar-nonpolar interfaces are described by z(1) and z(2). The unit vectors N(α) are normal to the monolayer surfaces and point toward the interior of the bilayer.The unit vector field n(α) points along the hydrocarbon chains. b(α)n(α) extend from z(α) to the surface separating the two leaflets, z(m). The mean height z+is the average of z(1) and z(2). Left: a bilayer in its minimal energy configuration at some given tension, in which λ(α) = 0, z(m) = z+, N(α) = n(α) = 0, thethickness is 2b∗ and the area per molecule is �∗ (dotted red). Since there are no protrusions, h(α) = z(α). Right: an arbitrarily deformed bilayer, exaggeratedfor illustrative purposes. On microscopic lengthscales, the polar-nonpolar interfaces h(α) are not smooth (dashed curves). The protrusion fields λ(α) displacethe interface in the normal direction, so that (−1)α λ(α)N(α) (dashed vectors) extend from z(α) to h(α). The fields z+ (blue) and z(m) (black) differ in general.Reprinted with permission from M. C. Watson, E. S. Penev, P. M. Welch, and F. L. H. Brown, J. Chem. Phys. 135, 244701 (2011). Copyright 2011, AmericanInstitute of Physics.



of the imposed tension. Although it is possible to relax thisassumption in principle, doing so in practice would signifi-cantly complicate an already complex theoretical model (seebelow). The assumption of lipid volume incompressibility iscommonplace within the literature related to continuum mod-eling of lipid bilayers14, 19, 34, 56, 58–62 and we will not attemptto remedy this shortcoming here. Our simulation results inSec. IV do suggest that lipid volume is not strictly conserved;further study along these lines is certainly warranted.

The superscripts α = 1 and α = 2 refer to the top andbottom leaflets, respectively. The quantity r = (x, y) denotesxy position. The fields z(α) (r) describe the macroscopic sur-face separating the polar heads and hydrocarbon tails of eachmonolayer (Fig. 1). The surface separating the top and bottommonolayers is denoted by z(m) (r). We also define

z+ ≡ z(1) + z(2)

2,

z− ≡ z(1) − z(2) − 2b∗2

, (4)

ε ≡ z(m) − z+.

z+ describes the undulating shape of the bilayer. z− capturesdeviations in the bilayer thickness from its mean value 2b∗.Perturbations of z(m) from the average shape z+ are describedin terms of ε.

The chain length b(α) describes the distance from thepolar-nonpolar interface to the end of the hydrocarbon chainas measured by a straight line. Though this quantity is onlywell-defined at positions where lipids are present, b(α) repre-sents a continuum field which is smoothly varying. To firstorder in small quantities

b(1) = z(1) − z(m), b(2) = z(m) − z(2). (5)

On lengthscales comparable to the bilayer thickness, thepolar-nonpolar interfaces are subject to microscopic displace-ments, or protrusions λ(α). We denote the actual position ofeach interface by h(α), which is defined to be h(α) = z(α)

+ λ(α) (see Fig. 1). Just like z+ and z−, the protrusions may bedecomposed into symmetric and antisymmetric contributions:

λ+ ≡ λ(1) + λ(2)

2, λ− ≡ λ(1) − λ(2)

2. (6)

The mean height of the bilayer and deviations in membranethickness are the quantities which we measure in simulations.Up to first order in small quantities,34

h ≡ z+ + λ+, t ≡ z− + λ−. (7)

The orientation of the lipids are described by continuumvector fields. We use the font c to denote a three-dimensionalvector, while c refers to its xy components. The vector calcu-lus operations of divergence, gradient, and curl are only ap-plied to vectors of the c variety and only act within the xyplane. For each monolayer, n(α) (x, y) represents the orienta-tion of the lipids and points along the length of the molecules.The vectors normal to the monolayer surfaces are denoted byN(α) (x, y) and point toward the interior of the membrane inour convention. Deviation of the lipid orientation away from

the monolayer normals is denoted by the tilt vector

m(α) = n(α) − N(α). (8)

Later it will be convenient to work with the symmetric andanti-symmetric tilt vectors

m ≡ m(1) + m(2)

2, m ≡ m(1) − m(2)

2. (9)

Since the polar-nonpolar interfaces are subject to protru-sions, the tilt vectors are also prone to microscopic perturba-tions. From Paper I,34 the actual symmetric and antisymmetrictilt vectors measured in simulations are given by

p ≡ m − ∇λ−, p ≡ m − ∇λ+. (10)

In Sec. II E, our theoretical expressions for fluctuations in h,t, p, and p will be compared with simulation data for mem-branes spanning a square periodic simulation box of area AP.

B. Thermodynamic ensemble

As discussed above, the mesoscopic “state” of ourcoarse-grained membrane is specified through the seven fields{z(α), z(m), m(α), λ(α)} ≡ {...}. This suggests that one couldcalculate the canonical partition function for the membraneas

Q(N,Ap, T ) =∑{...}

e− FkB T δ

(N − Vbilayer

v

), (11)

where N is the number of lipids in the membrane, AP is theprojected area of the membrane (i.e., the area of the simula-tion box), and T is temperature. F[z(α), z(m), m(α), λ(α)] is aLandau-Ginzburg Hamiltonian representing the effective freeenergy for the membrane obtained by averaging over the mi-croscopic degrees of freedom for the system, while constrain-ing the mesoscopic fields {...} to a particular state. The deltafunction is required to constrain lipid number to the value N;without such a restriction, the sum over configurations of thefields {...} includes all possible values of N ranging from zeroto infinity. Note that the summation expression is introducedas a notational convenience, the true configurational “sum” isactually a path integral.

Evaluation of Eq. (11) is not theoretically convenientdue to the presence of the volume constraint, but this can beavoided by considering the grand canonical partition functionat constant lipid chemical potential μ:

(μ,Ap, T ) =∞∑

N=0

eμN

kB T Q(N,Ap, T )

=∞∑

N=0

∑{...}

e− [F−μN]kB T δ

(N − Vbilayer

v

)

=∑{...}

e− [F−μN]kB T

=∑{...}

e− �GkB T . (12)

The last step simply defines �G[z(α), z(m), m(α), λ(α)] ≡ F− μN to be the effective free energy for the open sys-



tem, subject to the specified configurations of the mesoscopicfields. For the remainder of this paper, we will refer to �G asthe “effective Hamiltonian” for our membrane system.

Our prior publication34 also dealt with Eq. (12), but fo-cused exclusively on the case where μ was tuned to achievea condition of vanishing surface tension. By convention, thiscondition is often referred to as “vanishing chemical poten-tial,” however finding γ = μ = 0 at a given temperature re-lies upon an arbitrary definition of the absolute energy scaleto force this coincidence. Such a convention was adopted inPaper I;34 the free energy F introduced in that work coincideswith �G(μ = 0) introduced above, provided that we defineour energy scale such that μ = 0 at vanishing tension.

C. Effective Hamiltonian

Using the results from Appendix A of Paper I,34 the freeenergy per lipid in leaflet α, f (α), may be written in terms ofb(α), n(α), and m(α):

f (α)

�0= 1

2

(k�c − b0μ + kAb2

0

4

)[∇ · n(α)]2 + k�

c c0∇ · n(α)

+ kA

2

(b(α) − b0

b0

)2

+(kAb0

2− μ

) (b(α) − b0

b0

)∇ · n(α)

+ κθ

2|m(α)|2 + κtw

2(∇ × m(α))2 + kG det

(∂n

(α)j

∂rk

). (13)

The constants appearing above (k�c , μ, kA, c0, κθ , κtw, kG)

are identical to those introduced in Paper I34 and we refer thereader to that work (n.b. Table II of Paper I34) for further dis-cussion. The first two terms correspond to deviations of theeffective curvature −∇ · n(α) away from the total spontaneouscurvature c0. These contributions are related to the splay en-ergy of a nematic liquid crystal.63 Deviations in chain lengthfrom the preferred stress-free conformation, (b(α) − b0), arecaptured by the kA term. As a consequence of lipid volumeincompressibility, the chain length and area per lipid are in-versely related and kA can be identified with the membranearea compressibility modulus. The fourth term represents thecoupling between effective curvature and chain length. Notethat μ is a molecular constant adopted to conform with thenotation of Paper I34 and is not related to the lipid chemi-cal potential, μ. The κθ term corresponds to molecular tilt,in which the orientation of molecules deviate from the nor-mal to the surface.55, 64–67 The κtw term captures the effect ofmolecular twist.55, 63 The last term is related to the effectiveGaussian curvature. It is important to recognize that in writ-ing Eq. (13) without a constant term (i.e., f (α)[b = b0, n(α)

= 0, m(α) = 0] = 0) we are implicitly defining the absoluteenergy scale for our lipids such that μ(γ = 0, T) = 0, as willbecome clear in Sec. II D. This choice is theoretically conve-nient, but completely arbitrary. If one had reason to choose adifferent convention for the energy scale, μ(γ = 0, T) mustbe added to Eq. (13).

Neglecting protrusions for the moment, the totalmacroscopic free energy of the bilayer is obtained by sum-ming over the contributions of all lipids. To accomplish thistask in practice, we introduce the lipid volume density field

g(α)V (x, y). The infinitesimal volume of the lipid hydrocarbon

chains whose polar/nonpolar interface lies within dx dy isgiven by dV (α) ≡ g

(α)V dx dy (see Appendix A). Since chain

incompressibility guarantees v = b0�0 = constant, dV (α)

= dn(α)b0�0 where dn(α) is the lipid number density forchains crossing the interface within dx dy. Using the vari-able �macro

G to denote the macroscopic portion of the effectiveHamiltonian, this suggests

�macroG =

∫(1)

[f (1) − μ] dn(1) +∫

(2)[f (2) − μ] dn(2)

= 1

b0�0

∫ ([f (1) − μ]g(1)

V + [f (2) − μ]g(2)V

)dr.

(14)

An explicit expression for g(α)V is derived in Appendix A:

g(α)V

b0= 1 + b(α) − b0

b0+ b0

2∇ · n(α) + (

b(α) − b0) ∇ · n(α)

−1

2

∣∣∣m(α)∣∣∣2

+ 1

2

∣∣∣∇z(α)∣∣∣2

+ b20

3det

(∂n

(α)j

∂rk

). (15)

An equivalent expression was also derived in Paper I,34 how-ever we note a small typographical error in Eq. (A1) of thatwork; the last term of Eq. (A1) of Paper I34 should read(−1)α+1 n(α) · ∇z(α).

From Eqs. (13)–(15), the macroscopic free energy �macroG

contains a term proportional to μ(b(α) − b0). The minimumof �macro

G with respect to b(α) occurs at the mean monolayerthickness:

b∗ = b0

(1 + μ

kA�0

). (16)

Writing chain length terms in the free energy as a functionof (b(α) − b∗) rather than (b(α) − b0), the linear term in b(α)

is no longer present. Since all terms in the free energy in-volving (b(α) − b∗) are quadratic, �macro

G may be written interms of z(α) and z(m) using Eq. (5). Furthermore, the termsinvolving a divergence in n(α) may be expressed in terms ofz(α) and m(α) via the relationship ∇ · n(α) = (−1)α+1∇2z(α)

+ ∇ · m(α), which is equivalent to Eq. (8) to linear order insmall quantities.

After performing the above mentioned substitutions, re-taining all terms to second order in small quantities and sup-plementing the resulting �macro

G [z(α), z(m), m(α)] with protru-sion contributions to the free energy as outlined in Paper I,34

the resulting effective Hamiltonian �G[z(α), z(m), m(α), λ(α)]may be expressed as the sum of peristaltic fp and undulationfu modes:

�G =∫ (

fp(z−, m, λ−) + fu(z+, m, ε, λ+)

−[

2μ

�0+ μ2

kA�20

] )dr, (17)



where

fp(z−, m, λ−) = kA

b20

(z−)2 + kbc (∇2z− + ∇ · m)2

+ �(μ)

b0(∇2z− + ∇ · m)z− + κθ (μ)m2

+ κtw(∇ × m)2 + 2η(μ)(∇2z− + ∇ · m)

− μ

�0(∇z−)2 + 2kG(μ) det

(∂n

(α)j

∂rk

)

+ γλ(∇λ−)2 + 2γλ∇z− · ∇λ− + kλ(λ−)2, (18)

fu(z+, ε, m, λ+) = kbc (∇2z+ + ∇ · m)2 + kA

b20

ε2

+ κθ (μ) m2 − � (μ)

b0(∇ · m + ∇2z+)ε

+ κtw(∇ × m)2 − μ

�0(∇z+)2

+ 2kG (μ) det

(∂n

(α)j

∂rk

)

+ γλ(∇λ+)2 + 2γλ∇z+ · ∇λ+ + kλ(λ+)2, (19)

and

κθ (μ) ≡ κθ + μ

�0,

� (μ) ≡ � − 2b0μ

�0,

(20)

kG (μ) ≡ kG − μb20

3�0,

η (μ) ≡ k�c c0

(1 + μ

kA�0

)− μ

kA�0

(μ + b0μ

�0

).

For brevity of notation, we have defined � ≡ 2k�c c0 − 2μ

+ kAb0 and kbc ≡ k�

c + �b0/2 − kAb20/4, as in Paper I.34

These equations (Eqs. (17)–(20)) contain the primary theo-retical results of this paper. For the most part, the variousterms appearing in Eqs. (18) and (19) can be found in the μ

= 0 case studied in Paper I34 (see Eqs. (27) and (28) of Pa-per I34), however the coefficients appearing in this work havebeen generalized to account for non-vanishing chemical po-tential as detailed in Eq. (20). The saddle-splay terms contain-ing kG(μ) are expressed using the quantities n ≡ ∇z− + mand n ≡ ∇z+ + m. Integrating their contributions (as we willdo below), results only in boundary terms34 that disappear forthe periodic boundary condition geometries considered in thiswork, and will therefore not be considered in subsequent cal-culations.

The completely new terms appearing in the Hamiltonianwhen μ �= 0 are the (∇z±)2 terms familiar from Helfrichtheory (Eq. (2)) and the analogous contribution to the peri-staltic modes.59 The above expressions reduce to the resultsof Paper I34 when μ = 0 and we stress that all constantsappearing here have the same meaning as those introduced inPaper I.34 Note that the constant term in Eq. (17) was obtained

from Eq. (16). As discussed in Sec. II D, it corresponds to theapplied surface tension in the mean field approximation. Wenote that the λ± protrusion terms included here are identicalto those introduced in Paper I.34 No attempt is made in thiswork to predict the influence of μ on protrusions (e.g., in-troducing kλ(μ) or a similar scheme). Furthermore, we pre-viously found that approximating γ λ = 0 results in excellentagreement between the theory and simulations at zero chem-ical potential34 and we will only consider the case γ λ = 0 inthis paper. The general expression is provided to emphasizesimilarity to our prior treatment.

The current theory is limited to homogeneous bilayerswith a single lipid species in the liquid state. As a result,Eqs. (17)–(20) contain no contributions due to the inter-actions between opposing leaflets. Orientational coupling[n(1) − n(2)]2 between leaflets of the same composition hasbeen proposed68 for certain membrane phases at lower tem-peratures. We found this effect to be negligible in previoussimulations of the liquid state (see Paper I34). If the monolay-ers differ in composition, interleaflet interactions may provideadditional energetic contributions.69

D. Relation to applied tension

For practical reasons, simulations of lipid bilayers aretypically not carried out at constant chemical potential, butrather holding lipid number constant. It is also common insimulations, though not universal, to abandon the constantAP box size in favor of a constant applied tension(γ ) con-dition. The simulations we shall eventually compare to inSec. IV are run under constant (N, T, γ ) conditions. As indi-cated above, our theoretical model is not well suited to directanalysis within the canonical ensemble, nor is it well suited toanalysis at constant applied tension, making a fully rigorousand direct comparison between theory and simulation impos-sible. Although the various ensembles will become equivalentin the thermodynamic limit of infinite system size, there is noreason to expect that the modestly sized simulation boxes em-ployed in current studies approach such a limit. Indeed, this isa possible source of concern that has been discussed at lengthin the literature.14, 32, 33, 39, 48, 49, 70, 71

Notwithstanding the above admonition regarding smallsystem sizes and inequivalence of ensembles, we would stilllike to compare theoretical predictions based upon �G withour simulations carried out at constant (N, T, γ ). In principle,this could be done by computing the surface tension for ourfinite sized theoretical system at constant (μ, AP, T) via γ =−kBT∂ ln ()/∂AP and comparing to a simulation run at thesame tension. Performing this calculation exactly leads to un-wieldily expressions that are of little use in practice. Instead,we connect the two ensembles via an approximate mean-fieldlevel evaluation of �G. We assume that the minimum energyconfiguration of �G is given by the flat, constant thickness,tilt-free, and protrusion-free state displayed in the left-handpicture of Fig. 1. Were this not the case, our harmonic modelfor �G would be unstable. (This stability condition may beformally expressed in terms of inequalities satisfied by theelastic constants.56) The mean-field approximation to is



achieved by evaluating the path integral of Eq. (12) via thesaddle point approximation to give

−kBT ln MF = �(min)G

= AP

(kA(b∗ − b0)2

b20

− 2μ

�0

[1 + b∗ − b0

b0

])

= −AP

(μ2

kA�20

+ 2μ

�0

)(21)

= −AP

(kA(b∗ − b0)2

b20

+ 2kA(b∗ − b0)

b0

),

(22)

where the second expression is obtained by directly evaluat-ing Eq. (14) for the minimum free energy configuration andthe third and fourth expressions introduce Eq. (16) to expressthe result solely in terms of μ or solely in terms of b∗, re-spectively. Taking the derivative of Eq. (21) with respect toAP yields

γ = −(

2μ

�0+ μ2

kA�20

), (23)

which is easily inverted to provide lipid chemical potential asa function of surface tension

μ = −kA�0(1 −√

1 − γ /kA). (24)

Note that Eqs. (23) and (24) indicate simultaneous vanish-ing of μ and γ . As mentioned previously, this has arisendue to the lack of any constant term in Eq. (13), which setsthe lipid chemical potential to zero for the homogeneous flatmembrane geometry with monolayer thickness b0. Taking thederivative of Eq. (22) with respect to AP provides an alternateexpression for γ and leads to the prediction for bilayer thick-ness as a function of the applied tension

b∗ = b0

√1 − γ

kA

. (25)

Expressions similar to Eqs. (24) and (25) were recently pre-sented in Ref. 56 for a slightly different membrane model thatneglects the effects of lipid orientation. If our expressionsare truncated to linear order in γ , the results of Ref. 56 arerecovered.

It should also be mentioned that Eq. (23) can be derivedby directly integrating the Gibbs-Duhem equation

N dμ = −Ap dγ − S dT (26)

at constant temperature from μ = γ = 0 to finite μ andγ using N/AP = (2/�0)(1 + μ/kA�0) as implied by lipidvolume conservation and Eq. (16) if one assumes the homo-geneous flat membrane geometry inherent to the mean fieldapproximation. Equations (24) and (25) then follow imme-diately. The Gibbs-Duhem derivation is appealing becauseit shows why positive surface tensions correspond to nega-tive chemical potentials when we set the energy scale to fix

μ = 0 at γ = 0 for a given temperature. Physically, decreas-ing the chemical potential at fixed Ap encourages some lipidsto leave the membrane. The remaining lipids must becomestretched, corresponding to an increase in surface tension. Inthe case of vanishing tension, the forces within each lipid thatpromote and oppose the creation of interfacial area54 com-pletely balance each other. In this “saturated” state, the freeenergy per lipid is minimized with respect to its interfacialarea (or equivalently, chain length).65, 72, 73

For many experimental situations, the relationship be-tween surface tension and chemical potential is slightly morestraightforward. For a constant lipid number, biomembranescan only withstand perturbations in area up to a few percentbefore rupturing.41 Observable changes in membrane area arerather due to changes in the number of lipids within the bi-layer, while the area per lipid remains nearly constant. Thesurface tension may, therefore, be approximately regardedas a chemical potential when lipids are allowed to freelymove between the membrane and the solvent.73 However,thermal fluctuations introduce additional subtleties into theinterpretation.44, 70, 71, 74

E. Fluctuation spectra

This section will present predictions for the fluctuationspectra of various membrane properties observable in simu-lations. These predictions are obtained from Eqs. (17) to (19)and, strictly speaking, should only apply to simulations per-formed in the appropriate (μ, AP, T) ensemble. Compari-son to simulations performed at (γ , N, T), as in Sec. IV, isachieved by substituting Eq. (24) for μ into the theoreticalpredictions presented below and identifying the box dimen-sion L appearing in the definition of the Fourier wave vectorswith the average value 〈L〉 observed at a given γ .

For a homogeneous membrane in a square box of area Ap

= L2 and periodic boundary conditions, the effective Hamil-tonian may be rewritten in Fourier space. We use the Fouriertransform pair

gq = 1

L

∫g(r)e−iq·r dr,

g(r) = 1

L

∑q

gqeiq·r

for an arbitrary scalar function g(r). The values of the wavenumber are given by q = 2π (n,m)/L for the integers n,m

= {−M2 , . . . , 0, . . . M

2 − 1}, where M is dictated by a shortwavelength cutoff. The vector quantities ∇λ(α), m, and m arewritten in terms of components parallel and perpendicular toq. For any two-dimensional vector c (r), its longitudinal andtransverse components are given by

c‖q = q · cq

q, c⊥

q = (q × cq) · z

q.

We calculate the fluctuation spectra using the same methodas in Paper I.34 Explicit details of the calculations are given



in Appendix B. For the remainder of the paper, we will as-sume that the protrusions λ(α) and monolayer interfaces z(α)

are decoupled from each other, as suggested by Paper I.34 InSec. IV, we will show that this approximation results in a

good agreement between the theory and simulation results forthe systems we study. From the equipartition theorem, themagnitude of the height, thickness, and tilt fluctuations aregiven by

〈|hq|2〉 = kBT

2

[1

kc (μ) q4 − μ

�0q2

+ 1

κθ [q2 − μ

kc(μ)�0]

+ 1

kλ + γλq2

], (27)

〈|p‖q|2〉 = kBT

2

[q2

κθ [q2 − μ

kc(μ)�0]

− μ

κθ (μ) �0[kc (μ) q2 − μ

�0]

+ q2(kλ + γλq2

)], (28)

〈|tq|2〉 = kBT

2

[2(κθ (μ) + kb

c q2)

4 kA

b20

(κθ (μ) + kb

c q2) + q2

[4(kbc κθq2 − κθ (μ) μ

�0

)−

(�(μ)

b0

)2− 4�(μ)

b0κθ (μ)

] + 1

kλ + γλq2

], (29)

〈|p‖q|2〉 = kBT

2

[ 2(

kA

b20

+ q2(kbc q

2 − �(μ)b0

− μ

�0

))4 kA

b20

(κθ (μ) + kb

c q2) + q2

[4(kbc κθq2 − κθ (μ) μ

�0

)−

(�(μ)

b0

)2− 4�(μ)

b0κθ (μ)

] + q2

kλ + γλq2

], (30)

〈|p⊥q |2〉 = 〈|p⊥

q |2〉 = kBT

2

(1

κθ (μ) + κtwq2

). (31)

The effective bending rigidity appearing in the undulationspectrum is

kc (μ) =(

kbc − [� (μ)]2

4kA

) (κθ

κθ (μ)

). (32)

Equations (27)–(31) are more complicated than theirμ = 0 counterparts introduced in paper I. This might seemsurprising since only two new terms ([∇z±]2) appear inEq. (17) that were not present in Paper I.34 However, intro-duction of these terms leads to coupling between lipid tilt andlipid splay, which we previously demonstrated to be absentfrom this model under vanishing tension.75 This additionalphysics leads to more cumbersome expressions for the spectrainvolving height, thickness, and the longitudinal componentsof lipid tilting.

To understand the coupling between splay and tilt, con-sider the undulation portion of the free energy in which ε = 0and no twist is present:

fu(z+, m,∇ × m = 0, ε = 0)

= kbc (∇2z+ + ∇ · m)2 − μ

�0(∇z+)2 + κθ (μ)m2.

While the above equation is expressed in terms of z+ and m,reference to z+ can be eliminated using n ≡ ∇z+ + m:

fu (n, m,∇ × m = 0, ε = 0)

= kbc (∇ · n)2 − μ

�0(n − m)2 + κθ (μ) m2. (33)

When μ = 0, the splay (kbc term) is decoupled from the tilt

(m2).75 But when μ �= 0, the chemical potential term cou-ples tilt and splay together. For the three quantities {z+, m, n},the nature of the free energy in Eq. (33) ensures that anytwo of them will be coupled when μ �= 0 (all three cannotbe used since there are only two degrees of freedom in thissituation).

Since the protrusion contributions are decoupled from therest of the Hamiltonian, their variances add so that each quan-tity may be written as the sum of macroscopic and protrusioncontributions. Note that the protrusion terms are assumed (forsimplicity) not to be functions of the surface tension.

Though most of the expressions are complicated, someof their limiting behavior can be understood. The first term inthe height expression closely resembles the standard result ofEq. (3) (note that the factor of two originates from our defini-tion of elastic constants with respect to the monolayer, not thebilayer). The value of the effective bending modulus is, how-ever, a complicated function of the tension/chemical potential,and the constant multiplying the q2 term in the denominator(i.e., −2μ/�0) is not exactly the system tension, but doesagree with that interpretation to first order in γ .

The second term more strongly reflects the coupling be-tween height and tilt and is present even at zero surfacetension. Its contribution to 〈|hq|2〉 becomes negligible at longwavelengths. The third term is associated with protrusions.

Just like the zero surface tension case, the thickness de-viations 〈|tq|2〉 are proportional to

(b2

0/kA + 1/kλ

)at very



long wavelengths. At sufficiently long wavelengths, stretch-ing/compressing the membrane away from its preferred �∗state is equally costly regardless of the tension. This is a con-sequence of our molecular model (Eq. (13)) being completelyharmonic about b0. Despite modifying the position of the min-imum, tension cannot affect the curvature of the energy wellwithin our model.

Tilt deformations become more favorable as the tensionincreases. This can most clearly be seen in the (q → 0) limit,where all four tilt averages are equal to kBT/2κθ (μ) (fromEq. (20), κθ (μ) increases linearly with chemical potentialand thus decreases with tension). This makes sense from aphysical standpoint since a taught bilayer has a larger area permolecule, thus allowing the lipid heads and tails more free-dom to move relative to one another in the membrane plane.On length scales below the bilayer thickness, the predictionsfor 〈|p‖

q|2〉 and 〈|p‖q|2〉 eventually collapse to kBTq2/2kλ when

the protrusions are decoupled from the monolayer surfaces.Even when this coupling is included, 〈|p⊥

q |2〉 and 〈|p⊥q |2〉 are

still given by Eq. (31) since the transverse tilt is always de-coupled from the rest of the Hamiltonian.

F. Comments on the predictions

The tension dependent effective bending rigidity, kc (μ),appearing in the undulation spectrum requires some explana-tion. In Eqs. (18) and (19), kb

c corresponds to the cost of bend-ing the membrane via lipid splay, while the lipid chain lengthb is held equal to b∗ (see Paper I34 and Ref. 75). For a givenbilayer curvature, such a restricted deformation (Fig. 2) mustbe associated with a higher energy than could be achieved if bwere allowed to relax. Within the formulation of Eq. (17), re-laxation of b(α) between the two monolayers is accomplishedby displacement of the bilayer midplane field ε. If we con-sider a membrane undulation restricted to be free of tilt, thefree energy density is then

fu(z+, m = 0, ε)

= kbc (∇2z+)2 + kA

b20

ε2 − �(μ)

b0ε∇2z+ − μ

�0(∇z+)2.

For a given shape z+, the minimum free energy is given by

f minu (z+, m = 0) = k[no tilt]

c (μ)(∇2z+)2 − μ

�0(∇z+)2 (34)

with the effective bending modulus

k[no tilt]c (μ) = kb

c − [� (μ)]2

4kA

. (35)

At zero chemical potential, k[no tilt]c (μ) reduces back to the

neutral surface bending modulus described in Paper I.34 Whenμ = 0 there is no coupling between tilt and splay;75 re-stricting tilt to vanish has no effect on the bending modu-lus in such a limit. We also mention that the same expres-sion for k[no tilt]

c (μ) can be obtained by applying parallelsurface arguments.58 In the tilt free case, k[no tilt]

c (μ) is the

z+z(m)

z+= z(m)

FIG. 2. Two membrane configurations which share the same shape z+.Lipids are represented by boxes. For simplicity, no tilt is present (m = 0).Top: A configuration in which the local chain length is equal to b∗ and thesurface separating the two leaflets z(m) coincides with the midplane of thebilayer z+. Compared to the flat state, the area per molecule at the top andbottom interfaces decreases and increases, respectively. However, the averagearea per molecule [�(1) + �(2)]/2 remains unchanged. Bottom: z(m) is abovez+ so that ε∇2z+ > 0. For � > 0, the bottom case is energetically morefavorable than the one above (see Eq. (19)). The average area per molecule[�(1) + �(2)]/2 is also larger, lowering the free energy for μ < 0.

coefficient of q4 in the height fluctuation spectrum (Eq. (27)).But when the tilt degree of freedom is included, the coeffi-cient becomes kc (μ) = k[no tilt]

c (μ) [ κθ

κθ (μ) ]. Parallel surfacearguments break down in this case and, as mentioned previ-ously, the couplings between tilt and splay that appear in theHamiltonian make the full problem difficult to interpret. Notethat while k[no tilt]

c (μ) may be considered a parameter of theHamiltonian itself in the tilt-free case, kc (μ) is associatedwith fluctuations only.

To our knowledge, this is the first account of a chem-ical potential-induced renormalization of kc (μ) (Eq. (32))predicted within the context of a microscopic model. A longwavelength renormalization due to thermal fluctuations hasbeen previously proposed. Assuming a Hamiltonian givenby Eq. (2), Helfrich76 predicted that for non-vanishing ten-sions/chemical potentials, the rigidity measured in heightfluctuations (q4 coefficient in Eq. (3)) differs from the rigid-ity appearing in the free energy ([∇2h]2 coefficient of Eq.(2)). This fluctuation effect was negligible for the molecu-lar simulations studied by Farago and Pincus.32 The renor-malization we have described consists of two contributions:(1) a non-fluctuation-based renormalization of k[no tilt]

c (μ)(Eq. (35)) due to �(μ), and (2) a fluctuation-based effect,kc (μ) = k[no tilt]

c (μ) [κθ/κθ (μ)], originating from the tiltdegree of freedom. The chemical potential dependence ofkc (μ) may account for the change in the bending rigiditymeasured in monolayer systems.27–29, 52

Using the values of the elastic constants specific to theMARTINI model of DPPC,34 we find that for applied sur-face tensions as high as 19 mN/m, the corresponding effec-tive bending modulus deviates from kc (0) by less than 10%.Small changes in the measured bending rigidity at differenttensions were observed in the coarse-grained simulations ofShiba and Noguchi.31 However, the associated error bars werelarge compared to the differences in the rigidity at differenttensions.



The renormalization of the parameters �(μ) and κθ (μ)in Eq. (20) can be significant at moderate tensions. At a sur-face tension of 19 mN/m, �(μ) and κθ (μ) increase and de-crease by 37% and 19%, respectively, compared to their zerotension values.34 These changes are linear in μ/�0 and closeto linear in γ (Eq. (24)) at low to moderate tensions. As de-scribed in Sec. III B, a maximum tension of only 4 mN/mcould be applied to our other lipid system (CG) to avoid rup-turing. Within that range, kc (μ), �(μ), and κθ (μ) changeby less than 10%.

Bitbol et al.56 have recently considered a model some-what similar to that presented here, but neglecting lipid tilting.They advocate the need for “new terms” in their Hamiltonian(new relative to Refs. 19 and 57 and similar works that alsoneglect lipid tilting) in order to explain certain experimentaland simulation data related to the distortion of the membranein the presence of integral protein inclusions. In Appendix C,we demonstrate how tilt can be integrated out of Eq. (17) toprovide an effective Hamiltonian that does not explicitly in-clude tilt. The resulting Hamiltonian includes the “new terms”of Ref. 56 as well as additional terms that become increas-ingly important at small wavelengths.

Without resorting to the formalism presented here, the re-sults from Paper I34 could have been quickly extended to de-scribe finite tensions by adding gradient terms γ [∇z±]2 to theundulation (Eq. (2)) and peristaltic59 portions of the free en-ergy. However, the latter action would give a prediction whichdisagrees with our model and the thickness fluctuation datain Sec. IV. As with undulations, a tension term of the formγ [∇z−]2 suppresses thickness fluctuations. For both of thelipid systems we simulated, thickness fluctuations are foundto increase at moderate tensions (Figs. 6 and 8). This increaseis qualitatively (but not quantitatively) predicted by our modelvia the renormalization of �(μ). This coupling factor, whichis positive at zero tension, grows with increasing tension, andis responsible for the nonmonotonic fluctuation spectrum inthickness(〈|tq|2〉) and peristaltic tilt (〈|p‖

q|2〉)(see Paper I34).We close this section by noting that for a nonzero chemi-

cal potential, there are some subtleties involving the deriva-tion of Eqs. (17)–(19). In our derivation, we have retainedthe actual chain length variable b(α) through the point wherethe linear term in chain length is removed from �G by writ-ing all terms relative to b∗. Then, we write all variables, in-cluding b(α), in terms of the set retained in our final expres-sion [z(α), z(m), m(α), λ(α)]. This yields Eqs. (17)–(19). Alter-natively, one could introduce the final variables earlier in thederivation. In particular, b(α) could be immediately rewrittenin terms of the {z(α), z(m)} fields in Eqs. (13) and (15) using(see Eqs. (A3) and (A4) of Paper I34)

b(1) = z(1) − z(m) + b0

( |n(1)|22

− n(1) · ∇z(m)

),

(36)

b(2) = z(m) − z(2) + b0

( |n(2)|22

+ n(2) · ∇z(m)

).

All compression/stretch terms would be expanded about thevertical distance t∗(which is equal to b∗ within the mean fieldapproach of Sec. II D). Unfortunately, the two different pro-cedures yield two different results, due to the fact that the re-

FIG. 3. The top leaflet of a membrane in which no lipids have been de-formed. The volume dV of the lipids whose hydrocarbon-water interfaceslie within dx dy is shown in green. The volume can be written in two differ-ent ways: dV = b0

√1 + [∇z(1)]2 dx dy and dV = [z(1) − z(m)] dx dy.

lationship between z(α) and b(α) is non-linear and we alwaysdiscard terms of higher than quadratic order in �G. The sec-ond procedure, which we have not used, leads to final expres-sions for �G that do not contain the gradient terms |∇z±|2.This is a major problem as it is the gradient terms that capturethe most important aspects of applied tension!

We can partially illuminate this paradox by consid-ering an undeformed membrane which is slanted in ourreference frame (Fig. 3). The volume dV of the lipidswhose hydrocarbon-water interface lies inside dx dy can becalculated in two mathematically equivalent ways. In thefirst method, dV = b0 dA, where dA =

√1 + [∇z(1)]2 dx dy

is the membrane surface area within dx dy and b0

is the hydrocarbon chain length. Up to second order,dV = b0(1 + 1

2 [∇z(1)]2) dx dy. In this simple geometry, thevolume element has the shape of a brick sitting on an in-cline. In the second method, dV = (z(1) − z(m)) dx dy. Theshape of the volume element is a parallelogram and doesnot correspond to the actual volume occupied by the lipids.However, the parallelogram’s volume is equal to that ofthe brick. Though the expression dV = (z(1) − z(m)) dx dy iscompletely exact in this situation, it contains no reference tothe gradient |∇z(1)|2. Within our theory, the proper handlingof volume is crucial; the expression for dV (α) and the equa-tions v dn(α) = dV (α) ≡ g

(α)V dx dy65 enable us to transform

from a free energy per lipid (Eq. (13)) to a membrane freeenergy density in the xy plane (Eq. (14)). While multiplyingf (α) and g

(α)V in Eq. (14) merely renormalizes various elas-

tic moduli, the product μg(α)V determines the contribution of

chemical potential to the effective Hamiltonian; it is here thatthe different formulations of the volume element discussedabove affect our results.

It does not seem mathematically possible to prove one ofthe above approaches for calculating dV (α) (and g

(α)V ) correct

over the other, at least not within our current formulation ofthe problem. However, from a physical perspective, the ap-proach we have adopted has clear advantages. The calcula-tion of volume elements clearly reflects the physical spaceoccupied by lipid tails in this approach. Further, the valueof the chain length b(α) is physical and is independent of themathematical description of the surface. The same cannot besaid for the surfaces z(m) and z(α), which are referenced tothe plane of simulation box. It makes physical sense to re-tain b(α) throughout the calculation, only transforming to the



Surface Tension (mN/m)0 10 20 30 40

0.85

0.9

0.95

1

0 10 20 30 401

1.02

1.04

DPPC

FIG. 4. The average monolayer thickness b∗ as a function of applied surfacetension for the MARTINI model of DPPC. b0 is the average thickness forthe tensionless membrane. From Eq. (25), the predicted values are shown forkA = 260 mN/m (dashed) and kA = 200 mN/m (solid). Inset: The averagevolume per molecule b∗�∗ divided by the tensionless value b0�0 as a func-tion of surface tension. In both plots, the error bars are about the size of thesymbols.

z(m) and z(α) basis at the very end in order to facilitate compar-ison to the simulation data. We also note that Bitbol et al.56

use a variable analogous to b(α) in order to model thicknessperturbations under applied tension (in the absence of tiltfluctuations).

III. SIMULATION DETAILS

A. MARTINI force field model (DPPC)

We simulated the coarse-grained MARTINI model77, 78

for DPPC (dipalmitoylphosphatidylcholine) and explicit wa-ter. The simulations were run within the GROMACSpackage.79–81 Below, all time scales are quoted in simulationunits and are not rescaled to “real” time units. The integrationtime step was δt = 0.04 ps. The temperature (325 K) and pres-sure were maintained using the Berendsen coupling scheme.The pressure was maintained semi-isotropically, in which thepressure coupling is isotropic in the x and y directions, butadjusted independently in z. For all simulations, the pressurein the z direction was set to 1 bar. The applied surface ten-sion γ was controlled by varying the pressure in the lateraldirections:

γ = Lz

[Pzz − 1

2

(Pxx + Pyy

)], (37)

where Pii are the diagonal elements of the pressure tensor andLz is the box size in the z direction.

Systems consisting of 128 lipids were analyzed to studythe relationship between tension and membrane thickness(Fig. 4). A 128-lipid bilayer in the tensionless state was ob-tained from the MARTINI-related web resource.82 Surfacetension was turned on and followed by an equilibration pe-riod of 200 ns. The properties of the membrane were thenanalyzed over an additional 200 ns.

Thermal fluctuation spectra were obtained by simulatingsystems containing 2048 lipids. Preparation of the bilayer in

Surface Tension (mN/m)

CG

0 1 2 3 4 5

0.985

0.99

0.995

1

0 1 2 3 4 51

1.01

1.02

FIG. 5. The average monolayer thickness b∗ as a function of applied surfacetension for the implicit solvent model (CG). b0 is the average thickness forthe tensionless membrane. From Eq. (25), the predicted values are shown kA

= 190 mN/m (solid). Inset: The average volume per molecule b∗�∗ dividedby the tensionless value b0�0 as a function of surface tension.

the tensionless state was described in Paper I.34 Surface ten-sions of γ = {0, 5.0, 9.7, 19} mN/m were then applied, fol-lowed by 1.6 μs of equilibration. Production runs of 2 μs wereperformed with data saved every 25 000 steps.

B. Implicit solvent model (CG)

We conducted additional membrane simulations using acoarse-grained, implicit solvent lipid model (CG) presentedby Brannigan et al.20 All runs were conducted at a tempera-ture of kbT = 0.85ε with energy scale ε = 2.75 kJ/mol. Weadopted the same modifications to the original model as de-scribed in Paper I34 while using a bond potential of Ubend(θ )= cbendcos (θ ), where cbend = 7.0ε. A time step of δt = 0.01 pswas used. The temperature was controlled by implicit waterLangevin dynamics, and the pressure was controlled in the xyplane by a Nosé Hoover barostat. An applied negative pres-sure in the xy plane yielded surface tensions ranging from 0to 4 mN/m (Eq. (37)). The CG model was found to be unstablewith surface tensions of 6 mN/m and higher.

In Fig. 5, the mean thickness versus surface tension isplotted for 128-lipid simulations. For these systems, simula-tions were conducted for 2 μs , with data saved every 1 × 105

time steps. Larger 3200-lipid systems were analyzed to obtainthermal fluctuation spectra. For the large systems, surface ten-sions of γ = {0, 2, 4} mN/m were applied. Each simulationlasted for 10 μs, with data saved every 2 × 105 time steps.

IV. COMPARISON TO SIMULATION

For each surface tension, the average monolayer thick-ness b∗ was measured by calculating the mean bilayer thick-ness in the z-direction, averaging over all frames, and dividingby two. In Figs. 4 and 5, b∗ is plotted as a function of surfacetension. Though the data are for 128-lipid patches, we foundthat the value of b∗ is virtually independent of the system size.In Paper I,34 the best fit value of kA for DPPC was found tobe 260 (200–270) mN/m, where the numbers in parenthesis



0 1 2

Im〈p

‖ qh−

q〉(

nm3)

Im〈p

‖ qt −

q〉(

nm3)

〈|hq|2 〉

(nm

4)

〈|p‖ q|2 〉

(nm

2)

〈|tq|2 〉

(nm

4)

〈|p‖ q|2 〉

(nm

2)

DPPC Model, No Fits

0

2

4

6

8

10

0

0.04

0.08

0.02

0.04

0.06

0.08

0.02

0.04

0.06

0.08

0

0.1

0.2

0.3

0

0.01

0.02

0.03

0.02

0.04

0.06

0.02

0.04

0.06

0 1 2

FIG. 6. Thermal fluctuation spectra for the MARTINI force field of DPPC(©, �,♦, ) for average surface tensions of γ = {0, 5.0, 9.7, 19} mN/m,respectively. The values of the elastic constants were determined by fittingthe measured spectra at zero tension to Eqs. (27)–(31) with γ λ = 0. Thecurves at finite tensions are based on our theoretical predictions (Eqs. (27)–(31) with γ λ = 0), without using any additional fits (i.e., “method A,” seetext for details). The value of γ used in Eqs. (27)–(31) is dictated by thesimulation settings. 〈|hq|2〉 is shown on a log scale in q. In the fourth row, the

purely imaginary cross correlations 〈hqp‖−q〉 and 〈tqp

‖−q〉 are displayed.

denote the 95% confidence interval. From Eq. (25), the pre-dicted thickness is shown for kA = 260 and 200 mN/m. Thelatter value shows good qualitative agreement with the sim-ulations. From Eq. (25), the nonlinear behavior of the data iscaptured as well. For CG, the best fit value of kA = 190 mN/mfrom Paper I34 is in excellent agreement with the data inFig. 5.

In addition to the monolayer thickness, we also measuredthe average volume of the hydrophobic tail region. The vol-ume was computed by multiplying the mean bilayer thicknessby the total membrane area in the xy plane. The results areshown in Figs. 4 and 5. In both simulations, the volumeincreases with surface tension by a few percent, whichcontradicts the assumption of lipid volume incompressibilitywithin our theoretical treatment. This effect is quite substan-tial compared to the relative changes in membrane thickness.

0 1 2 0 1 2

Im〈p

‖ qh−

q〉(

nm3)

Im〈p

‖ qt −

q〉(

nm3)

〈|hq|2 〉

(nm

4)

〈|p‖ q|2 〉

(nm

2)

〈|tq|2 〉

(nm

4)

〈|p‖ q|2 〉

(nm

2)

DPPC Model With Fits

0

0.1

0.2

0.3

0

0.01

0.02

0.03

0

2

4

6

8

10

0.02

0.04

0.06

0.08

0.02

0.04

0.06

0

0.04

0.08

0.02

0.04

0.06

0.08

0.02

0.04

0.06

FIG. 7. The simulation data are the same as Fig. 6 but the fitting methodis different. The curves are based on our theoretical predictions (Eqs. (27)–(31)) with γ λ = 0. The values of the elastic constants were determined byfitting the measured spectra to Eqs. (27)–(31) at each different tension (i.e.,“method B,” see text for details). The spectra obtained via method C are notsignificantly different from those displayed here and are not included.

The deviation between b∗/b0 as predicted by Eq. (25) and asmeasured in the simulations is comparable to the degree ofvolume non-conservation observed in the simulations. Notealso that compared to Fig. 4, the total volume for CG rapidlyrises over a relatively small range in tension. We will returnto this point in Sec. V.

The fluctuation spectra were measured by mapping thepositions and orientations of the lipids onto an M × M gridand using this raw data to generate the following spectra datasets (see Paper I34 for technical details):

S = {〈|hq|2〉, 〈|tq|2〉, 〈|p⊥q |2〉, 〈|p⊥

q |2〉, 〈|p‖q|2〉, 〈|p‖

q|2〉}.(38)

Three methods were used to compare this data to the theorypresented in Sec. II and the results are displayed in Figs. 6–9:

1. In method A, the values of the elastic constants{kλ, kc, kA, �, κθ , κtw} were determined by fitting the



0 1 2 0 1 2

Im〈p

‖ qh−

q〉(

nm3)

Im〈p

‖ qt −

q〉(

nm3)

〈|hq|2 〉

(nm

4)

〈|p‖ q|2 〉

(nm

2)

〈|tq|2 〉

(nm

4)

〈|p‖ q|2 〉

(nm

2)

CG Model, No Fits

0

0.2

0.4

0.6

0

0.05

0.1

0

2

4

6

8

10

0

0.1

0.2

0

0.05

0.1

0

0.1

0.2

0

0.1

0.2

0

0.05

0.1

FIG. 8. Thermal fluctuation spectra for the implicit solvent model (©,�, ♦)for average surface tensions of γ = {0, 2, 4} mN/m, respectively. See Fig. 6for more details.

spectra of the zero tension simulation; these are the samevalues presented in Paper I.34 (Here, kc ≡ kc(μ = 0) isthe same as the neutral surface bending rigidity called“kc” in Paper I.34) The values of these constants wererenormalized as predicted by Eqs. (20) and (32), usingthe values of γ taken from the simulation settings andEq. (24) to convert γ to μ. These renormalized con-stants were used directly within Eqs. (27)–(31) to pre-dict the spectra at finite tensions. We stress that methodA involves no fitting of constants beyond the determina-tion of the bare elastic constants at zero tension. If thetheory presented in Sec. II were 100% accurate, methodA would predict the tension dependence of spectra inperfect agreement with simulation, with no adjustableparameters.

2. For method B, the procedure is similar to method A ex-cept that {kλ, kc, kA, �, κθ , κtw} were used as fittingparameters for each simulation, while the value of γ wastaken from the simulation settings. This method allowsfor the possibility that the functional form of our predic-tions is correct, but that the curvatures associated with

0 1 2 0 1 2

Im〈p

‖ qh−

q〉(

nm3)

Im〈p

‖ qt −

q〉(

nm3)

〈|hq|2 〉

(nm

4)

〈|p‖ q|2 〉

(nm

2)

〈|tq|2 〉

(nm

4)

〈|p‖ q|2 〉

(nm

2)

CG Model With Fits

0

0.05

0.1

0

2

4

6

8

10

0

0.05

0.1

0

0.1

0.2

0

0.05

0.1

0

0.3

0.6

0

0.1

0.2

0

0.1

0.2

FIG. 9. The simulation data for the implicit solvent model (Fig. 8) with dif-ferent fitting parameters for each tension. See Fig. 7 for details.

our harmonic model for lipid energetics (Eq. (13)) varyas a function of tension, which is outside the scope ofthe present model.

3. In method C, both the constants {kλ, kc, kA, �, κθ , κtw}and γ were treated as parameters and used to fit the datafor each simulation. This method is presented essentiallyas a control, to verify that the fits of method B are mean-ingful and cannot be considerably improved by allowingthe tension to unphysically vary from the imposed value.

As previously discussed, although Eqs. (27)–(31) includethe possibility of nonzero γ λ for generality and to coincidewith the analogous expressions in Paper I,34 all of our meth-ods A, B, and C assume that γ λ = 0. It was verified that relax-ing this restriction did not lead to significant improvements offit relative to the results that are reported.

The best-fit values of the extracted elastic constants forall methods are shown in Table I and the best-fit approxima-tions to the simulation data are displayed in Figs. 6–9. Aspredicted by our theoretical model (method A), the values ofthe macroscopic elastic constants do not strongly vary over



TABLE I. The material parameters for our coarse-grained implicit solvent model (CG) and the MARTINI force field simulation (DPPC) as extracted from thesimulations. Each row corresponds to a different simulation. The first number in the surface tension (γ ) column is the tension applied over the course of a givensimulation, whereas the adjacent number in parentheses corresponds to the best-fit tension determined via the method C fits described in the text. All quantitieswere obtained through fitting Eqs. (27)–(31) to the data while assuming γ λ = 0. For method A, the elastic constants were obtained by fitting the fluctuationspectra at zero tension (γ = 0) using {kλ, kc, kA, �, κθ , κtw} as adjustable parameters. Predictions at all other tensions were obtained directly from these zerotension numbers with no further fitting, as described in the text. For method B, a different set of elastic constants {kλ, kc, kA, �, κθ , κtw} was obtained for eachapplied tension, assuming the value of γ taken from the simulation settings. The resulting values from the method B fits are listed with no parenthesis. Notethat in the two rows corresponding to zero applied tension, the numbers appearing without parentheses correspond to both the method A and method B results.For method C, {kλ, kc, kA, �, κθ , κtw} and γ were all treated as adjustable parameters for each applied tension. The corresponding values are listed insideparentheses. The final column of the table reports the renormalized bending rigidity appearing in the undulation spectrum (see Eq. (32)). This quantity is not apart of the fitting procedure, but follows directly from the other constants appearing in the table. It is included to indicate the extent of renormalization of themeasured bending rigidity as inferred from thermal undulations (i.e., Eq. (3)).

System γ( mN

m

)kλ

(10−20J

nm4

)kc

(10−20J

)kA

(10−20J

nm2

)�

(10−20J

nm

)κθ

(10−20J

nm2

)κtw

(10−20J

)kc (μ)

(10−20J

)DPPC 0.0(0.0) 43(43) 6.7(6.7) 26(26) 9.9(9.9) 5.4(5.4) 0.77(0.77) 6.7(6.7)

5.0(5.6) 43(43) 6.6(6.1) 24(24) 9.8(9.9) 5.4(5.4) 0.69(0.69) 6.8(6.2)9.7(9.4) 43(43) 6.2(6.7) 24(23) 10(10) 5.4(5.4) 0.64(0.64) 6.7(7.0)19(19) 43(43) 5.7(6.4) 23(22) 10(10) 5.4(5.4) 0.56(0.56) 6.4(6.5)

CG 0.0(0.0) 20(20) 16(16) 19(19) 16(16) 2.6(2.6) 1.1(1.1) 16(16)2.0(0.0) 20(20) 13(16) 17(16) 18(18) 2.6(2.5) 1.1(1.1) 13(17)4.0(5.3) 7.0(6.9) 14(12) 16(16) 20(19) 2.6(2.7) 1.0(1.0) 15(13)

the range of tensions explored. However, the slight variationsthat do occur within the method B/C frameworks lead tosignificantly improved fits relative to the pure theoretical(method A) results. In all cases, the predicted curves usingmethod C are very close to those from method B andthese redundant plots are not shown. The cross-correlationsbetween height and tilt are also displayed on the bottom rowof Figs. 6–9. As explained in Paper I,34 these quantities arepurely imaginary. We emphasize that the data for 〈hqp

‖−q〉

were not used to influence the fit parameters; the plots areincluded solely to indicate the behavior of additional physicalobservables that were not explicitly fit to, but which aremodified by the values of the physical parameters extractedfrom the fitting procedure.

We first consider the case of the DPPC simulations andassociated analysis. The method A results do quite a goodjob of fitting the simulation data (Fig. 6) and indicate thatthe theory is doing reasonably well in explaining the simu-lated results. The shortcomings of our theory that stand outare its inability to quantitatively capture the increase in thethickness spectrum peak as a function of tension (althoughthe upward trend is predicted) and the behavior of the tiltspectra at high wavevectors. These deficiencies in the fits arepartially resolved by methods B and C (Fig. 7), however thethickness peak is still not fully captured, nor is the protru-sion regime of the longitudinal peristaltic tilt 〈|p‖

q|2〉. It is im-portant to emphasize that although the quality of the fits cer-tainly improves on moving from the pure theoretical results(method A) to the method B/C procedures, the values of thephysical parameters themselves change only on the order of10% when given the freedom to deviate from the theoreticalpredictions (i.e., that {kλ, kc, kA, �, κθ , κtw} remain constantover all applied tensions). Most of these variations fall withinthe confidence intervals associated with assignment of valuesto {kλ, kc, kA, �, κθ , κtw} from the zero tension simulation(see Paper I34 for the confidence intervals on the physical pa-rameters). The lone exception to this agreement with theory

is the twist modulus, κtw, which decreases by about 30% atthe highest applied tension. We have no explanation for this,but note that the energetics associated with lipid twisting areinherently small83 and derive from inter-lipid interactions asopposed to the remaining contributions to the Hamiltonianwhich can be attributed to lipid shape through the opposingforces model54 (see Paper I34). It is perhaps unsurprising thatthe twist modulus shows the strongest dependence on tension(and hence the area per lipid) of all the reported quantities.Finally, we comment that method C naturally chooses theapplied tensions as the best fit values of γ and faithfully re-produce the method B results. This indicates that γ is servingits expected theoretical role and that our procedure to map ap-plied tensions into our constant chemical potential theoreticalframework is valid.

The results for the CG model are bit more difficult to in-terpret. Although the quality of the method A fits are com-parable to those for the DPPC model, the method B/C fitsappear to do better than in the DPPC case. However, thereis a troubling inconsistency seen in the method C results inthat the best-fit tensions do not closely correspond to the ap-plied tensions. Furthermore, the method C fits, which exploitunphysical values of tension, do not yield fits that look byeye to be any better than the method B fits. The resolutionto this paradox lies in the fact that tension is not as criticalto fitting the CG data as it is to fit the DPPC data. The rea-son for this is twofold. First, the range of accessible tensionsfor the CG model is relatively small, due to the fact that themodel becomes unstable at only moderate tensions. Second,the bending rigidity for the CG model is more than twice aslarge as for DPPC. This is problematic because the most pro-nounced impact of tension on the fluctuation spectra is in thelong wavelength behavior of the undulations; this is clearlyseen in the DPPC data. The problem with the CG model is thatthe combination of small tensions and high bending rigiditiesmeans that the simulations never reach the long wavelengthregime. A practical definition for “long wavelength” comes



from considering Eq. (3). The undulations are dominated bytension when γ q2 beats out the kcq4 in the denominator, i.e.,when L > 2π

√kc/γ . Using the largest accessible CG tension

(γ = 0.4 × 10−20 J/nm2) and kc = 16 × 10−20 J we obtain acrossover length of 40 nm, which is considerably larger thanthe 30 nm box dimension; even at the longest wavelength ob-served in the simulations, tension is not playing a dominantrole in the undulations and this is clearly seen in the simu-lation data. For comparison, the cutoff length for DPPC is12 nm at γ = 1.9 × 10−20 J/nm2 as compared to a 25 nm sim-ulation box. Without the undulation data serving as a strongconstraint on γ , the method C fits go a bit haywire since sim-ilar quality (and identical looking) fits to the method B resultscan be obtained for a range of different γ values. We believethat the small value of kλ observed at γ = 0.4 × 10−20 J/nm2

is an indication that this system is at the margin of stability.An increase in tension beyond this point results in membranerupture and the abnormally high protrusions would seem tobe a forerunner to this.

Common to both simulations is the fact that the tilt fluctu-ations slightly increase with tension. To our knowledge, this isthe first direct observation of this effect, although the behav-ior is in agreement with measurements of the nematic orderparameter at moderate tensions in a different coarse-grainedlipid model.14 Despite the imperfect quantitative agreementbetween our theory and the simulations (i.e., the disparity be-tween model A and the data) the theory does correctly predictthe observed trends related to the tilt fluctuations. For both CGand DPPC, the bare value of κθ is in good agreement withtheoretical estimates.55, 64, 67 Deviations in the bilayer thick-ness also increase with tension for both sets of simulations,as predicted theoretically. Again, the theoretical results pre-dict the correct trend, but are not successful in quantitativelyreproducing the simulations. The amplified thickness fluctua-tions can also be seen in Fig. 12 of Neder et al.14 for moder-ate tension (at much higher tensions, they found that thicknessfluctuations eventually decrease when the two leaflets becomeinterdigitated).

V. DISCUSSION AND CONCLUSION

While the energetics associated with bending and surfacetension are well known (Eq. (1)), the framework presented inthis paper provides a general theory which includes the effectsof surface tension, bending, thickness, molecular orientation,and microscopic noise. The expressions were found to be inreasonable overall agreement with molecular simulations forsurface tensions up to roughly 20 mN/m.

A theoretically interesting consequence of our model isthat the effective bending rigidity kc (μ) is predicted to de-pend upon the applied tension (Eq. (32)). This result is notsurprising, since the packing of the lipids is altered in thepresence of tension. For the systems and surface tensions(0–20 mN/m) we simulated, however, kc (μ) and kc (0) dif-fer by less than 10%, which would be difficult to unambigu-ously measure in simulations or experiments. The standardtheory (Eq. (3)) contains a constant value of the bending rigid-ity kc and has been sufficient to explain the limiting behaviorof micropipette aspiration experiments.44–47 However, these

experiments continue to be reinterpreted with growing lev-els of sophistication44, 47 in order to account for their behav-ior across all surface tensions. The effect of surface tensionon the bending rigidity may be the easiest measure in mono-layer systems, in which a wide range of tensions can be ap-plied. While such effects have been observed in molecularsimulations,27–29, 52 experimental verification would be morechallenging. Grazing-incidence diffuse X-ray scattering onLangmuir monolayers has been used to determine the value ofthe bending rigidity at various tensions.84 However, the exper-imental error was too large to probe any rigidity dependenceon γ .

In qualitative agreement with simulation data, our theorypredicts that thickness fluctuations increase at low to moder-ate tensions. This behavior is contrary to a more simplisticmodel which assumes that tension always opposes the cre-ation of interfacial area.

Despite the generally good agreement between theoryand simulation, it is clear that there is room for improvement.The results of our method A fits are not in perfect agreementwith simulation and it is natural to speculate on the possiblesource(s) of the observed inconsistencies. In Sec. II D we pre-sented thermodynamic arguments to establish a connectionbetween our constant (μ, AP, T) theory and simulations runat constant (N, γ , T). The arguments revolved around a mean-field treatment and it is reasonable to question the validity ofthis approximation. However, our method C fits to the DPPCdata returned surface tensions in good agreement with the ten-sions applied to the simulation box. Further, the small dispar-ities between the best fit γ values and applied tensions appearto be completely non-systematic. It seems that the proposedconnection between chemical potential and tension holds atthe level of precision available to this study. While the resultsof the CG simulations appear problematic with regard to thebest-fit tensions, as discussed above this is due to features in-herent to the CG model that effectively remove the influenceof tension from the observed undulation spectrum. Althoughthe deviations between applied and best-fit tensions are largefor CG, they are also non-systematic and leave no reason toquestion our procedure for mapping between ensembles.

An additional approximation inherent to our theoreticaltreatment is the assumed form of the lipid energy used inEq. (13). While we view the quadratic form of the energy tobe essential to the entire approach used in this paper, it is notessential that we expand about the tension-free b∗ = b0 stateas done in Eq. (13). Indeed, a more consistent approach wouldbe to expand about b∗ (which varies with applied tension) andthere is no a priori reason to assume that the curvature of theenergy well around b∗ need be the same as that around b0. Itis possible to generalize our effective Hamiltonian by expand-ing the opposing forces model34, 54, 65, 66 about b∗ as opposedto expanding about b0 (expanding the opposing forces modelabout b0 leads to Eq. (13) as shown in Paper I34). Althoughthis approach does lead to modified values of the physicalparameters (e.g., kA becomes kA(μ)) the predicted changesare very slight for physical tensions and cannot account forthe inconsistencies seen between the method A fits and thesimulation data. Furthermore, when the parameters associatedwith Eq. (13) are allowed to vary with tension in arbitrary



fashion via the method B/C fits, the agreement between fitsand data is still imperfect. So, while it is possible that start-ing from a microscopic picture other than the opposing forcesmodel might be able to enhance the agreement between the-ory and simulation data while retaining the overall functionalform of our effective Hamiltonian, we know that this agree-ment will remain imperfect because arbitrary adjustment ofthe model parameters is unable to perfectly match the data.

The preceding two paragraphs suggest some flaw withour underlying theoretical approach that cannot be accountedfor by imprecision in our theoretical definition of tension orthe other physical parameters appearing within our model. Wesuggest that our assumption of lipid volume incompressibility,which is a standard approximation invoked in the modelingof membrane systems at the continuum level,14, 19, 34, 56, 58–62

may be partially responsible for the observed disparities be-tween theory and simulated spectra. The volume compress-ibility modulus for lipid bilayers can be measured experimen-tally and is comparable to that of bulk water.41 The insets ofFigs. 4 and 5 clearly indicate that lipid volume is not strictlyconserved in our simulations. Although the change in volumeis only a few percent for both the DPPC and CG models overthe entire range of applied tensions, this is comparable to theobserved changes in thickness as a function of tension. It isalso important to recognize that the measured fluctuations inbilayer thickness are small relative to the average thicknessof the bilayer, so it seems likely that the unaccounted fluctua-tions in volume may explain features in the thickness spectrathat the current theory misses. Given the interplay betweenarea per molecule and lipid tilting, it also seems possible thatvolume fluctuations contribute to the tilt spectra. The incom-pressibility assumption may in principle be relaxed by allow-ing volume to depend on quantities such as chain length, tilt,and splay. Introducing lipid volume perturbations to our freeenergy expressions would result in additional cross-terms andnew elastic moduli. This generalized formulation lies outsidethe scope of this work, but should be an interesting topic forfuture investigations.

ACKNOWLEDGMENTS

Some computing time was provided by the Los AlamosNational Laboratory Institutional Computing Program. Fi-nancial support was provided by the Los Alamos NationalLaboratory Institute for Multiscale Materials Studies, oper-ated under the auspices of the National Nuclear Security Ad-ministration of the U. S. Department of Energy under Con-tract No. DE-AC52-06NA25396 and the National ScienceFoundation (NSF CHE-0848809, CHE-1153096, and CNS-0960316). F.L.H.B. is a Camille Dreyfus Teacher-Scholar.

APPENDIX A: DETERMINATION OF g(α)V

In Paper I,34 a derivation of Eq. (15) was presentedbased on a geometric construction involving decompositionof the monolayer volume element into several tetrahedra. Thatmethod, though correct, is algebraically tedious and an alter-native derivation is provided here.

FIG. 10. The infinitesimal volume element dV (1) for the top monolayer. Forgraphical clarity we write x′ ≡ x + dx and y′ ≡ y + dy. The surfaces X(1)

0 (x, y)

and X(1)1 (x, y) are shown in green and purple, respectively. Reprinted with

permission from M. C. Watson, E. S. Penev, P. M. Welch, and F. L. H. Brown,J. Chem. Phys. 135, 244701 (2011). Copyright 2011, American Institute ofPhysics.

In general, we can write the equation for a surface asX(s1, s2) where s1 and s2 are coordinates of the surface. Xis a position in 3D space and each doublet of surface coordi-nates {s1, s2} specifies a location the surface passes through.For example, adopting the Monge gauge representation usedelsewhere in this work, we have X(x, y) = (x, y, h(x, y)) forheight field h(x, y). If we agree to take the surface associ-ated with height field z(α) as the canonical monolayer sur-face for leaflet α (we could equally well have chosen z(m)

or any surface residing between these two extremes) wehave X(α)

0 (x, y) = (x, y, z(α)). At each point on the surfaceX(α)

0 (x, y) lipid tails point in direction n(α)(x, y) and the tailsextend completely linearly a distance b(α)(x, y) (see Fig. 10).We define a progress variable w along the lipid tails with0 ≤ w ≤ 1 such that w = 0 is associated with the surface X(α)

0and w = 1 is associated with the bilayer midplane. It is thenpossible to specify each point within the monolayer via thetriplet {x, y,w} and a family of intermediate surfaces withinthe monolayer by X(α)

w (x, y) = (xw, yw, zw) with

xw = x + b(α)(x, y)n(α)x (x, y)w,

yw = y + b(α)(x, y)n(α)y (x, y)w, (A1)

zw = z(α)(x, y) + b(α)(x, y)n(α)z (x, y)w.

Note that only for w = 0 do the variables x and y correspondto actual Cartesian positions in space on the surface Xw. Thesevariables refer to the Cartesian origination point of the lipidtail passing through position (xw, yw, zw).

The monolayer volume element associated with the lipidsoriginating within dx dy on the surface X(α)

0 (x, y) is definedto be dV = g

(α)V dx dy. g

(α)V is conveniently calculated by

integrating the infinitesimal volume element associated withthe displacements dx, dy, and dw over the progress variablew. Using the standard volume formula for non-orthogonal co-ordinate systems the infinitesimal volume element is given by∣∣(U1(x, y,w) × U2(x, y,w)) · n(α)(x, y)

∣∣ b(α)(x, y) dx dy dw,



where U1 dx ≡ ∂xX(α)t (x, y) dx and U2 dy ≡ ∂yX(α)

t (x, y) dy

are the two local tangent vectors to the surface X(α)t associ-

ated with displacements dx and dy. Writing the integrationexplicitly and making use of Eq. (A1), we find

g(α)V (x, y)

= b(α)(x, y)∫ 1

0dw

∣∣n(α)(x, y) · (U1(x, y,w)×U2(x, y,w))∣∣ ,

where

U1 = (1, 0, ∂xz(α)) + w∂x(b(α)n(α)),

U2 = (0, 1, ∂yz(α)) + w∂y(b(α)n(α)).

Equation (15) follows by carrying out the vector multiplica-tions indicated above and performing the required elementaryintegrals in w, retaining all terms up to second order in smallquantities.

APPENDIX B: CALCULATION OF FLUCTUATIONSPECTRA

In order to determine the amplitudes of the thermal fluc-tuation spectra, the Hamiltonian must first be expressed inFourier space. It is convenient to define the vectors

fu(q) = (z+q , m‖

q, λ+q , m⊥

q , εq),

fp(q) = (z−q , m‖

q, λ−q , m⊥

q ),

the undulation and peristaltic portions of the free energy maybe written as

Fu =∑

q

fu(−q) ⊗ A ⊗ fTu (q),

Fp =∑

q

fp(−q) ⊗ B ⊗ fTp (q),

where “⊗” denotes matrix multiplication. The matrices A andB are given by

A =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎝

− μ

�0q2 + kb

c q4 −ikb

c q3 γλq

2 0 �(μ)2b0

q2

ikbc q

3 κθ (μ) + kbc q

2 0 0 iq�(μ)

2b0

γλq2 0 kλ + γλq

2 0 0

0 0 0 κθ (μ) + κtwq2 0�(μ)

2b0q2 −iq

�(μ)2b0

0 0 kA

b20

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎠

, (B1)

B =

⎛⎜⎜⎜⎜⎝

kA

b20

− q2[�(μ)b0

+ μ

�0] + kb

c q4 iq

�(μ)2b0

− ikbc q

3 γλq2 0

−iq�(μ)

2b0+ ikb

c q3 κθ (μ) + kb

c q2 0 0

γλq2 0 kλ + γλq

2 0

0 0 0 κθ (μ) + κtwq2

⎞⎟⎟⎟⎟⎠ . (B2)

Note that since∫∫L2

∇2z− dr =∫∫

L2(∇ · m) dr = 0

under periodic boundary conditions, the terms with theη(μ) prefactor in Eq. (18) do not appear in the Fourierrepresentation.

The fluctuation spectrum is determined directly from theequipartition theorem. In general, when the free energy

H = 1

2

∑q

f (−q) ⊗ D ⊗ fT (q)

can be written in terms of a Hermitian matrix D, the fluctua-tions satisfy36, 58, 85

〈fT (q) ⊗ f(q′)〉 = kBT δq,−q′D−1,

which is simply a statement of the equipartition of energy forthe Fourier components of a real-space observables.

As discussed in Sec. II, the quantities measured in simu-lations are modeled as a sum of macroscopic and microscopiccomponents (Eqs. (7) and (10)). Their corresponding thermalfluctuation spectra contain macroscopic and microscopic con-tributions as well. For example, 〈|hq|2〉, 〈|p‖

q|2〉, and 〈|p⊥q |2〉

are calculated as follows:

〈|hq|2〉 = 〈(z+q + λ+

q )(z+−q + λ+

−q)〉

= kBT

2

(A−1

11 + A−133 + A−1

13 + A−131

),

〈|p‖q|2〉 = 〈(m‖

q − iqλ+q

) (m

‖−q + iqλ+

−q

)〉

= kBT

2

(A−1

22 + iqA−123 − iqA−1

32 + q2A−133

),

〈|p⊥q |2〉 = kBT

2

(A−1

44

).



Similar expressions hold for the peristaltic fluctuations. Notethat 〈|p⊥

q |2〉 is not affected by protrusions since the observedtilt is only altered by gradients in λ± (see Eq. (10)).

APPENDIX C: INTEGRATING OUT THE TILT DEGREEOF FREEDOM

In this section we give an expression for the free en-ergy after integrating out the tilt degree of freedom in Fourierspace. Assuming γ λ = 0, protrusions are decoupled from z±

and can be integrated out immediately. The same follows form⊥

q and m⊥q . For the peristaltic modes, the partition function

Qp associated with z−q is given by

Qp ∝∏q>0

∫ ∞

−∞e− Hp

kB T dRe[z−q ] dIm[z−

q ] dRe[m‖q] dIm[m‖

q],

(C1)

where

Hp =∑q>0

{[kbc q

2 + κθ (μ)](Re[m‖

q]2 + Im[m‖q]2)

+[kA

b2∗−

(μ

�0+ � (μ)

b0

)q2 + kb

c q4

]

× (Re[z−

q ]2 + Im[z−q ]2

)+

[iq

� (μ)

b0− 2iq3kb

c

]

× (Re[z−

q ]Re[m‖q] + Im[z−

q ]Im[m‖q]

)},

and q > 0 signifies integration over the upper half plane ofq-values.58 Integrating over Re[m‖

q] and Im[m‖q], Eq. (C1)

becomes

Qp ∝∏q>0

(π

kBT(κθ (μ) + kb

c q2)) 1

2

×∫ ∞

−∞e− Hp

kB T dRe[z−q ] dIm[z−

q ],

where

Hp =∑q>0

{(2kb

c q3 − q

�(μ)b0

)2

κθ (μ) + q2kbc

+ kA

b20

−(

μ

�0+ � (μ)

b0

)q2 + kb

c q4

} (Re[z−

q ]2 + Im[z−q ]2

).

(C2)

The first term contained within the curly brackets of Eq. (C2)is due to the integration over tilt, while the remaining termswere present before the integration. While the first term doesnot reduce to a simple form in real space, it can be expanded

about q-values much less than qc � √kbc /κθ (μ):

Hp =∑q>0

{kA

b20

+(

� (μ)2

b20κθ (μ)

− μ

�0− � (μ)

b0

)q2

+(

kbc − kb

c � (μ) (� (μ) /b0 + 4κθ (μ))

b0κθ (μ)2

)q4

+O(q6

) } (Re[z−

q ]2 + Im[z−q ]2

).

On long length scales, integrating out tilt results in a renor-malization of the q2 and q4 terms. One may also integrate outtilt for undulations in the same manner as above. However,no coefficients below O(q6) are renormalized. While thecollection of terms in front of q4 corresponds to the bendingmodulus (kb

c before the integration), renormalizing the q2

coefficient (−μ/�0 + �(μ)/b0] before the integration)may correspond to a [∇z−]2 or z−∇2z− term in the realspace Hamiltonian. For homogeneous membranes with noboundaries, these terms are energetically indistinguishable.For convenience, we show how the constants in Bitbol et al.56

{K ′a,K

′′a } are related to ours:

K ′a

2= −� (μ)

b0+ � (μ)2

b20κθ (μ)

,

(C3)K ′′

a

2= kb

c − kbc � (μ) (� (μ) /b0 + 4κθ (μ))

b0κθ (μ)2 .

When �(μ) = 0, integrating out the tilt degree of freedomfrom our model does not lead to the creation of any en-ergetically distinguishable terms in the Hamiltonian, sincethe renormalization of the q2 term, � (μ)2 /b2

0κθ (μ), isproportional to �(μ). Aside from the renormalization ofconstants, the Hamiltonian with tilt integrated away and theHamiltonian with zero tilt have the same functional forms.

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Thermal fluctuations in shape, thickness, and molecular ... · I. INTRODUCTION Surface tension...

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