Europhysics Letters
PREPRINT
Stabilization of Membrane Pores by Packing
D. J. Bicout 1,2 , F. Schmid 1,3 , O. Lenz 3 and E. Kats 1,4
1
Institut Laue-Langevin, 6 rue Jules Horowitz, B.P. 156, 38042 Grenoble, France
Biomathematics and Epidemiology, ENVL - TIMC, B.P. 83, 69280 Marcy l’Etoile,
France
3
Fakultät für Physik, Universität Bielefeld, D-33615 Bielefeld, Germany
4
L. D. Landau Institute for Theoretical Physics, RAS, 117940 GSP-1, Moscow, Russia
2
PACS. 87.16.Dg – Membranes, bilayers, and vesicles.
PACS. 87.15.Aa – Theory and modeling; computer simulation.
PACS. 82.45.Mp – Thin layers, films, monolayers, membranes.
Abstract. – We present a model for pore stabilization in membranes without surface tension.
It is shown that an isolated pore is always unstable irrespective of its size, shape and bilayer
material parameters (i.e. it shrinks tending to re-seal the pore). However, excluded volume
interactions in a system of many pores can stabilize pores of a given size in a certain range
of model parameters. In mean field approximation we calculate the pore size distribution,
and discuss the pore lifetime. For large negative line tensions, we predict the formation of a
sieve-like porous membrane state, a state also observed in computer simulations.
Pores can form and grow in membranes in response to thermal fluctuations and external
influences. Pore growth enhances the transport of biomolecules across the membranes and its
biological relevance can bring new prospective biotechnological applications (see, e.g., Ref. [1,
2]). Holes appear in the membrane via a thermally activated poration process, and their
subsequent growth is controlled by the effective line tension (assuming a negligible small
surface tension).
Consider the free energy f (n) of a single pore in a membrane without surface tension. In
the simplest approximation, it is the result of two contributions: a purely energetic part as
suggested by Litster [3], and an entropic part as modeled by Shillcock and Seifert [4],
0
; 0 ≤ n < n0
(1)
f (n) =
F0 + kB (Tm − T) [n − n0 ] + (2 − α) kB T ln(n/n0 ) ;
n ≥ n0
where F0 is the free energy required to create or initiate a pore of smallest size n0 , kB T =
1/β the thermal energy, kB Tm = λ/ ln(z) with λ the line tension of the pore edge, z the
connectivity constant of the matrix medium, and α an exponent related to the entropy of the
possible pore contour conformations. α = 1/2, corresponds to the entropy of self-avoiding
random walks in two dimensions and α < 2 for most pores. Simple inspection of Eq.(1) above
shows that for membranes without surface tension, in which we are interested here, the free
energy in Eq.(1) increases monotonically with growing pore size at low temperatures T < T m
c EDP Sciences
2
EUROPHYSICS LETTERS
and, therefore, pores of any size are unstable. In contrast, for high temperature T ≥ T m , the
pores grow without limit destabilizing hence the membrane, because the effective line tension
becomes negative.
A number of works have investigated mechanisms for the stabilization of single pores.
These include, for instance, membrane bending fluctuations, renormalization of linear and
surface tension coefficients [5], area exchange in tense membranes [6], osmotic stress [7], hydrodynamics [8, 9], orientational ordering [10], and others (see Ref. [11] for more details and
references). In this paper, we propose a new mechanism of pore stabilization that could become important in membranes containing many pores. We will show that pores of a given
size can be stabilized collectively by excluded volume interactions.
We consider an ensemble of many pores and assume that the pores do not interact directly
(as confirmed recently in Ref. [12]). Of course, different pores cannot overlap without losing
their identity. Hence we basically have an ideal gas of pores with varying number of “particles”
(a grandcanonical ensemble) and an excluded volume constraint. In the mean field van der
Waals approach, the total free energy of this system is given by
X
X
e(A − A0 )
N (n) ln
N (n)f (n) − kB T
(2)
F =
a0 N (n)
n
n
where N (n) counts the number of pores of size n, and A − A0 is the accessible area for pores
such that A represents the total area and A0 is defined as,
X
N (n)a(n) .
(3)
A0 =
n
Here a(n) is the difference between the excluded and the actual pore area, which is nonzero
due to the fractal shape of the pores and scales like a(n) = a0 (n/n0 )2ν with the Flory exponent
ν ≈ 3/4 for self-avoiding walks in two dimensions. This also defines a0 . Minimizing the free
energy in Eq.(2) with respect to N (n), we find that the equilibrium distribution of pore sizes
is given by
N (n)
exp {−βG(n)}
P∞
=
,
(4)
Q
N
(n)
n=1
where the effective free energy of the system reads as, G(n/n0 ) = f (n) + kB T Q(n/n0 )2ν , and
the partition function Q, which is a P
sort of measure for the pore packing density, is given
self-consistently by the relation Q = n a0 N (n)/(A − A0 ). In what follows, we will neglect
in Eq.(1) the entropic logarithmic term (2 − α) ln(n/n0 ). (It only slightly renormalizes the
results). Using ν = 3/4, the effective free energy G(x) of a pore interacting with an ensemble
of pores now reads,
(
0
h
i ; 0≤x<1
G(x) =
(5)
Tm
3/2
F0 + kB T −Q0 1 − T (x − 1) + Q x
;
x≥1
where Q0 = n0 ln(z) and the partition function Q is given self-consistently by,
Z ∞
Z ∞
Tm
3/2
−βG(x)
βF0
(x − 1) − Q x
.
Q=
e
dx = e
dx exp Q0 1 −
T
1
1
(6)
As already mentioned above, single or independent pores destabilize the membrane for temperature T ≥ Tm as the effective line tension becomes negative. In a multiple pore system,
:
D. J. Bicout, F. Schmid, O. Lenz and E. Kats
2.0
exp{β[G(1)-G(x)]}
βF0 = 2
T > Tc
1.5
1.0
0.5
T = Tm
0.0
0
2
4
x
6
8
10
Fig. 1 – Reduced equilibrium distribution, e−βG(x) e−βG(1) , of pore size x for βF0 = 2, Q0 = 4 and
two temperature
regimes. The free energy G(x) is given in Eq.(5) with Q = 0.215 for T = T m and
√
Q = 8/(9 3) = 0.513 for T > Tc (i.e., for T = 3Tm /2 and Tc = 1.082Tm ). The dashed vertical line
indicates the maximum x1 = 3 for T > Tc .
however, membranes remain stable even beyond Tm . Moreover, for temperatures such that
Tm < Tc ≤ T, G(x) in Eq. (5) admits a minimum at x = x1 such that,
√
Tc
T − Tm
Qc
x1 =
,
(7)
Q
Tc − T m
T
where Q(T = Tc ) = Qc is given by 3Qc = 2Q0 (1 − Tm /Tc ), and the critical temperature Tc
is defined from
Z ∞
2Q0 (1 − z)
Tm
2Q0
−zβm F0
3/2
;z=
=e
(1 − z) x
. (8)
dx exp Q0 (1 − z) (x − 1) −
3
3
Tc
1
Two important conclusions can be drawn from these results: First, the excluded volume
interaction between pores can stabilize membranes even in parameter regions where the effective line tension of the membrane is negative. Second, the pore size distribution in the case
of negative effective membrane line tension may have a maximum at nonzero contour length
n as illustrated in Fig. 1.
Three regimes can be distinguished: (i) At low temperatures, T < Tm , the line tension
is positive, and the pore size distribution N (n) drops monotonically. (ii) At intermediate
temperatures, Tm < T < Tc , the line tension is negative, and the pores are stabilized by the
presence of other pores. The pore size distribution still drops monotonically. (iii) At high
temperatures T > Tc , a maximum emerges in the pore size distribution, i.e., pores have a
most probable size. Figure 1 shows the reduced pore size distribution for two temperatures
below and above Tc , with a maximum at finite x for T > Tc (here, x = 3 or n = 3n0 ).
The width or window of the intermediate regime depends sensitively on the bare pore
energy, F0 , which describes the free energy required to create a single pore of minimum size
x = 1. This is illustrated in Fig. 2 where the temperature Tc appears to approach Tm if one
increases either F0 or the effective smallest pore size Q0 .
In the packing stabilized regime, beyond Tc , one should thus expect that the membrane
contains many long-lived pores with a size distribution exhibiting one preferred contour length.
Stabiliza
4
EUROPHYSICS LETTERS
1.0
Q0 = 10
0.9
Q0 = 4
0.8
Tm/Tc
0.7
Q0 = 2
0.6
0.5
0.4
0
2
4
6
8
10
F0/kBTm
Fig. 2 – Ratio Tm /Tc in Eq.(8) as a function of the reduced energy of pore formation.
Indeed, such structures can been observed in membranes close to the disintegration temperature. This was recently reported by Cooke, Deserno, and Kremer [13], and also found by us
in simulations of simple model membranes. In our simulations, we have used a coarse-grained
lipid model, where the lipids are represented by chains of (N+1) beads – one purely repulsive
“head” bead, and N “tail” beads with an attractive interaction of Lennard-Jones type. These
lipids are exposed to (computationally cheap) “phantom solvent” particles, which interact like
repulsive spheres with the beads, but have no interactions among each other. Details of the
model can be found in Ref. [14]. More specifically, the length of the lipid chains here was
N=2, and the head bead was chosen slightly larger than the tail bead (factor 1.2) in order to
reduce the line tension.
At low temperatures, the lipids self-assemble spontaneously into membranes. At high
temperatures, the membranes disintegrate. A snapshot of a configuration close to the disintegration temperature is shown in Fig. 3: The membrane contains many pores. The pore sizes
vary, but contour lengths in the range n ∼ 10 − 20 bead diameters are clearly favored over the
others. Packing stabilization is a candidate mechanism, which could possibly explain partly
the formation of this structure. Of course, much work (systematic simulations and a detailed
data analysis) still needs to be done in order to prove the connection.
From a fundamental point of view, the appearance of the porous membrane state can
be rationalized as follows: At negative line tensions, the membrane tries to create as much
pore rim as possible. In a sense, the resulting state is similar to the droplet microemulsion
structure in amphiphilic systems, where the fluid is macroscopically homogeneous, but filled
with internal interfaces on the microscopic scale [16].
From a practical point of view, the porous membrane state is interesting because it should
have peculiar permeability characteristics. Apart from the number and size of membrane
pores, another quantity that determines the permeability is the lifetime of pores, or, more
specifically, the time that a pore stays open. Unfortunately, lifetime estimations are difficult,
because the lifetime depends not only on the free energy landscape G(x) for existing pores, but
also on the dynamical model. In our case, the dynamical processes contributing to the lifetime
of pores are pore opening and closing, pore growth and shrinking, and pore coalescence and
splitting.
The first process, pore opening and closing, is governed by the unknown potential barrier
D. J. Bicout, F. Schmid, O. Lenz and E. Kats
:
Fig. 3 – Simulation snapshot of a porous membrane. For clarity, only the tail beads are shown.
that one must overcome in order to create a pore: The amphiphiles must change their orientation, and the free energy in the intermediate state is different from the corresponding energy
when a pore already exists. However, the characteristic time for this process does not depend
on the pore packing. Therefore, we shall not discuss it further.
The third process, pore coalescence and splitting, shall also be disregarded for simplicity.
This is of course highly questionable, but we are only interested in a very rough estimation
here. Moreover, one can argue that coalescence and splitting events do not affect pore opening
times, since the pores do not disappear, they just lose their identity.
Hence we are left with the characteristic time τ for pore growth and shrinking. Neglecting
hydrodynamic effects [15], we can approximate the dynamics of pore growth and shrinking by
a random walk or diffusion process in the potential G(x) with a diffusion constant D (assumed
to be independent of x). The characteristic time τ is then given by the average time that it
takes an existing pore to first reach the minimum pore size n0 , corresponding to x = 1.
At temperatures below Tc , the random walk has a drift towards x = 1, i. e., all pores of
any size will on average shrink with time. In the stabilized regime above Tc , the pore size
diffuses towards the minimum x1 > 1 of the effective pore potential G(x), i. e., pores of size of
about x1 are stabilized and the pore size x = 1 can only be reached via an escape process over
the barrier height from the minimum, ∆ = G(1) − G(x1 ). According to the first passage time
theory, the mean shrinking time of such a pore can be estimated from the relation [17–19]
Z ∞
2
Z ∞
βG(x)
−βG(y)
dx e
e
dy
n2
x
Z ∞
τ= 0 1
.
(9)
D
e−βG(x) dx
1
At temperatures well above Tc , this expression can be approximated by τ ∝ exp(β ∆).
The pore shrinking times τ are shown for different values of F0 as a function of temperature
in Fig. 4. At Tm , they begin to increase considerably. Therefore, we conclude that pores remain
open much longer in the packing stabilized regime than in the regime with positive line tension.
We emphasize again that this is only a very rough estimate. The correct description of pore
dynamics must of cause include coalescence and splitting events as mentioned above.
Stabiliza
6
EUROPHYSICS LETTERS
8
βF0 = 2
6
Dτ/n0 4
2
1
0.5
2
0
-1.5
-1.0
-0.5
0.0
0.5
1.0
1.5
Q0(1-Tm/T)
Fig. 4 – Reduced pore shrinking time in Eq.(9) as a function of the reduced temperature Q 0 (1−Tm /T )
for different βF0 . The dotted vertical line indicates the temperature T = Tm above which the
line tension becomes negative. The dashed vertical lines indicate the crossover point T = T c [i.e.,
Q0 (1 − Tm /Tc ) = 0.975, 0.771, 0.463 for βF0 = 0.5, 1, 2, respectively] where the pore size distribution
starts to exhibit a maximum like in Fig. 2.
To summarize, we have analyzed the statistics of multiple pore formation membranes
without surface tension within a simple van-der Waals type approach. We find that nonoverlap pore-pore interactions can keep the membranes stable even when the effective line
tension becomes negative. In certain parameter ranges, long-lived holes of finite size can be
stabilized. This leads to the formation of a nano-porous membrane state, where the membrane
has the structure of a sieve. Such a state has also been observed in simulations.
Although the model we have studied is a toy model, when properly interpreted it can
yield reasonable pore size distributions and multipore structures. Owing to the diversity
of biological systems (all the more artificial lipid bilayers) and the wide range of accessible
membrane parameters, we expect that stable multipore membranes can exist at physiological
conditions within a range of pore sizes of (1 − 10) nm.
Our result illustrates the rich diversity of membrane structures that can form when chemically or physically tuning the line tension λ. In order to assess the value of this parameter,
one can approximate it very crudely by the bare line tension, which is proportional to γ 0 h
(γ0 is the surface free energy per unit area between coexisting regions of hydrophobic and
hydrophilic molecules, and h the membrane thickness). Note that though this estimation is
valid for so-called hydrophobic pores, the same scaling ∝ h holds as well for hydrophilic pores,
where λ should scale as the bending modulus κ ∝ h2 times the membrane pore curvature
1/h. Hence one would expect that porous membrane states will more likely be observed in
thin membranes. However, we have to keep in mind that the effective line tension can be very
different from the bare line tension. For example, various mechanisms exist that reduce the
line tension in mixed membranes or membranes with additives.
In conclusion, we have proposed an excluded volume mechanism that stabilizes membrane
structures with many pores and seems to have been overlooked in the literature up to now.
Under certain circumstances, this mechanism promotes sieve-like membrane structures. This
invites speculations on possible applications of such structures, e.g., membranes with selective
permeability for controlled drug delivery, or to promote biomolecule translocation through
D. J. Bicout, F. Schmid, O. Lenz and E. Kats
pores.
REFERENCES
[1] Abidor I. G., Arakelyan V. B., Chernomyrdik L. V., Chizmadzhev Y. A.,Pastushenko
V. A. and Tarasevich M. R., Biolectrochem. Bioenergy, 6 (1979) 37.
[2] Safran S. A., Kuhl T. L. and Israelachvili J. N., Biophys. J., 81 (2001) 859.
[3] Litster L., Phys. Lett. A, 53 (1975) 193.
[4] Shillcock J. C. and Seifert U., Biophys. J., 74 (1998) 1754.
[5] Farago O. and Santangelo C. D., J. Chem. Phys., 122 (2005) 044901.
[6] Sens P. and Safran S. A., Europhys. Lett., 43 (1998) 95.
[7] Levin Y. and Idiart M. A., Physica A, 331 (2004) 571.
[8] Moroz J. D. and Nelson P., Biophys. J., 72 (1997) 2211.
[9] Zhelev D. V. and Needham D., Biochim. Biophys. Acta, 1147 (1993) 89.
[10] Golo V. L., Kats E. I. and Porte G., JETP Lett., 64 (1996) 575.
[11] Lipowsky R. and Sackmann E. (Editors), Structure and Dynamics of Membranes (Elsevier,
Amsterdam) 1995.
[12] Loison C., Mareschal M. and Schmid F., J. Chem. Phys., 121 (2004) 1890.
[13] Cooke I., Deserno M. and Kremer K., poster presented at the NIC winter school in Computational Soft Matter, (2004) .
[14] Lenz O. and Schmid F., J. Mol. Liquids, 117 (2005) 147.
[15] The hydrodynamic time scale τh ∝ ηh2 /λ (η is the effective viscosity, including the membrane
itself and the surrounding liquid, h is the membrane thickness, and λ the line tension), which
roughly demarcates the crossover from the random walk regime to the hydrodynamic regime, is
of the order of (1 − 102 ) s and thus presumably much larger than the typical life time (msec) of
unstable or metastable pores.
[16] Gompper G. and Schick M. , Self-Assembling Amphiphilic Systems (Academic Press, London)
1994.
[17] Risken H., The Fokker-Planck Equation (Springer, Berlin) 1989.
[18] Szabo A., Schulten K. and Schulten Z., J. Chem. Phys., 72 (1980) 4350.
[19] Bicout D. J. and Szabo A., J. Chem. Phys., 106 (1997) 10292.
:
Stabiliza