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Concerted diffusion of lipids in raft-like
membranes
Touko Apajalahti,
a
Perttu Niemel€
a,
b
Praveen Nedumpully Govindan,
c
Markus S. Miettinen,
c
Emppu Salonen,
c
Siewert-Jan Marrink
d
and Ilpo Vattulainen*
efg
Received 23rd January 2009, Accepted 25th March 2009
First published as an Advance Article on the web 15th August 2009
DOI: 10.1039/b901487j
Currently, there is no comprehensive model for the dynamics of cellular
membranes. The understanding of even the basic dynamic processes, such as
lateral diffusion of lipids, is still quite limited. Recent studies of one-component
membrane systems have shown that instead of single-particle motions, the lateral
diffusion is driven by a more complex, concerted mechanism for lipid diffusion
(E. Falck et al.,J. Am. Chem. Soc., 2008, 130, 44–45), where a lipid and its
neighbors move in unison in terms of loosely defined clusters. In this work, we
extend the previous study by considering the concerted lipid diffusion
phenomena in many-component raft-like membranes. This nature of diffusion
phenomena emerge in all the cases we have considered, including both atom-scale
simulations of lateral diffusion within rafts and coarse-grained MARTINI
simulations of diffusion in membranes characterized by coexistence of raft and
non-raft domains. The data allows us to identify characteristic time scales for the
concerted lipid motions, which turn out to range from hundreds of nanoseconds
to several microseconds. Further, we characterize typical length scales associated
with the correlated lipid diffusion patterns and find them to be about 10 nm, or
even larger if weak correlations are taken into account. Finally, the concerted
nature of lipid motions is also found in dissipative particle dynamics simulations
of lipid membranes, clarifying the role of hydrodynamics (local momentum
conservation) in membrane diffusion phenomena.
I. Introduction
The dynamics of membranes have been studied extensively over several decades, see
e.g. ref. 1–3. Yet, one of the outstanding issues regarding cell membrane properties is
the fact that we know far too little about membrane dynamics, and in particular about
the underlying molecular mechanisms through which the dynamic processes take
place. Experimentally, clarifying this issue is very challenging due to the very short
time and length scales associated with many of the dynamic membrane processes,
since what one should deal with are processes taking place over time scales of the
a
Department of Physics, Tampere University of Technology, Finland
b
VTT Technical Research Center of Finland, Espoo, Finland
c
Department of Applied Physics, Helsinki University of Technology, Finland
d
Groningen Biomolecular Sciences and Biotechnology Institute, Zernike Institute for Advanced
Materials, University of Groningen, The Netherlands
e
Department of Physics, Tampere University of Technology, Finland
f
Department of Applied Physics, Helsinki University of Technology, Finland
g
MEMPHYS – Center for Biomembrane Physics, University of Southern Denmark, Denmark.
E-mail: Ilpo.Vattulainen@tut.fi.
PAPER www.rsc.org/faraday_d | Faraday Discussions
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 411
order of tens or hundreds of nanoseconds, and length scales of the order of molecular
size. For example, while it is well known that lipid translocations (flip-flops) from one
membrane leaflet to another are on average very slow processes, typical rates being
one event per minute or an hour, the actual flip-flop event may happen in tens of
nanoseconds through formation of transient membrane defects such as pores.
4,5
The limited understanding of membrane dynamics is in contrast to structural proper-
ties of membranes which in turn are reasonably well understood, highlighted by
a number of structural models that have been proposed during the last few decades
and reviewed very recently.
6–11
Particular interest has been directed to understanding
the roles of membrane heterogeneity and lipid rafts, the latter being highly ordered,
biologically relevant membrane domains rich in cholesterol and saturated lipids.
7–11
The limited understanding of membrane dynamics is quite problematic.
Membrane dynamics is central to a variety of cellular processes such as the forma-
tion of lipid rafts, assembly of membrane–protein complexes as well as their gating
mechanisms, and signaling. Understanding of cellular functions is largely incom-
plete without the proper understanding of membrane dynamics. Rather surprisingly,
however, even mechanisms of seemingly simple dynamic processes such as the
motion of lipids in the membrane plane (lateral diffusion) are weakly understood.
It would be appealing to think that lipids in membranes diffuse through single-
particle motions in terms of nearly instantaneous discrete jumps, where a lipid moves
out of its cage, moving a distance comparable to its own size. However, it is not
obvious that such a simplified scheme that is typical in solid systems would take
place also in soft matter. What is known reliably is the fact that the rate of lipid
motion depends on the time scale at which it is observed. At short times of the order
of 1 ns, large lateral diffusion coefficients have been reported: in the fluid phase, the
reported ones are usually about 10
ÿ6
cm
2
s
ÿ1
.
12–14
At longer times, there is a crossover
to different dynamic behavior characterized by substantially smaller diffusion coef-
ficients, typically about 10
ÿ8
to 10
ÿ7
cm
2
s
ÿ114–17
in the fluid liquid-disordered phase.
It is commonly thought that short-range techniques such as quasi-elastic neutron
scattering (QENS) measure the rapid rattling-in-a-cage motion, characterizing the
motion of a lipid confined to a cage formed by its neighbors, while long-range tech-
niques such as fluorescence recovery after photobleaching (FRAP) and fluorescence
correlation spectroscopy (FCS) probe the slower motion arising from random-walk-
like displacements due to lipid–lipid collisions over much larger scales. The nano-
scale rattling-in-a-cage motion has been validated by atomistic simulations,
18
, and
also the random walk of lipids has been confirmed over macroscopic scales.
19
Yet
what happens at intermediate times, and what is the actual mechanism through
which lipid diffusion takes place, have remained unsolved.
Thus, is it possible that lipids would diffuse in terms of nearly instantaneous,
single-particle jumps? Data from QENS backscattering experiments
12,20
have been
interpreted in terms of a ‘‘jump model’’, providing some indirect support for this
view. Atomistic simulations of Moore et al.
21
have identified ‘‘jump’’ like diffusion
events from one cage to another, but the found events were very rare. Summarizing,
no direct experimental or simulation evidence for single-particle jumps as a domi-
nating mechanism exist. Meanwhile, simulations of Ayton and Voth
22
have proposed
that density fluctuations and other collective effects may have a role to play in lateral
diffusion. Recent studies by Falck et al.
23
showed that this is indeed the case. They
considered one-component membranes through atomistic simulations and found the
diffusion of lipids to be highly collective. Instead of discrete single-particle jumps,
Falck et al. observed clusters of lipids moving in unison. The concerted motions
were driven by thermal fluctuations and resulted in intriguing flow-like patterns.
However, due to the major computational cost associated with atomistic simula-
tions, the understanding of the length and time scales associated with the concerted
lipid motions remained largely incomplete.
In this work, we extend the work by Falck et al.
23
in quite a number of ways. As in
ref. 23, we focus on the mechanisms of lipid lateral diffusion. To overcome the
412 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
barriers related to nanoscales, we use molecular simulations instead of experiments
to elucidate how lipids diffuse in complex membranes. Our main objective is to
clarify how the concerted diffusion phenomena take place in many-component rafts
over a multitude of scales ranging from nanometers to tens of nanometers, and over
times from nanoseconds to microseconds. To cover all these scales, we combine
information from atomistic and coarse-grained simulations. We use atom-scale
molecular dynamics simulations to characterize lateral diffusion within rafts,
24
and the coarse-grained MARTINI model
25
to elucidate diffusion phenomena in
membranes characterized by the coexistence of raft and non-raft domains over
larger scales. Additionally, we use dissipative particle dynamics
26
simulations to
clarify the importance of hydrodynamic conservation laws for the observed diffusion
patterns. The results highlight the emergence of concerted lipid motions in all cases
we have studied, and allow us to identify typical length and time scales associated
with the correlated lipid motions. The findings emphasize the collective nature of
lipid dynamics at mesoscopic scales, stressing the importance of better under-
standing the role of collective motions in cellular membranes, in particular in the
context of protein–lipid dynamics.
II. Models and methods
We performed atomic-scale and coarse-grained (CG) simulations for a number of
single- and many-component membranes using three different simulation
approaches. First, we conducted dissipative particle dynamics (DPD) simulations
for single-component lipid bilayers to consider the influence of hydrodynamic modes
(local as well as global momentum conservation) on lipid diffusion. Second, we
employed atomistic molecular dynamics (MD) simulations to lateral diffusion
within raft-like membranes. Finally, third, we used coarse-grained MD simulations
to consider lipid diffusion in the regime where raft and non-raft domains coexist,
focusing on diffusion over large scales in time and space.
A. DPD simulations
We used the DPD method
26
to simulate a large single-component lipid bilayer in the
fluid phase in the NVT ensemble. The simulation parameters were obtained from
a previous study,
27
in which the lipids were constructed from four DPD particles,
one representing the hydrophilic head group and three representing the acyl chain.
The whole system consisted of 5500 single-chained lipids of type ‘‘A’’, together with
350 000 water beads in total. The interaction potentials, the parameters used in the
Hamiltonian, and further details of the model including its validation are described
in ref. 27.
The simulations were carried out in reduced units with a time step of Dt¼0.05
and a total of 20 000 time steps. The size of the cubic simulation box was (50 r
c
)
3
,
where r
c
is the cut-off radius of interactions that follow the standard DPD form
for interactions: the weight function for random forces was chosen as u
R
(r)f
(1 ÿr/r
c
), if r<r
c
, and zero otherwise. To convert the reduced units into real units,
we determined the lateral diffusion coefficient (see eqn (1) below) through DPD
simulations for the model under consideration and compared that with the diffusion
coefficient D¼10
ÿ7
cm
2
s
ÿ1
that is typical for lipid bilayers in the fluid phase.
14
This
comparison together with a reasonable estimate for the area of 0.25 nm
2
per single-
chained lipid results in real units of approximately Dt¼0.04 ns and r
c
¼0.5244 nm,
thus the total simulated time scale is about 800 ns. We will use these units when dis-
playing the results.
B. Atomistic simulations of rafts
The data for the atomic-scale raft-like bilayer were obtained from our previous
study.
24
In particular, here we concentrate on the analysis of the simulation
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 413
trajectory of a bilayer consisting of 1024 lipids in total, with a 2:1:1 molar mixture of
palmitoyl-oleoyl-phosphatidylcholine (POPC), palmitoyl-sphingomyelin (PSM),
and cholesterol (Chol), in respective order. This composition and the thermody-
namic conditions we have used correspond to a membrane domain in the liquid-
ordered phase, and experimental studies of a similar system have concluded the
system to be raft-like, displaying coexistence of liquid-ordered and liquid-disordered
phase domains.
28
The temperature of the study was T¼310 K and the simulation
time scale was 100 ns. Details of the simulation protocol, force fields, starting config-
urations and other relevant simulation conditions are described elsewhere.
24
Important to the interpretation of the results is to be aware of the analyzed bilayer
structures. First, the system size is about 15 nm in linear dimension in the bilayer
plane, thus what our bilayer corresponds to is the membrane structure within
a raft domain. Second, while there is nanoscale membrane heterogeneity with regard
to in-plane distribution of cholesterol (see Fig. 1 and 2 in ref. 24), the lipid distribu-
tions are yet rather homogeneous. Though, the most relevant point is that the atom-
istic simulation data can not provide an insight into the large-scale diffusion
phenomena over the entire membrane domains, or across membrane domain bound-
aries. To this end, we employ CG simulations.
C. Coarse-grained MARTINI simulations of rafts
To access larger scales, we carried out CG simulations on eight different systems. All
models are based on the MARTINI force field,
25
see below. The lipids used in the
simulations were diarachidonoyl-phosphatidylcholine (DAPC), which is a phospho-
lipid with four double bonds in both chains, the chain length being 20 carbons;
dilinoleyl-phosphatidylcholine (referred to as DUPC), which is a phospholipid
with two diunsaturated chains with 18 carbons in each chain; dipalmitoyl-phospha-
tidylcholine (DPPC), a phospholipid with two saturated chains composed of
16 carbons; and cholesterol.
To create initial configurations for the systems, first a pure DAPC system of 1152
lipids (SA1) was created from the end result of a previous (unpublished) simulation of
pure DPPC. This was achieved by changing the beads of DPPC to those of DAPC,
adding one extra bead needed by DAPC to each of the lipid chains, and moving
the leaflets slightly apart to make room for the additional beads. After this, energy
minimization was performed on the system. Using this as the base system, three other
systems with DAPC were created: one by replacing randomly chosen DAPC mole-
cules from each leaflet with DPPC and Chol molecules so that the system would
have molar concentrations of DAPC : DPPC : Chol ¼2 : 1 : 1 (S
A3
), and another
similarly but having concentrations of DAPC : DPPC : Chol ¼8 : 1 : 1 (S
A2
). The
fourth system with DAPC was achieved by joining four copies of S
A1
and then replac-
ing randomly chosen DAPC molecules within a chosen radius from the center of the
system with DPPC and Chol, thus creating a raft-like circular initial structure, having
molar concentrations of DAPC : DPPC : Chol ¼2 : 1 : 1 (S
A4
).
Four more systems were created in the same manner, except using DUPC instead
of DAPC. The only difference is that unlike DAPC, DUPC has the same number of
beads as DPPC, so there was no need for moving leaflets apart. These systems (S
B
)
are named analogously to the S
A
-systems.
Molar concentrations and initial configurations of all systems are given in Table 1,
and snapshots of the initial and final configurations of all ternary systems are
depicted in Fig. 1 (for membranes with DAPC) and 2 (for bilayers with DUPC).
All simulations were carried out using the GROMACS software package version
3.3.3,
29,30
except for the systems S
A4
and S
B4
which were run using the development
version of GROMACS 4
31
because of the substantial system size.
All models are based on the MARTINI force field.
25
Force fields for DAPC and
Chol are the same as in ref. 32, and for DPPC and water the same as in ref. 25. The
force field for DUPC is described in ref. 33.
414 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
For integrating the equations of motion, we used a timestep of 20 fs. The coordi-
nates were written every 0.5 ns. Temperature and pressure coupling were performed
using the Berendsen thermostat and barostat.
34
A temperature of 323 K was used in
all simulations to match the temperature used in a related study by Falck et al.
23
Each simulation was run over 8 ms of real time (using a factor of four to scale simu-
lation time to real units
25
).
D. Analysis and analysed quantities
The definition of the lateral self-diffusion coefficient of lipids is
D¼lim
t/N
1
4tN X
N
i¼1
½riðtÞ ÿ rið0Þ2(1)
where the sum runs over all Nlipids, and {r
i
(t)} are the center-of-mass (CM) posi-
tions of lipids, projected to the plane of the bilayer. The brackets h i denote an
average over different origins of time. We calculated the diffusion coefficients by
following the position of a molecule with respect to the CM of that monolayer,
unless mentioned otherwise. In this manner, the influence of the drift of the two
monolayers relative to one another is accounted for.
35–37
To facilitate addressing the issue of diffusion mechanisms, we plotted in-plane
(two-dimensional (2D)) displacement vectors of the center of mass (CM) positions
of each lipid in a monolayer over different time intervals Dt. Examples are shown
in Fig. 3 and 4. The point here is to illustrate the local lateral correlations in lipid
motions and to qualitatively address the time dependence of these correlations.
The displacement figures mostly serve for qualitative purposes. To gain more infor-
mation on the character of these correlated motions, we determined a 2D displace-
ment correlation map as follows. First, we fixed the time interval Dt. Then, for
a tagged lipid, the displacement vector determined over Dtis centered in the box, lying
along the x-direction, pointing to the right (+x), and the displacement vectors of other
lipids determined over the same Dtare moved respectively, taking periodic boundary
conditions into account. Then, the box surrounding the central lipid is divided into
64 64 bins; additional tests with 128 128 bins yielded consistent results. Next,
dot products between the unit central displacement vector and the unit displacement
vectors of other lipids within half of the box width are calculated, and the values are
allocated to bins according to their relative position to the central lipid; the results are
averaged by repeating this procedure for all lipids and over the whole trajectory. For
visual purposes, the resulting 2D displacement correlation maps (example shown in
Fig. 5 (left)) were smoothed by bilinear interpolation before plotting.
The map depicts dynamical correlations between the lateral motions of the lipid
being considered and the other lipids in its vicinity. Clearly, the above approach is
Table 1 Molar compositions (in units of mol%) and initial configurations of the simulated
systems
Lipid system DAPC DUPC DPPC Chol No. of lipids Initial configuration
S
A1
100 — — — 1152 Random
S
A2
80 — 10 10 1152 Random
S
A3
50 — 25 25 1152 Random
S
A4
50 — 25 25 4606 Raft-like
S
B1
— 100 — — 1152 Random
S
B2
— 80 10 10 1152 Random
S
B3
— 50 25 25 1152 Random
S
B4
— 50 25 25 4608 Raft-like
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 415
just one of a possible number of means to gauge these correlations, but for the
present purpose it serves well. When Dtis small, we expect some correlations to
emerge. On the other hand, when Dtincreases and exceeds some characteristic
time, we expect the correlations between lateral motions of diffusing lipids to vanish
since in the long-time limit the lipids should act like independent random walkers.
The results discussed below show that this is indeed what happens.
Fig. 1 Snapshots of the (left) initial and (right) final configurations for systems with DAPC
based on the coarse-grained MARTINI simulations. In the left column, from top to bottom,
there are snapshots from above for the initial configurations of the systems S
A2
, S
A3
, and
S
A4
. On the right-hand side, there are respective data for the final configurations at the end
of the simulations. DAPC beads are shown in grey, Chol in orange, and DPPC in purple.
The small empty spots in the initial configurations are due to system preparation before any
dynamics has taken place. Regarding scales of final configurations, the linear system size is
about 21.3 nm in S
A2
, 19.2 nm in S
A3
, and 42.1 nm in S
A4
.
416 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
When studying correlated movement in systems involving rafts, a couple of addi-
tional points need to be taken into account. As a raft diffuses in the membrane as an
entity, it leads to a situation where movements of raft-forming lipids are highly
correlated at long time scales. This obscures the correlation pattern within the
raft, leading to a displacement correlation map completely different from the one
in Fig. 5 (left). Instead, the result looks more like Fig. 5 (right). This effect can be
Fig. 2 Snapshots of the (left) initial and (right) final configurations for systems with DUPC
based on the coarse-grained MARTINI simulations. In the left column, from top to bottom,
there are snapshots from above for the initial configurations of the systems S
B2
, S
B3
, and
S
B4
. On the right-hand side, there are respective data for the final configurations at the end
of the simulations. DUPC beads are shown in grey, Chol in orange, and DPPC in purple.
Regarding scales of final configurations, the linear system size is about 20.1 nm in S
B2
,
18.3 nm in S
B3
, and 39.5 nm in S
B4
.
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 417
Fig. 3 Lateral displacement vectors of lipid CM positions over different time intervals (DPD
simulation). Each arrow/vector describes the in-plane lateral displacement of a single lipid
during a given time interval. The plots are from one of the two monolayers in the DPD-simu-
lation.
Fig. 4 Three consecutive snapshots of lateral displacement vectors for the same monolayer in
the membrane system modeled by DPD, with a time interval of 6 ns. A subset of molecules in
the same transient cluster with corresponding displacement vectors is shown in red to better
illustrate the time dependence of local spatial correlations in the movements.
Fig. 5 Left: An example of a correlation plot for DUPC over Dt¼2 ns simulated through the
CG MARTINI model (system S
B1
). Colour scale represents the strength of correlation. One
finds lipids especially in front of and behind the tagged lipid to be positively correlated with
the motion of the tagged particle. The lipids beside it and farther away are negatively corre-
lated, moving on average in the opposite direction. Right: An example of a correlation plot
for DPPC in a raft over Dt¼2000 ns (system S
A3
). Note that in this case the motion of the
raft as an entity is not removed from the displacement of particles in a raft. See text for further
discussion.
418 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
removed by modifying the trajectory so that the center of mass movement of raft-
forming lipids is removed, essentially neutralizing the effect due to the motion of
the raft as a whole. When this is accounted for, the nature of movement inside the
raft can be studied more concretely. However, the non-corrected figure is not useless
either. As it depicts the collective movement of the whole raft, it can be used to assess
some key quantities of the raft, such as its size and shape. For example, the shape of
the raft in Fig. 5 (right) ranges from stripe-like to circular with a radius of about
6 nm.
The correlation map yields information about the shape of the flow-like patterns
for a chosen time interval. To assess the lifetimes of these correlations we calculate
the average positive and negative values of the dot products as a function of the time
interval. It is performed by first calculating the 2D displacement correlation map for
a given time interval and then computing the average value of positive and negative
bins, successively increasing the interval and repeating the calculation. This yields
information on the time evolution of positive and negative correlation in the system.
The data are then fitted with the decay function
C(t)¼C
0
exp(–t/s) + C
n
(2)
where C(t) is the average (positive or negative) correlation at time t,C
0
is a constant
describing the magnitude of correlation in the system, C
n
is the random correlation
that remains when t/N, and sis the decay constant of the correlation. Using s, we
can compare lifetimes of correlated motions of different lipids in different systems.
Note that due to thermal fluctuations, the decay functions C
+
(t) and C
ÿ
(t) of posi-
tive and negative correlations (in respective order) do not decay to zero at long times.
However, their sum is expected to vanish for t/N.
III. Results for DPD simulations
Fig. 3 illustrates how the lipid CMs move over different time intervals. It is clear that
on the short time scale (0.4 ns) the lipids hardly move at all, whereas on the longer
time scales there are correlations among lipid motions persisting over a distance of
10 nm. The dynamic correlations in the present system in the liquid-disordered phase
are evident for time scales of about 5 ns or more, in agreement with previous find-
ings.
23
Further, the largest correlated areas seem to be queue-like in shape. This
observation is also in accordance with the earlier report by Falck et al. on atom-scale
bilayers.
23
Fig. 4 illustrates an important aspect of the observed local spatial correlations.
The flow patterns are different in each of the consecutive 6-ns time windows, which
means that the arrows do not represent conventional fluid flows, but they are rather
short-term snapshots of transient lipid motions that vary in time. Therefore, an
appropriate way to describe these patterns is to talk about local short-term correla-
tions in lipid diffusion, or concerted transient lipid motions.
The fact that DPD simulations reproduce the recent atom-scale simulation find-
ings
23
is an important checkpoint for the suggested mechanism, since in hydrody-
namic flow phenomena the local conservation of momentum drives the formation
of flow-like patterns such as vortices.
38
In DPD simulations, the underlying descrip-
tion of pairwise forces especially for the dissipative component guarantees the
conservation of local momentum in every particle–particle collision (see, e.g., ref.
39), thus leading to hydrodynamic behavior in the spirit of Navier–Stokes. This
suggests that the (local) conservation of momentum could be the principal reason
for the emergence of the concerted lipid motions in Fig. 3 and 4. However, in ref.
23 the Nose–Hoover thermostat was employed where local momentum conservation
is violated: individual particle–particle collisions do not conserve momentum. In
many simulations, the global momentum is made to conserve through the removal
of the total system’s CM motion, but this does not imply local momentum
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 419
conservation. More generally, local momentum conservation is violated in basically
every thermostat (including Berendsen, Andersen etc.) and barostat
39
that are
commonly used in atomistic simulations.
The bottom line is that the momentum conservation is not the principal reason for
the concerted lipid motions, rather it is due to other factors such as thermal and
density fluctuations. The rapidly changing transient lipid motions in Fig. 4 are
consistent with this view: if hydrodynamic flows due to conservation of momentum
were dominant, the lifetimes of cluster motions would probably be considerably
larger.
IV. Results for atomistic raft simulations
The diffusion coefficient for the selected atom-scale raft-like bilayer was around D¼
0.08 10
ÿ7
cm
2
s
ÿ1
for all three components,
24
which is in rough agreement with
pulsed field-gradient experiments in the liquid-ordered phase.
40
Due to the limited
time-scale (100 ns) of the study, only small-scale molecular re-arrangements were
observed rather than formation of domains.
The results in Fig. 6 reveal that the previously observed local correlations in lipid
movements are not limited to one-component bilayers in the liquid-disordered
phase. The figure illustrates that the correlations are not dependent on lipid types
or the liquid phase of the bilayer, but that the concerted lipid motions emerge
also in complex many-component membranes, reflecting the general nature of the
phenomenon under study.
V. Results for coarse-grained raft simulations
To consider phenomena over larger scales, we used the MARTINI model. We found
cholesterol to prefer saturated lipids, which gave rise to formation of Chol- and
DPPC-rich raft domains. The rafts in the two leaflets were clearly coupled to one
another (registration), and the conformational order in raft domains was distinctly
larger than in the domains enriched in polyunsaturated lipids. This was evident also
in cholesterol tilt, which among the S
A
systems was smallest in S
A4
, followed by S
A3
and S
A2
. Cholesterol also played a role in membrane thickness, which was larger in
rafts compared to polyunsaturated membrane regions. Overall, the results that we
found for membrane structural properties, domain formation, and lipid flip-flops
are consistent with the very recent results discussed by Marrink et al. in ref. 33.
Thus, here we will not consider those issues any further but rather concentrate on
the lateral dynamics of lipids in the membrane plane.
Snapshots of the final configurations at 8 ms are shown in Fig. 1 and 2. All systems
except for S
B2
are clearly raft-like. In S
A3
the raft has a stripe-like shape extending
Fig. 6 Lateral displacement vectors of lipid CM positions over different time intervals of the
atom-scale raft simulation. The colors indicate different lipid types: POPC (gray), PSM (blue),
and Chol (red).
420 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
across the system, whereas in S
A2
and S
A4
the rafts are circular. In S
B3
and S
B4
the
shapes are less clear, though S
B4
seems to retain the round shape of the initial config-
uration. In S
B2
one can find temporary clustering, though no permanent phase sepa-
ration into raft-like behavior takes place.
A. Lateral diffusion
The lateral diffusion coefficients of different lipid types in the systems are given in
Table 2. Note that for each of the lipid types in a given system, the diffusion coeffi-
cient has been computed over all molecules of that type by following the position of
a molecule with respect to the CM of the lipids of the same type in that monolayer.
In practice, this means that in single-component systems the positions of lipids in
a given monolayer are considered with respect to the CM of that monolayer, as in
the studies discussed earlier in this paper. In many-component systems with rafts,
the approach used here essentially yields diffusion coefficients that describe diffusion
inside a raft; diffusion with respect to the CM motion of the raft. This is most evident
for DPPCs that are highly enriched in rafts, being found only rarely outside them.
Finally, it is worth pointing out that since cholesterols flip-flop rather often,
33
the
lateral diffusion coefficients have been computed only from those Chols that do
not flip-flop during the simulation.
The lateral diffusion coefficients found in this work range from 10
ÿ8
to 10
ÿ7
cm
2
s
ÿ1
, which are in line with experimental
40
and simulation studies
24
of three-compo-
nent PC/SM/Chol raft systems. Further, the lateral diffusion data is consistent
with experiments, which generally indicate the lateral diffusion coefficient to be
about 10
ÿ7
cm
2
s
ÿ1
in the fluid phase for single-component membranes,
15–17,41,42
and to decrease for an increasing concentration of cholesterol.
16,17,41,42
One finds the lateral diffusion coefficients to follow a number of trends. First, let
us bring about the observation that the raft forming tendency follows a pattern (in
order of significance) S
A4
> S
A3
> S
A2
> S
A1
, and similarly for S
B
. The diffusion
results then show that the more raft-like the system is, the slower is diffusion of
DPPC and Chol.
The effect of rafts is the clearest when looking at the diffusion of DPPC. In the
systems S
A3
and S
A4
the rate of DPPC diffusion is ten times slower than the diffusion
of other lipids in all systems. What this means is that in these systems DPPC move-
ment is confined into the raft and almost no movement of DPPC outside the raft is
observed. DPPC in the raft-system S
B3
undergoes faster diffusion, suggesting a less
strong phase separation in that system.
When looking at diffusion of Chol, the effect of rafts can also be noticed.
However, because individual cholesterols that reside outside raft domains diffuse
rather fast, increasing the average diffusion rate, the difference is not so pronounced.
Table 2 Diffusion coefficients determined from eqn (1) in units of 10
ÿ7
cm
2
s
ÿ1
. Error estimates
of the coefficients are of the order of a few percent
System S
A1
S
A2
S
A3
S
A4
D
DAPC
4.15 3.83 2.16 4.10
D
DPPC
— 0.33 0.16 0.16
D
Chol
— 3.39 1.25 1.13
S
B1
S
B2
S
B3
S
B4
D
DUPC
4.06 3.58 1.99 1.93
D
DPPC
— 2.45 1.02 0.36
D
Chol
— 4.61 2.13 0.93
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 421
Still, comparison of systems S
B3
and S
B2
, of which the latter does not contain rafts,
shows a 60% difference in the diffusion coefficients of Chol.
Both S
A
and S
B
show the same tendency, that is, diffusion slows down upon
increasing Chol and DPPC concentration. The change of concentrations from
8:1:1 (S
A2
and S
B2
) to 2:1:1 (S
A3
and S
A4
, and S
B3
and S
B4
) decreases the diffusion
rate roughly by 50% for all lipids. Even though the polyunsaturated lipids enter
the rafts rarely, the presence of raft domains, where diffusion is slow and order is
high, slows down their diffusion considerably, too.
When comparing S
A3
with S
B3
and S
A2
with S
B2
one observes only a slight differ-
ence in the diffusion coefficient of the polyunsaturated lipid. Diffusion of DUPC is
a bit slower than that of DAPC in corresponding systems, which is logical given that
DUPC is less unsaturated than DAPC, thus being more likely to interact favorably
with Chol. Also, the phase separation is less clear or even non-existent in DUPC-
systems, which implies that more DUPC are in contact with Chol than in the
strongly phase separated systems with DPPC-Chol rafts.
B. Concerted diffusion patterns
Fig. 7 and 8 show the concerted diffusion patterns over chosen time intervals in one
of the leaflets of the system S
A3
used as an example to illustrate the main features.
Fig. 7 Three consecutive snapshots of displacement vectors in the system S
A3
(see also Fig. 1
(2nd row)) studied through the CG MARTINI model. DAPC is shown in grey, Chol in red,
and DPPC in blue. To take possible flip-flops into account, the spots where a molecule leaves
the leaflet in the beginning of the interval are marked with ‘‘o’’, and the spots where the mole-
cule enters the leaflet in the end of the interval are shown with ‘‘x’’.
Fig. 8 Snapshots of in-plane displacement vectors in the system S
A3
(see also Fig. 1 (2nd row))
over time intervals ranging from 10 ns to 1000 ns, studied through the CG MARTINI model.
DAPC is shown in grey, Chol in red, and DPPC in blue. To take possible flip-flops into
account, the spots where a molecule leaves the leaflet in the beginning of the interval are marked
with ‘‘o’’, and the spots where the molecule enters the leaflet in the end of the interval are shown
with ‘‘x’’.
422 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
The figures show a similar kind of behaviour as that found by Falck et al.
23
Move-
ment of lipids seems to happen in streams or flows. As discussed above, the analogy
to flows is somewhat problematic, though, since the flow patterns are short lived.
For example, when looking at Fig. 7, which shows consecutive snapshots of
displacements over a period of 2 ns, one sees that flow patterns change their direction
considerably during such a short time scale. Thus, when talking about flows, what
we mean are indeed temporary, short-lived collective movements.
Due to the different diffusion rates, the strength of flows depends on the phase.
Fig. 7 depicts this clearly for the raft region of the system: lipid displacements in
the DAPC-rich liquid-disordered phase are significantly larger than in the liquid-
ordered raft region that is in the lower third of the system (see the 2nd row in
Fig. 1). This is in line with the diffusion results, as the raft-forming lipids were found
to undergo slower diffusion than the polyunsaturated non-raft lipids. Within rafts,
the motion of lipids tends to be slower in the middle of the raft than at the edges. The
motion is largely circular, which is also reflected in the motion of the non-raft lipids
near the boundaries of rafts, as they tend to move more often along the raft
boundary than perpendicular to it.
Another concrete finding that can be observed from Fig. 8 is that for times of
about 1 ms, the flows are quite large when compared to the size of the system: in quite
a few cases there are flows ranging from half of the bilayer size all the way up to the
system size. Again, these flows are short-lived; there is no continuous stream
revolving in the system.
Over times of several hundred nanoseconds, one finds the diffusion of the raft as
a whole. Circular flow patterns transform towards linear movement, and the
displacements of in-raft lipids are highly correlated, as at the same time the outside
matrix has lost all correlation. In the larger systems S
A4
and S
B4
this behavior
emerged only at the longest time intervals from 0.5 to 1 ms (data not shown). In
smaller systems the lateral diffusion of rafts already started to dominate at 200 ns.
Overall, the flow patterns yield an interesting insight into how membrane
dynamics look on varying time scales. However, they provide a mostly qualitative
insight. For a more quantitative understanding, we carried out a more concrete anal-
ysis on the observed correlated nature of lipid motion.
C. Two-dimensional displacement correlation maps
Fig. 9 depicts the 2D displacement correlation maps for DPPC in S
A2
, when the
center of mass movement of all DPPC molecules is removed. Fig. 10 shows the
results obtained for the same system when no CM movement correction is made.
These results for DPPC in the S
A2
system largely describe the essence of dynamic
correlation patterns in raft domains. That is, we focus in the following discussion on
DPPC since the ordered domains are highly enriched in this saturated lipid. The S
A2
system is chosen for discussion since it is one of the appropriate ones where one finds
a clear raft-like domain.
Overall, the results show that the nature of correlated movement in rafts is inde-
pendent of the system in question. The basic pattern is the same in all raft systems:
the largest positive correlation is found just in front of and behind the tagged
particle, and the largest negative correlation is found in about a quarter of the simu-
lation box width to the side of the molecule under study. Significant positive corre-
lation is observed roughly 3 nm both forward and backward with respect to the
tagged lipid, and this distance varies only slightly between different systems and lipid
types. Further, the concerted motions mostly take place in the direction of move-
ment, complemented by the negative backflow correlation found along the sides.
The negative correlation describing backflows on the sides is much weaker than
the positive correlation found in the direction of movement, meaning that backflows
do not take place directly opposite to the direction of the central flow. Rather, the
large area of negative correlation seems to reflect multiple different backflows
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 423
with varying directions. This is in line with what one can observe from the flow
patterns in Fig. 7 and 8. That is, considering any of the instantaneous flows, one
cannot find a corresponding backflow which would go exactly to the opposite
direction.
Fig. 10 2D displacement correlation maps of DPPC in the system S
A2
without center of mass
movement correction (CG MARTINI model). Colour scale is the same in all figures.
Fig. 9 2D displacement correlation maps of DPPC in the system S
A2
with center of mass
movement of DPPC removed (CG MARTINI model). Colour scale is the same in all figures.
424 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
The correlation patterns remain relatively constant for times below 100 ns, after
which the correlation starts to get notably weaker in time, the correlations fading
and the patterns becoming smoother and smoother. The correlation has practically
vanished by 1000 ns. The results for Chol (data not shown) in S
A2
(and in the other
raft systems) reflect those of DPPC—not really a surprise, as Chol is the pair of
DPPC in raft formation.
Fig. 10 illustrates the case where one essentially considers the motion of a raft
domain as a whole. At long times, there is a circular area of very large positive corre-
lation, reflecting the size and shape of the raft in the system, accompanied by
a surrounding ring of negative correlation. The data describes the movement of
the raft: all lipids in the raft move mainly similarly at long times, causing major
correlation, and the surrounding area of negative correlation is due to the outside
matrix giving room to the moving raft. It is worth pointing out that Cicuta et al.
have recently studied the lateral diffusion of entire micrometer-sized membrane
domains in giant unilamellar vesicles, considering, e.g., the size dependence of
domain diffusion. They found the results to be consistent with membrane-domi-
nated drag in viscous liquid-ordered phases and bulk-dominated drag for less
viscous liquid-disordered phases.
43
When similar 2D displacement correlation maps were constructed for the polyun-
saturated lipids, we found the general features to be largely identical to those
described above. There are strong forward flow effects in front of and behind the
tagged polyunsaturated lipid, and weaker backward flows on the sides.
In essentially all of the CG-systems we have studied, the size of the positively
correlated regime is about 10 nm 6 nm. The total size of the correlated region,
including the surrounding negatively correlated area, is roughly 10 nm 15 nm,
being comprised of about 200 lipids. While these correlated regions are relatively
large in size, they shrink in time. In every system and for every lipid we considered,
the correlations vanished in a few microseconds or less.
Due to the extensive amount of data needed to explain the time dependence of the
2D correlation patterns for all systems and lipid types, we do not deal with them here
any further. Instead, we prefer to employ a more quantitative means to characterize
the correlation times of the observed collective motions.
D. Correlation times of concerted motions
To gauge the characteristic times of correlated motions, we determined the temporal
behavior of positive and negative correlations in the 2D displacement correlation
maps and used eqn (2) to fit the data to exponential form. This was performed for
each lipid type in each system, resulting in the correlation time sthat describes
the lifetime of concerted motions within a loosely bound cluster of lipids.
The resulting plots are shown in Fig. 11. We find that not all systems express expo-
nential decay, see below for discussion. Nonetheless, the results of all systems are
qualitatively consistent with regard to typical correlation times, which are found
to be of the order of 1 ms.
For more quantitative insight, we focus on those systems and lipid types where
exponential decay was most evident. Decay constants sextracted from exponential
fits are shown in Tables 3 and 4. They show quantitatively that the characteristic
times of concerted motions are indeed of the order of 1 ms, typical values ranging
from about 200 ns to 1.5 ms. These times constitute a major fraction of the simula-
tion time of 8 ms, which readily explains why the data in Fig. 11 fluctuate rather
strongly especially in raft systems where sampling is the weakest. Difficulties to
establish exponential decay are also related to undulations, which in some of the
systems were pronounced, thus the 2D lateral projections used in the analysis inev-
itably affect the results. Further, for some of the lipids, and especially cholesterols,
flip-flops are rather frequent.
33
Since Chol molecules that did flip-flop during an
analysis for a given time interval were not included in the computation, the sampling
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 425
gets weaker and weaker for increasing time in Fig. 11. Despite these limitations,
exponential decay is quite apparent and in favor of the view that temporal correla-
tions vanish over a microsecond time scale.
For polyunsaturated lipids DAPC and DUPC (see Table 3), the characteristic
correlation time is smaller the more ordered the system is. The largest correlation
Fig. 11 Time evolutions of positive (+) and negative (ÿ) correlation (CG MARTINI model).
In the left-hand columns, from top to bottom, there are data for the systems S
A1
, S
A2
, S
A3
, and
S
A4
. In the right-hand columns, there are corresponding data for the S
B
-systems, respectively.
The computed data are shown with dots, and the exponential fit is shown with a solid line.
DPPC data is presented in purple, Chol in orange, and DAPC/DUPC in grey. In every case,
the 2D displacement correlation maps used in the analysis have been computed such that the
lipid displacements are determined with respect to the CM of all the lipids of the same type
in that monolayer.
426 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
time is found in the pure one-component system, and the smallest correlation times
are observed in the raft systems (S
A3
and S
A4
, and S
B3
). This trend likely stems from
cholesterol, which is partly partitioned in the liquid-disordered phase rich in polyun-
saturated lipids, see Fig. 1 and 2. Thus, increasing the amounts of Chol strengthen its
local ordering effect in the liquid-disordered phase, suppressing lipid diffusion and
emergence of correlated motions. In the same spirit, the most long-lived correlations
in the polyunsaturated lipid rich phases are found in the pure single-component
systems (S
A1
and S
B1
), where disorder is the strongest.
As for correlations in rafts, consider data for DPPC which partitions almost
completely into the raft domain. The results (see Table 4) show that correlations
in a raft are somewhat more long-lived than in the neighboring matrix composed
of polyunsaturated lipids. However, this view should be taken with caution, since
the correlation times in Table 4 are almost identical within the error bars. Clearly,
the sampling should be substantially more extensive for drawing firmer conclusions.
At the moment, what can be said with certainty is that the correlations associated
with concerted lipid motions die out in the microsecond time scale.
VI. Concluding remarks
In this work, we have shown that lateral diffusion in many-component lipid
membranes (that are characterized by formation of raft and non-raft domains) takes
place through a complex concerted diffusion mechanism, a feature that was earlier
found in one-component lipid bilayers.
23
Instead of single-particle motions, the
lateral diffusion of lipids occurs through the movement of dynamically correlated
lipids moving in unison as loosely defined clusters. While the observed mechanism
is news considering that it has not been previously observed in many-component
lipid membranes, the actual value of the present study lies in the scales of concerted
motion that we have observed. The coarse-grained simulations bring about a view
that the correlated lipid motions take place over length and time scales that are
considerably larger than the nanoscales describing the motion of individual lipids.
As for length, the dynamic clusters were found to contain typically a few hundred
lipids, the sizes of these transient clusters being roughly 10 nm 15 nm. As for
Table 3 Decay constants of positive (+) and negative (ÿ) correlation in units of nanosecond
[ns]. The notion ‘‘NA’’ indicates that no reasonable exponential decay fit could be made. Error
estimates are based on 95% confidence limits
Lipid system S
A1
S
A2
S
A3
S
A4
s
DAPC(+)
752 12 600 16 406 15 170 38
s
DAPC(ÿ)
930 40 992 99 261 43 NA
S
B1
S
B2
S
B3
S
B4
s
DUPC(+)
1139 23 603 9 441 10 NA
s
DUPC(ÿ)
1523 84 573 11 572 33 NA
Table 4 Decay constants of positive (+) and negative (ÿ) correlation in units of nanosecond
[ns] for the system S
B3
. Error estimates are based on 95% confidence limits
s
DUPC(+)
s
DUPC(ÿ)
s
DPPC(+)
s
DPPC(ÿ)
s
Chol(+)
s
Chol(ÿ)
441 10 572 33 618 19 560 29 493 13 355 17
This journal is ªThe Royal Society of Chemistry 2010 Faraday Discuss., 2010, 144, 411–430 | 427
time scales, the lifetime, which characterizes the decay time of correlations in these
clusters, was found to be about a microsecond.
The lifetime of dynamic correlations indicates that the motion of a tagged lipid is
influenced by the motion of nearby lipids over this time interval of 1ms. Assuming
a lateral diffusion coefficient of 10
ÿ7
cm
2
s
ÿ1
, an individual lipid moves a lateral
distance of about 6 nm during this time window. This length scale and also the
time scale of 1 ms are much smaller than the scales probed by most experimental
techniques for lateral diffusion, such as FRAP, FCS, and NMR. However, in
current atomistic simulations the studied time scales are much smaller than the
1ms time scale. This implies that the interpretation of lateral diffusion data extracted
from atomistic simulations is not straightforward. Further work is clearly warranted
to clarify this matter.
The results presented in this work are consistent with experiments concerning
values of lateral diffusion coefficients and their trends for varying lipid composition.
However, comparison of simulation data with experiments with regard to the diffu-
sion mechanism, the size of the dynamically correlated lipid clusters and their life-
times is more subtle. This is due to the fact that currently there are no published
experimental studies that would have focused on the same topic. Experiments
dealing with the concerted diffusion mechanism are likely to be possible through
QENS and X-ray scattering, and we are looking forward to forthcoming results,
but currently further comparison is not possible.
Are the observed concerted motions specific to lipid membranes, or has similar
behavior been observed in other soft matter and liquid systems? At the moment,
we can only address this question in part. Similar flow-like features have been
observed in simple liquids modeled in terms of 2D Lennard-Jones fluids.
44
Also,
collective patterns of the same type have been found in 3D supercooled liquids.
45
However, these studies
44,45
were carried out through simulations in the NVE
ensemble, where energy conservation and especially the lack of a thermostat imply
conservation of momentum. In the simulations presented and discussed in the
present work, however, we found that momentum conservation is not a crucial
condition for the emergence of diffusion flows and collective diffusion phenomena
that we have found. Studies of Lennard-Jones fluids in the absence of momentum
conservation would indicate whether transient collective flow patterns would emerge
also in simple liquids under the conditions used in this work. However, as we are not
aware of such studies, this issue remains open. It seems yet likely that the concerted
diffusion mechanism observed here is an inherent feature found rather generally in
soft matter systems.
There is reason to assume that the concerted diffusion phenomena could affect the
molecular mechanisms of several processes in membranes. They probably influence
membrane fusion
46
and local pore formation in membranes.
47
They are also expected
to affect the structure and function of membrane proteins by contributing locally to
lateral pressure profiles in the membrane.
48,49
Of particular interest would be to
understand the role of these concerted lipid motions on the dynamics of rafts and
the joint dynamics of lipid–protein complexes.
7
The latter topic is of profound
importance, since while the dynamics of membrane proteins is not well understood,
the understanding of the joint dynamics of lipids complexed to a membrane protein
is even more limited. Atomistic and coarse-grained simulations such as those pre-
sented in this work will be essential in future efforts to clarify the complex dynamic
membrane phenomena over a variety of scales in time and space.
Acknowledgements
The authors would like to thank Emma Falck and P. B. Sunil Kumar for discussions
and correspondence. This work was supported by the Academy of Finland and the
Netherlands Organization for Scientific Research (NWO). The Finnish IT Centre
428 | Faraday Discuss., 2010, 144, 411–430 This journal is ªThe Royal Society of Chemistry 2010
for Science and the HorseShoe (DCSC) supercluster computing facility at the
University of Southern Denmark are thanked for computer resources.
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