On measuring the bending modulus of lipid bilayers with cholesterol
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On measuring the bending modulus of lipid bilayers with cholesterol
John F. Naglea)
1
Department of Physics, Carnegie Mellon University, Pittsburgh, PA 15213,
USA
(Dated: 11 August 2021)
Regarding the effect on the bending modulus of adding cholesterol to lipid bilayers, recent results using
neutron spin echo and nuclear magnetic resonance relaxation methods that involve linear transport properties
have conflicted with earlier results from purely equilibrium experiments that do not involve linear transport
properties. A general discussion indicates how one can be misled by data obtained by methods that involve
arXiv:2108.04653v1 [physics.bio-ph] 10 Aug 2021
linear transport properties. It is then shown specifically how the recent neutron spin echo results can be
interpreted to agree with the earlier purely equilibrium experimental results, thereby resolving that conflict.
Regarding the nuclear magnetic resonance relaxation method, it is noted that current interpretation of the
data is unclear regarding the identity of the modulus that is involved, and an alternative interpretation is
explored that does not disagree with the results of the equilibrium experiments.
Keywords: lipid membranes, elasticity, bending modulus
I. INTRODUCTION data in addition to the static elastic moduli. While this
potentially allows NSE and NMRR to obtain more in-
A decade ago, four independent groups reported the formation than the purely equilibrium methods, it also
unanticipated and still surprising result that the bending means that there are more quantities that have to be de-
modulus KC of DOPC does not increase with addition of termined from limited data. Most importantly, it means
cholesterol. Bassereau’s group1 and Baumgart’s group2 that the non-equilibrium statistical mechanical theory for
studied tubules, Dimova’s group3 analyzed shape fluctu- interpreting the data has to be correct. Given that non-
ations (SA) in giant unilamellar vesicles as well as vesicle equilibrium theory is more complex and less certain than
deformation in electric fields, and my group4,5 analyzed equilibrium theory, it is clearly a warning that something
X-ray diffuse scattering (XDS) from stacks of bilayers. may be wrong when those conclusions do not agree with
My group also systematically studied the effect of choles- results from four purely equilibrium methods.
terol on other standard lipids with the expected result The NSE and NMRR data are appealing because they
that KC increased dramatically for bilayers of DMPC clearly show that cholesterol slows down measured rates
lipid that has saturated hydrocarbon chains. For mono- substantially. However, this alone says nothing directly
unsaturated SOPC, KC increased, but less. And KC about the bending modulus. Section II reminds biophysi-
did not increase for C22:1PC, which like DOPC, is di- cists of this. Section III reviews details of the NSE inter-
unsaturated. pretation. This interpretation has gradually evolved and
In striking contrast, a recent paper reports that the it is accepted in this paper up to the final equation. That
bending modulus of DOPC increases threefold with in- equation had been modified some time ago for cholesterol
creasing cholesterol concentration up to 50%.6 The two in a non-NSE context.5 That modification fully reconciles
experimental techniques that pertained to the bending the NSE data with the equilibrium results. In contrast,
modulus were neutron spin echo (NSE) and nuclear mag- the NMRR data are not so easily understood as is dis-
netic resonance relaxation (NMRR). This development cussed in Section IV. The NMRR data interpretation is
has made this into a controversial topic.7,8 This paper forty years old. Arguments are given that, like the NSE
will attempt to reconcile this controversy. interpretation, it too needs to evolve, and a suggestion
The four methods in the first paragraph are quite is discussed for what that might look like. In Section V
different from each other. But they share the feature there is a brief discussion of molecular dynamics (MD)
that the interpretations of their data do not involve non- simulations. A simulation that was performed to support
equilibrium quantities and therefore rely only on the firm the NSE and NMRR experimental results6 disagrees with
principles of equilibrium statistical mechanics. In con- an earlier simulation that agrees with the results of the
trast, NSE and NMRR data basically report decay rates equilibrium methods.9
which necessarily involve non-equilibrium quantities. It
is usually assumed that the lipid bilayer system can be
treated in the linear transport regime of non-equilibrium II. PROPAEDEUTICS
statistical mechanics, and then viscosity and friction are
quantities that have to be included for interpreting the The bending modulus is a static equilibrium quantity.
That may sound strange to biophysicists because it is of-
ten measured from fluctuations, as in the XDS and SA
methods, until it is appreciated which aspects of fluc-
a) E-mail: nagle@cmu.edu tuations yield purely equilibrium quantities and which2
likely difference in molecular nano scale systems, which
can make spring 1 dynamically slower than spring 2, is
viscous friction. Viscosity is a non-equilibrium quantity
in the linear transport category. A system with higher
1 2
viscosity that is pulled from its mean value will relax back
A(t) more slowly than one that has the same energy landscape
but with lower viscosity. When the system spontaneously
0 undergoes a thermal fluctuation from its average value,
it likewise takes longer for the system to fluctuate back
towards its average value so the dynamics are stretched
out as in Fig.1.
Of course, it should be remembered that the funda-
mental fluctuation-dissipation theorem11 importantly re-
lates fluctuations to non-equilibrium transport quanti-
Time t ties, but those are not the time-independent equal-time
fluctuations h(A(t))2 it used for the equilibrium mod-
uli. Instead, they are the time dependent correlations
FIG. 1. Schematic of the fluctuations A(t) in the dynamic hA(t + τ )A(t)it at different times τ . The faster dynamics
amplitude for system 1 (left) and system 2 (right). in system 2, namely the faster decay of hA(t + τ )A(t)it
with τ would be related by the fluctuation-dissipation
theorem to a non-equilibrium transport quantity such as
require non-equilibrium quantities. Of course, any sys- a smaller viscosity in system 2 than in system 1.
tem in equilibrium in a thermal environment at tem- This is a good place to comment on misconceptions
perature T must, over time, repeatedly sample all its about time scales of various experiments. It was writ-
configurations ξ with Boltzmann relative probabilities ten that the time scale of the XDS experiment is long,6
exp(−E(ξ)/kBT ) where E(ξ) is the energy of state ξ. but the actual XDS time scale is of the order of fem-
A system in thermal equilibrium with its environment toseconds, far shorter than the nanosecond time scale of
at non-zero temperature cannot just sit in its lowest en- NSE, and therefore a priori superior for obtaining the
ergy state; it must have equilibrium fluctuations. That is relevant τ = 0 correlations for the equilibrium bending
consistent with the system being in its lowest free energy modulus. As in any experimental method data are ac-
state. In the case of the bending modulus of membranes, cumulated over many events; XDS data are accumulated
KC is determined by the SA and XDS methods from the over a long time to obtain good statistics from many pho-
average mean square equal-time equilibrium fluctuations tons, each with its femtosecond time scale. It should also
in the Fourier coefficients of the height-height correla- be emphasized that there is no relevant time scale for the
tion functions. The tubule method1,2 and the electro- tubule method or the electro-deformation method, both
deformation method3 are actually more direct in that of which are performed in the steady state.
they measure bending under steady applied force rather It has also been suggested that NSE and NMRR are at
than relying on fluctuations. the appropriate molecular length scale,6 but there is no
It is important that biophysicists appreciate how and separate bending modulus for shorter length scales. The
why the equilibrium fluctuations used to measure the classical Helfrich bending modulus is defined as the lead-
bending modulus are distinct from dynamics. Fig.1 ing order of energy that is dominant at the longest length
shows two dynamical traces corresponding to two sys- scale. At shorter length scales, consideration of other
tems, labelled 1 and 2, for the fluctuations in any gen- types of energy can become important. It has become
eral amplitude A as a function of time. The two traces clear that molecular tilt plays a role at shorter length
are identical except that the slow trace occurs at half scales and that requires a separate tilt modulus.12–15
the speed of the fast trace. Therefore, the time aver- Disentangling these two moduli experimentally is non-
age of the mean square fluctuations hA(t)2 it is identical. trivial, although it has been accomplished by XDS.16,17
Those are the fluctuations whose Fourier transforms are Let us also introduce another example that might help
directly related to the bending modulus by the standard, illuminate the concepts. Suppose that one increases the
well-known formula KC = kB T /hA(q)2 iq 4 for bilayers mass of a system in such a way that its energy landscape
under zero tension,10 so the bending modulus of the two does not change. Again, the relaxation time is longer
systems must also be equal despite having unequal dy- and the thermal time-dependent fluctuations are slower
namics. because the random hits from the thermal environment
Another example of this fundamental distinction is result in smaller changes in velocity because of greater
if the trace amplitudes illustrated in Fig.1 were to re- inertia of the system, but the average mean square equal-
port time-independent fluctuations in the lengths of two time fluctuations will not change over time and any equi-
springs. Then the spring constants, which are also equi- librium modulus will have the same value. This suggests
librium quantities like moduli, must also be equal. A a macroscopic example that may be intuitively useful as3
it can be perceived at the personal level. Suppose you around used a value of η roughly three times greater than
approach an unlatched door that is in the closed position. the viscosity of water.19,20
If the door has a large mass, it will appear “stiffer” to you That problem was resolved for NSE analysis when
when you push to go through it than a less massive door. Watson et al. developed a non-trivial quantitative theory
This perception of stiffness has nothing to do with the on top of the previous theory to take into account inter-
energy landscape of the doors; the corresponding mod- nal friction in the membrane.21 The NSE theory then
uli can be made equal by the usual (though somewhat evolved to
complicated in practice) door closure hardware. Instead,
1/2
perceived door stiffness is related to non-equilibrium per-
kB T kB T
turbations to the system, in this case your pushing the Γ(q) = 0.025 q3 , (2)
KCD η
door. Likewise, the more massive door will relax to clo-
sure more slowly with equal closure hardware. Of course, This replaced KC in Eq.(1) with a dynamical bending
the more rapidly you open the door, the more work you modulus in the NSE time regime KCD given by22
do, and that extra work is dissipated irreversibly.
These examples should alert biophysicists to an im- KCD = KC + h2 KA , (3)
portant distinction between physical quantities of inter-
est that can be obtained from a system in equilibrium. where KA is the area compressibility modulus of the bi-
Some quantities are static equilibrium moduli that can be layer and h is the distance from the bilayer midplane to
obtained straightforwardly from time-independent quan- the neutral surface of either monolayer of a symmetric
tities like hA(t)2 it . The static moduli suffice for describ- bilayer. The neutral surface, closely related to the piv-
ing cellular subsystems that effectively operate near equi- otal plane, is an important conceptual property in mem-
librium. Other non-equilibrium quantities like viscos- brane mechanics used by theorists.21–24 , It is the location
ity and friction are related, rather less straightforwardly in each monolayer where stretching is decoupled from
and limited to the linear regime, to measurement of the bending.25 It has not been experimentally measured in
τ time-dependent hA(t + τ )A(t)it auto-correlation func- bilayers, but is widely assumed to be located near the
tion. When a cell process deviates from reversibility, non- interface between the hydrocarbon chain region and the
equilibrium dissipative quantities become important, and headgroups.
the relative importance of the two types of quantities de- It is interestingly deceptive that all the quantities on
pends upon how far the process deviates from reversibil- the right hand side of Eq.(3) are equilibrium quantities.
ity. Because different biological processes span a large Even though the derivation21 involves non-equilibrium
range of time and length scales, the details of the actual internal viscosity ηs and a friction coefficient b, in the
biological process matter. Assigning a single “effective” time and length scale regimes of NSE (but not in
bending modulus from experiments that are unrelated to those of dynamic light scattering), these non-equilibrium
a specific biological process6 is likely to be misleading and transport quantities drop out of Eq.(3), rather like the
impede physical understanding of biological processes. Cheshire cat leaving behind only an equilibrium smile.
It is illuminating to consider the relative magnitudes
of the terms in Eq.(3). For typical values of h and KA ,26
III. RECONCILIATION OF NSE WITH EQUILIBRIUM the h2 KA term in Eq.(3) is an order of magnitude greater
DOPC/CHOLESTEROL RESULTS than KC . This is consistent with the previously noted re-
sult that KC was an order of magnitude too large when
The NSE story is quite interesting. In its experimen- the viscosity of water was used in Eq.(1). Given this weak
tally available time window NSE provides the q depen- dependence of KC on the NSE measured KCD , Eq.(3)
dent relaxation rates for bending fluctuations Γ(q) which suggests that a more appropriate and important way to
rather non-trivial non-equilibrium theory interprets as use NSE data would be to provide the first experimental
1/2 measurement of the neutral surface h in bilayers. Values
of KA only have 10% uncertainties.26 There are larger un-
kB T kB T
Γ(q) = 0.025 q3 , (1) certainties of 30% in KC between different techniques,17
KC η
but that matters even less because KC itself makes such
where η is the solvent viscosity.18 The presence of the a small contribution to KCD in Eq.(3). (It should also
non-equilibrium transport property η in this interpreta- be noted that, in contrast to the magnitude of KC , the
tion is expected and reiterates the fact that NSE mea- uncertainties in the cholesterol dependence of KC be-
sures dynamics in a time window that does not permit di- tween different methods are much less than 30%.) If the
rect measurement of hA(t)2 it from which the static bend- theory21 giving Eq.(2) and Eq.(3) is accurate, then NSE
ing modulus could be directly obtained. While it was could provide experimental values for the fundamental h
historically assumed that KC in Eq.(1) was the static quantity.
equilibrium bending modulus, when the water value of η Instead, NSE practitioners have been prone to inter-
was used, the extracted value of KC was much greater pret their data as yet another way to obtain KC . To do
than any other measurement of KC and for a time a work- this, a popular relation between KC and KA has been4
invoked27 Combining with Eq.(2) gives
KC KCD
KA = β , (4) KC = , (7)
(2Dc )2 1 + 12(h/δ)2
where 2DC is the hydrocarbon thickness of the bilayer. If we assume that h is the same as the monolayer hy-
The number β has been variously given as 12 for coupled drocarbon thickness DC = 15.3 Å at 50% cholesterol,5
monolayers, 48 for uncoupled monolayers and 24 for the then the denominator on the right had side of Eq.(7) is
polymer brush model.26 Combining Eq.(4) with Eq.(3) about 35 for 50% cholesterol. In contrast, the denomina-
then yields the formula tor of Eq.(5) is 13 for no cholesterol. This would require
KCD to increase essentially by a factor of 3 in order for
KCD KC to increase at all. Instead, it was noted5 that an
KC = , (5)
1 + β( 2DhC )2 increase in KA in Eq.(6) would be consistent with no
increase in KC if δ were somewhat larger than 9 Å. A
This equation has been used to obtain KC by assuming purely phenomenological calculation of δ that satisfies
values of β, h and using experimental values for 2DC . Eq.(6) obtained a cholesterol dependence that sensibly
Although it was stated that the polymer brush model interpolated to pure DOPC and Eq.(4). And the same
was used,27 the actual values used were β = 48 and h = phenomenology appeared to apply to SOPC bilayers with
DC . Using these values in Eq.(5) and using the result to cholesterol.5
eliminate KCD in favor of KC , one obtains an equation Although it would be good to have a more fundamen-
that has exactly the same form as Eq.(1) except that tal microscopic theory to gain further insight into the
the numerical prefactor 0.025 is replaced by 0.0069.6,27 effect of cholesterol on the mechanical properties of lipid
The point of recounting this history is to emphasize that bilayers, this section shows that there is no conflict be-
the recent experimental determinations of KC using NSE tween the NSE results and the older experimental results
rely on (i) Zilman-Granek theory,18 (ii) Watson-Brown that reported no increase in KC as cholesterol is added
theory,21 and most importantly for what follows, (iii) an to DOPC bilayers.
assumed relation between KC and KA in Eq.(4).
We are now ready to return to the controversial case
of cholesterol in DOPC. The NSE results that have led IV. NMRR AND THE DOPC/CHOLESTEROL
to the controversy have used Eq.(4) with essentially the CONTROVERSY
same values of β and the ratio h/DC as cholesterol was
added. Instead, let us bypass Eq.(4) and examine what Nuclear magnetic resonance relaxation reports6 relax-
Eq.(3) can tell us directly from experiment and by assum- ation time τ which is a dynamical quantity involving
ing plausibly constant values of h. Then, the cholesterol hA(t + τ )A(t)it . As such, the data are related to a non-
dependence of KCD mainly depends on KA . The exper- equilibrium quantity as is indeed shown by the presence
imental KA of DOPC upon adding cholesterol increases of viscosity η in the formula6 used to interpret the mea-
by a factor of 3 with 50% cholesterol.28 The threefold sured quantity C
increase with cholesterol in the dynamical bending mod- √
ulus KCD that is obtained by NSE6 is therefore entirely 3kB T η
C= √ . (8)
consistent with the threefold experimental increase in KA 5πSS2 2K 3
and requires no increase in the minor contribution from
the static equilibrium bending modulus KC . This essen- Interestingly, this η was described as a membrane vis-
tially resolves the controversy regarding NSE data. cosity in contrast to the solvent viscosity in the NSE
But this controversy is insightful because the different formula in Eq.(1). The presence of the molecular order
behavior of KA and KC with cholesterol concentration parameter SS emphasizes the small length scale of the
in DOPC implies that Eq.(4) does not hold for choles- method. The K modulus in Eq.(8) was declared to be
terol. This was already appreciated some time ago. Evan essentially equal to the bending modulus KC divided by
Evans informed the authors of the XDS study that the the membrane thickness, although it was noted that K
basis of the polymer brush model was not satisfied when was a single elastic constant, consistent with there being
a rigid molecule like cholesterol is added and he suggested other possibilities. The original paper that introduced
an alternative model for high 50% cholesterol.5 Evans’s K declared that it could be derived from the general liq-
cholesterol model postulated that the mechanical prop- uid crystal literature, but the specific references given
erties are dominated by a stiff region in each uncoupled were for the nematic phase.29 Since the nematic phase
monolayer that has a length δ = 9 Å corresponding to has no layers to undulate with a related bending modu-
the length of the rigid hydrocarbon rings of cholesterol. lus KC , this prompts one to ask about the meaning of K
This theory replaced Eq.(4) with in the context of the continuum description that actually
defines the moduli of membranes. It might also be men-
KC tioned that Eq.(8) is based on the hypothesis that the
KA = 12 (6) slow NMR dynamics are due to collective motions; that
δ25
hypothesis has been challenged by fitting NMR data to NMRR formula that hasn’t yet been derived, but they
a non-collective model.30 support the tentative suggestion that the tilt modulus
Both nematic and smectic phases have molecular tilt is something to be considered in the interpretation of
degrees of freedom. This has become increasingly recog- NMRR data.
nized in the membrane literature and the corresponding What is needed for the interpretation of NMRR data
modulus in the collective continuum description is the is a proper derivation of a relation to the moduli that
tilt modulus Kt that accompanies the tilt vector m.12 are fundamentally defined in the continuum description
The relevant part of the free energy in this description is in Eq.(9). Such a formula would likely include the tilt
modulus, the bending modulus, the bulk modulus B
1
Z
2 for multilamellar dispersions, and maybe even the new
X
H= d2 r ((KC ∇2r un + ∇r · mn +
2 A n (9) splay-tilt coupling term.34 It had been previously noted
that the single-modulus K in Eq.(8) might consist of
Kt mn 2 + B(un+1 − un )2 ),
more than one modulus, but the suggested formula (Eq.
4.9)32 assigned the same q dependence to all moduli. A
where the average bilayer normal is along the z axis with
proper connection to the continuum energy definition in
fluctuations un from the average position of the nth bi-
Eq.(9) would necessarily assign different powers of q to
layer. For multilamellar systems with more than one
the different moduli.13 Of course, viscosity and/or fric-
membrane n, B is the bulk compression modulus. No
tion would also appear; note that there are both mem-
derivation of the connection between this defining de-
brane viscosity ηm and a separate friction term b for in-
scription of the moduli and the K that appears in Eq.(8)
termonolayer slippage.21,35 It might also be noted that
has been published.
a major improvement in the analysis of NSE data oc-
It has been recognized that B plays a role in NMRR
curred when it was realized that the time dependence of
analysis,31 although it has been argued that it does
the relaxation required a stretched exponential.18 This
not affect the analysis in the MHz range of multil-
possibility does not appear to have been considered in
amellar smectic systems because the length scale is lim-
the analysis of NMRR data.
ited to that of membranes undulating freely from their
neighbors.32 Ignoring concerns of being able to discern It would likely be a daunting task for NMRR analysis
gradual crossover between these length scales, let us fo- to obtain the numerical values of all the properties that
cus on tilt in Eq.(9) which has not yet been considered affect the data. This task would be easier if some values
in Eq.(8).6,29,32,33 It is well recognized that the effect of could be fixed to those obtained from other experimental
tilt on the Fourier undulation spectrum increases relative methods. Measurements that only focuses on viscosity,
to the effect of bending as the length scale decreases to- like a newly developed one,36 could provide that value.
ward the smaller molecular length scale with a crossover Equilibrium methods that only focus on KC could pro-
length (KC /Kt ) of order 1 nm.13 Given the recent em- vide that value. For multilamellar systems, the B value
phasis placed on the small length scale of NMRR,6 it is obtained by XDS.5 At this time, only the XDS method
seems appropriate to consider that K may be more re- obtains experimental values for Kt and that is empirically
lated to the tilt modulus Kt than to the bending modulus challenging because the lower end of the x-ray length
KC . scale is a bit larger than the molecular length scale.16,17
The added value of NMRR could be that it might be a
Let us consider the possibility that a proper derivation
second method for measuring Kt as well as tying together
might confirm Eq.(8) but with a different interpretation
the other quantities. Whether or not that will transpire,
for K. The previous interpretation is that K equals KC
it is clear that the present assertion that cholesterol in-
divided by hydrocarbon thickness 2DC , and it will now
creases KC in DOPC bilayers6 is not firmly based on the
be considered that K = Kt times 2DC . Using typical
NMRR data, so it is quite insufficient to overturn the
values of 10−19 J for KC , 2.5 nm for 2DC , and 0.05 N/m
original equilibrium results for the effect of cholesterol
for Kt would replace K = 4x10−11 N by 12.5x10−11 N
on the KC of DOPC lipid bilayers.1–5 According to the
in Eq.(8). For the same measured values of C and SS
tentative suggestion presented here for re-interpreting K
in Eq.(8), this increase in K by a factor of 3.1 would in-
in Eq.(8), the NMRR data would indicate a plausible
crease the membrane viscosity η by a factor of 3.13 = 30.
threefold increase in the tilt modulus Kt as cholesterol is
The recent paper did not report a value for viscosity ob-
added to DOPC.
tained from NMRR to compare to the value for DOPC of
1.5x10−8 Pa.s.m obtained from NSE.6 However, an ear-
lier NMRR study33 reported a bulk viscosity for DMPC
of η = 0.0707 Pa.s, which converts, after multiplying by V. MOLECULAR DYNAMICS
the hydrocarbon thickness, to a membrane viscosity of
1.8x10−10 Pa.s.m, two orders of magnitude smaller than It would be satisfying if the remarkably good agree-
the NSE value for DOPC. This unlikely large difference ment of the molecular dynamics simulations6 with the
would be alleviated by multiplying by the factor of 30 dynamical NSE KCD is simply because both are dynam-
obtained above by replacing K = KC /2DC by Kt 2DC . ical methods. Unfortunately, that is not the case. Molec-
Of course, these numbers are rough and depend upon an ular dynamics is truly dynamical in the sense that an6
MD simulation obtains trajectories in time. By calculat- fication to the last part of the recent NSE data anal-
ing relaxation and time dependent autocorrelation func- ysis completely resolves that inconsistency. It is to be
tions, MD can obtain dynamical non-equilibrium prop- hoped that consistency for the NMRR method can also
erties by looking at τ -dependent hA(t + τ )A(t)it fluctu- be achieved by updating the formula used for interpreting
ations. But MD can also obtain equilibrium properties those data.
and many molecular dynamics studies go to consider- Although it is clear experimentally that the equilib-
able effort to ensure that simulations run long enough rium bending modulus KC of DOPC bilayers does not
to achieve and then sample the equilibrium fluctuations. increase with up to 50% cholesterol, this result remains
Then, time averaged mean square equal time (τ =0) fluc- surprising, especially since the bilayer becomes thicker as
tuations hA(t)2 it can be extracted to provide equilibrium cholesterol is added.5 There is no ab initio theory that ex-
thermodynamic quantities that do not involve time or plains why KC does not increase, either for DOPC or for
dynamics. Like Monte Carlo methods, this use of molec- the longer chain di-unsaturated C22:1PC lipid, whereas
ular dynamics is a powerful way to sample the equilib- KC does increase with added cholesterol in SOPC that
rium ensemble and provide static equilibrium quantities has one saturated chain and quite dramatically in DMPC
of the h(A(t))2 it type. The reason for emphasizing this that has both chains saturated.5 While this is not well un-
is to avoid semantic confusion when referring to molec- derstood theoretically, it might be biomedically relevant
ular dynamics simulations that obtain the equilibrium because animals require cholesterol and membranes have
bending modulus in a way that does not use dynamical to be flexible, so a way to satisfy both requirements is to
hA(t + τ )A(t)it quantities. have lipids with unsaturated chains rather than having
The atomistic MD simulations that report an increase short saturated chains.42 Asserting that there is a uni-
in bending modulus of DOPC with increasing choles- versal effect of cholesterol on all lipid bilayers6,8 imposes
terol similarly obtain non-dynamical time averaged mean an artificial consistency that is likely to impoverish our
square equal time fluctuations hA(t)2 it .6 Therefore, this understanding of membrane mechanics. Instead, it is to
simulation should be obtaining KC not KCD , so it is be hoped that the additional perspectives that can be
mysterious why the simulated KC cholesterol depen- brought to bear by a variety of properly interpreted ex-
dence agrees so well with the NSE KCD results. An- perimental techniques will further our understanding of
other perhaps related mystery is that this MD simula- the effect of cholesterol on the biophysical properties of
tion group has used similar analysis methods to study membranes.
the area compressibility modulus KA of each monolayer Acknowledgements: I thank Markus Deserno, Rumi-
in a bilayer.37 The counter intuitive result was reported ana Dimova, Michihiro Nagao, Elizabeth Kelley and
that each monolayer in a symmetric bilayer has the same Evan Evans for helpful comments.
value of KA as the bilayer itself instead of the intuitively
obvious and rigorously proven result that KA for a bi-
layer is the sum of the KA of its monolayers.38 The anal- REFERENCES
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VI. SUMMARY AND AFTERTHOUGHTS
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25 S. A. Safran, Statistical thermodynamics of surfaces, interfaces,You can also read
