Acta Mechanica Sinica (2012) 28(4):1180–1185
DOI 10.1007/s10409-012-0090-y
RESEARCH PAPER
Fluctuation pressure on a bio-membrane confined
within a parabolic potential well
L. B. Freund
Received: 2 May 2012 / Revised: 14 June 2012 / Accepted: 14 June 2012
©The Chinese Society of Theoretical and Applied Mechanics and Springer-Verlag Berlin Heidelberg 2012
Abstract A compliant bio-membrane with a nominally flat
reference configuration is prone to random transverse deflections when placed in water, due primarily to the Brownian
motion of the water molecules. On the average, these fluctuations result in a state of thermodynamic equilibrium between the entropic energy of the water and the total free energy of the membrane. When the membrane is in close proximity to a parallel surface, that surface restricts the fluctuations of the membrane which, in turn, results in an increase
in its free energy. The amount of that increase depends on
the degree of confinement, and the resulting gradient in free
energy with degree of confinement implies the existence of
a confining pressure. In the present study, we assume that
the confinement is in the form of a continuous parabolic potential well resisting fluctuation. Analysis leads to a closed
form expression for the mean pressure resulting from this
confinement, and the results are discussed within the broader
context of results in this area. In particular, the results provide insights into the roles of membrane stiffness, number of
degrees of freedom in the model of the membrane and other
system parameters.
Keywords Statistical mechanics · Membrane fluctuations ·
Parabolic confinement · Confining pressure
1 Introduction
We report on a study of the statistical fluctuations of a biomembrane immersed in water when those fluctuations are
confined in some way. The fact that a membrane fluctuating
L. B. Freund (¬)
Department of Materials Science and Engineering,
University of Illinois at Urbana-Champaign,
1304 West Green Street,
Urbana, IL 61801 USA
e-mail: [email protected]
in close proximity to a parallel surface experiences an effective pressure resulting from confinement was first considered
quantitatively by Helfrich [1]. The source of fluctuations is
impact of the membrane surface by water molecules undergoing random flight trajectories, that is, Brownian motion,
as a manifestation of the prevailing temperature. The confining surface may be an actual physical surface, perhaps a
surface to which the membrane is to be bonded, for example,
or it may be a reflective symmetry surface between identical membranes nominally parallel to each other, in which
case confinement arises through direct contact interaction
between adjacent membranes. In either case, the confinement of a membrane results in a decrease in its entropy or
an increase in its free energy. Furthermore, the magnitude of
this effect typically increases as the distance to the confining
surface becomes smaller. The resulting gradient in free energy in the direction normal to the membrane reference plane
defines a pressure tending to increase that spacing. The dependence of this pressure on the membrane-surface spacing
is the topic central to the present discussion.
It is important to make clear which features of mechanical behavior are intended when describing a deformable
sheet as a membrane in this context. Traditionally, in the mechanics literature, the term membrane is used to describe a
thin sheet of material for which the bending resistance is negligibly small but which resists in-plane extension. The term
is interpreted in a somewhat different way when considering
a bio-membrane such as, for example, the phospho-lipid bilayer membrane that serves as the outer enclosure surface in
a cell [2]. In this case, the term membrane is understood to
describe a thin layer that resists out-of-plane bending, but it
does so in such a way that no in-plane shear stress is generated within the membrane layer. In addition, the term is
commonly understood to imply that the total membrane area
is conserved. The nature of bending resistance is made more
specific below when the elastic energy due to bending of a
membrane is calculated.
The existence of the pressure identified above, com-
Fluctuation pressure on a bio-membrane confined within a parabolic potential well
monly called a steric pressure, was first described by Helfrich in a pioneering paper [1], and it has been the focus of
active interest ever since, both theoretically and experimentally. The system considered in Ref. [1] was a membrane,
planar as its reference configuration, positioned midway between parallel rigid confining planes and at a distance c from
each. The reference plane of the membrane is assumed to
be the xy-plane, and the reference configuration occupies the
portion 0 < x < L, 0 < y < L of that plane. The transverse
deflection of the material point initially at x, y in the reference configuration is denoted by h(x, y) and the condition of
hard confinement is then
−c ≤ h(x, y) ≤ c,
0 ≤ x ≤ L,
0 ≤ y ≤ L.
(1)
This system is illustrated in Fig. 1.
Fig. 1 A schematic diagram of a square L× L membrane positioned
between rigid confining surfaces, each at a distance c from the reference plane of the membrane, which restrict the thermal fluctuations
of the membrane
Recognizing the difficulty in enforcing this inequality
constraint in a statistical analysis, Helfrich turned to the
weaker condition that the statistically expected value of the
mean membrane deflection squared, that is, the statistical
variance of the mean deflection, must equal c2 . In effect, the
weaker condition implies that the hard constraints implied by
Eq. (1) have been replaced by a soft constraint and that the
constraint takes the form of a requirement that the variance
of the mean deflection equals c2 . Based on a combination
of analysis and heuristic reasoning, this was interpreted to
imply that the free energy change due to confinement varies
with c in proportion to c−2 and that the resulting confining
pressure varies with c as the gradient of the free energy or as
p ∼ c−3 .
The purpose here is to approach the issue of fluctuations within a parabolic confining potential directly. In particular, a mathematically exact statistical solution for fluctuation of the square membrane within a parabolic energy well
is provided. The variance of mean deflection implied by this
solution is then set equal to c2 and implications for the dependence of free energy on c are examined. It is found that
the result differs in several respects from the behavior anticipated by Helfrich in Ref. [1].
1181
The study of membrane fluctuations subjected to a onesided harmonic constraint or within a soft harmonic well has
been pursued for a number of purposes. Rädler et al. [3]
studied the fluctuations of a giant vesicle over the flat region
where it pressed against a glass surface. This configuration
made it possible to view the fluctuations directly by means
of interference microscopy and to draw conclusions on membrane behavior from those observations and an accompanying statistical analysis. Another interesting study involving
one sided constraint was reported by Gov et al. [4]. This
work was focussed on fluctuations of the bounding membrane of a red blood cell, specifically, on how those fluctuations are influenced by the cytoskeleton. In this discussion, the role of the cytoskeleton was modeled by means of
a parabolic confining potential. A very thorough study of
fluctuation spectra of confined or supported membranes was
reported by Merath and Seifert [5]. Here, the systems involving continuously supported membranes were modeled
by assuming that the support was introduced in the form
of a harmonic resisting potential. The analysis in Ref. [5]
case is closely related to that reported here, but the focus
was on different aspects of the phenomenon. More recently,
Farago [6] examined fluctuation of a square membrane near
a rigid plane for the case when the corners of the membrane
are firmly attached to the plane surface. The interactions between the membrane and the plane surface were represented
by a parabolic confining potential.
2 The model
The configuration of the system of interest, with a nominally
planar membrane extending over a square L×L area and confined between two surfaces at distance c from the reference
plane of the membrane is illustrated in Fig. 1. The notation
follows that adopted in the above description of the model.
The energy wells representing “hard” and “soft” confinement are illustrated schematically in Fig. 2. The middle
line on the left side of that diagram represents the reference
plane of the membrane, and the parallel lines on either side
of the membrane at a distance c represent the confining surfaces. The dependence of energy per unit area acquired by
the membrane on the transverse deflection h is illustrated on
the right side of the figure. The curve labelled “hard” is the
square energy well corresponding to the condition stated in
Eq. (1). In effect, there is no resistance to membrane fluctuation in this case as long as the deflection is less in magnitude
than c. The sides of the square well are indefinitely high and
deflection beyond the magnitude c is impossible. This is the
standard circumstance of hard confinement.
On the other hand, the parabolic potential illustrated in
Fig. 2 is an example of “soft” confinement. In such a case,
any small patch of membrane experiences resistance as soon
as the deflection becomes nonzero in either direction. As the
deflection increases in magnitude, the resisting pressure increases linearly with deflection or, equivalently, the energy
1182
of confinement increases parabolically. In the present case,
attention is limited to parabolic wells with reflective symmetry with respect to the plane of zero deflection and which are
characterized by the root curvature γ as illustrated in Fig. 2.
L. B. Freund
random variables by adopting the Helfrich expression [7] for
bending energy per unit area
1
κ(∂ xx h(x, y) + ∂yy h(x, y))2 ,
(3)
2
where κ is the elastic bending modulus; its physical dimensions are force×length. Integration over the (undeformed)
area of the membrane then yields
Ub =
Fig. 2 A schematic diagram illustrating the position of the membrane with respect to the confining surfaces on the left side, and
the corresponding energy wells, shown on the right side, that define
the nature of the constraint imposed by each type of confinement
considered in this discussion. The main distinction drawn is that
between hard and soft confinement
This configuration is analyzed within the framework of
classical statistical mechanics. Accordingly, accessible configurations are represented by means of a suitable set of random variables. The probability of finding any particular configuration is given by the Boltzmann distribution which incorporates the internal energy of the configuration for any
choice of random variables. In the present circumstances,
the internal energy is the sum of the elastic bending energy
Ub and the energy of confinement Uc . Conclusions relevant
to macroscopic behavior are accessible through the partition
function for the system which is the normalizing factor of the
Boltzmann probability distribution.
A modal description of the membrane is adopted
whereby the deflected shape is described by the finite Fourier
series
n
n 2π jy
2πix
cos
, L = nλ, (2)
h(x, y) = λ
ai j cos
L
L
i=1 j=1
where n is the number of modes in each direction, L is the
edge dimension, λ = L/n is the smallest wavelength included
in the description of deformation, and ai j is a square n × n
matrix of random variables adopted to represent mode amplitudes. Each component of the matrix ai j can take on any positive or negative real value. The influence of those values for
which deformations are large in some sense are suppressed
by the Boltzmann factor, so that only values corresponding
to relatively small deformations consistent with the underlying assumptions are incorporated in a consequential way
within the result. The steps toward extraction of the partition
function are outlined in the next section.
3 Analysis
The total elastic bending energy is expressed in terms of the
n
n
8κπ4 2 2
a (i + j2 )2 ,
n2 i=1 j=1 i j
(4)
for the total bending energy. Note that this representation is
a quadratic form in the components of the matrix ai j .
Similarly, the total energy of confinement within the
parabolic well is expressed in terms of the random variables.
As indicated in Fig. 2, the confinement energy per unit area
1
at location x, y is γh(x, y)2 where the dimensions of the
2
physical constant γ are force/length3 . Again, integration over
the area of the membrane yields the result
L L
n
n
1
γλ4 n2 2
γh(x, y)2 dxdy =
a ,
(5)
Uc =
8 i=1 j=1 i j
0
0 2
for the energy of confinement. The sum of Eqs. (4) and (5)
is the internal energy of the system in terms of the random
variables.
With the total internal energy U = Ub + Uc expressed
in terms of the random variables, the partition function for
the system is expressed as
∞ ∞
Z(n) =
· · · e−U(a11 ,a12 ,··· )/kT da11 da12 · · · ,
(6)
−∞
−∞
where the product of the Boltzmann constant k and the absolute temperature T is the reference thermal energy unit.
With the deformation restricted to small deflections and the
confinement being parabolic, this multiple integral can be
evaluated in closed form to yield
n
n 8n2 πkT
,
(7)
Z(n) =
16π4 κ(i2 + j2 )2 + γn4 λ4
i=1 j=1
which expresses the partition function in terms of the number
of modes n and the system parameters κ, γ, λ and T .
The partition function is the link between random microscopic fluctuations of the physical system and the mean
values of thermodynamic quantities of physical interest
which can be observed macroscopically. In particular, the
system free energy A is determined from the partition function through the relationship A(n) = −kT ln Z(n) which, in
view of Eq. (7), yields the result
n
n
8n2 πkT
1 A(n) = − kT
ln
,
(8)
2
16π4 κ(i2 + j2 )2 + γn4 λ4
i=1 j=1
in terms of the number of modes considered and the system
parameters.
Fluctuation pressure on a bio-membrane confined within a parabolic potential well
1183
4 Discussion
In general, the confining pressure can be deduced from the
dependence of the free energy of the membrane on the degree
of confinement. In the case of confinement of the membrane
motion by a nominally rigid surface that is parallel to the
reference plane of the membrane, the degree of confinement
is represented by the normal distance between the confining
plane and the membrane reference plane. The reason is that
the confining force—the resultant force of the pressure—is
the quantity that is work-conjugate to the spacing with respect with respect to the free energy.
On the other hand, in the case of parabolic confinement,
there is no natural length parameter that is work conjugate to
the confining force with respect to free energy. One possible choice of a length parameter to adopt in seeking the
confining force is the root mean square of the fluctuation
displacement or, equivalently, the statistical variance of the
mean fluctuation. However, this does not arise naturally in
the analysis.
A more readily accessible choice of length parameter
can be defined by dividing the parabola into two parts having reflective symmetry with respect to the reference plane
of the membrane. If these parts are slid apart a prescribed
distance c, as indicated in Fig. 3, then the change in free
energy associated with that process could be used to define
energy. In this instance, the dependence of the free energy
on confinement within a divided parabolic potential is not
represented by Eq. (8) nor is it otherwise known.
Fig. 4 An illustration of the correspondence between particular
values of curvature γ of the confinement energy well and c which
denotes the value of h at which the potential strength is the particular value Γ
Because the value of Γ is fixed at some level, this relationship specifies a direct relationship between the root curvature γ of any particular parabola and the length parameter c,
namely, γ = 2Γ/c2 . Therefore, if γ in the expression for A is
replaced by 2Γ/c2 then the free energy becomes an explicit
function of the distance c. This makes it possible to define
the resultant force of confinement P simply as the potential
gradient P = −∂c A, with the result that this force is
−1
n n
Pλ c c 3 4 κ (i2 + j2 )2
=
+
8π
,
kT
λ
λ
Γλ2
n4
i=1 j=1
(10)
in nondimensional form. This expression for force is mathematically exact within the framework established.
1
Fig. 3 Plot of the split confinement potential γ(h2 − c2 ) versus h
2
for |h| ≥ c
Here, we turn to a simpler and more explicit choice of
length parameter. Instead of describing the one parameter
family of confining parabolic potential well shapes in terms
of the root curvature of each parabola, suppose they are described in terms of the distance from the symmetry plane, say
c, at which the energy per unit area represented by a particular parabola is a specific value, say Γ. This alternate means
of describing the parabolas is illustrated in Fig. 4. In that
case, the parameters c and Γ are related through the curvature γ for that particular parabola according to
1
(9)
Γ = γc2 .
2
It can be seen that the second term within square brackets in the definition of force is the ratio of the two terms in
the denominator of the partition function. As was observed
above, the internal energy is quadratic in the random variables ai j . It then follows from the equipartition theorem for
that the statistically expected value of total internal energy is
1 2
n kT . This is the expected value of the sum Ub + Uc or,
2
equivalently, the sum of the expected values of the two energies. It appears to be necessary to turn to numerical evaluation in order to better understand the influence of degree of
confinement on the confining pressure.
A few qualitative features of this result for force are immediately evident from Eq. (10). For example, if the bending stiffness of the membrane is indefinitely small then the
resultant force reduces to P = n2 kT/c which is the ideal gas
limit for n2 degrees of freedom. On the other hand, if the
magnitude of κ is of the same order of magnitude as that of
Γλ2 then the role of the second term in the square brackets is
much more difficult to ascertain. To gain some insight into
the implications of the result for force, one aspect of the behavior of Pλ/kT on c/λ is illustrated numerically for a range
of values of κ/Γλ2 but a fixed value of n in Fig. 5. Similarly,
1184
aspects of the dependence of the non-dimensional force on
c/λ for a range of values of n but a fixed value of κ/Γλ2 are
illustrated in Fig. 6.
Fig. 5 Plots of resultant force of confining pressure versus the
degree of confinement represented by the distance parameter c for
n = 10 and several values of the ratio of membrane bending stiffness to energy of confinement per area λ2
L. B. Freund
In most reports which have appeared in the literature, it
has been presumed that the lowest modes, or longest wavelength modes, are the principal contributors to the confinement pressure. This has been justified by noting that the
higher (or shorter wavelength) modes have a fluctuation that
is too small to matter in any significant way. For the problem
at hand, the results appearing in Fig. 6 call this supposition
into question. It is seen that the magnitude of pressure shows
strong dependence on the value of n for the range included
there. Furthermore, values of n in excess of 1 000 are commonly realized in experimental membrane configurations.
To purse this matter quantitatively, suppose that the
dimensionless confining pressure p varies with the dimensional spacing c = c/λ according to p = αc −m for c ≥ 1
where α is a constant. If this expression is differentiated
with respect to c, then each side of the resulting equation
is divided by the corresponding side of the undifferentiated
equation, and the result is evaluated for c = 1, it is found that
p (c) m=−
,
(11)
p(c) c→1+
where the prime denotes a derivative of the function with respect to its argument.
Fig. 6 Plots of resultant force of confining pressure versus the
degree of confinement represented by the distance parameter c for
κ/Γλ2 = 1 and several values of the n, the measure of the number
of degrees of freedom incorporated into the statistical description
Much of the discussion of confining pressure in the literature is focussed on the dependence of pressure or mean
force on the separation distance c. This distance is a macroscopic distance, and its range is restricted to values greater
than the largest dimension at which quantum mechanical descriptions of behavior are pertinent, say the deBroglie wavelength. The “microscopic” dimension λ will serve this role
for purposes of the present discussion, so that the range of c
is restricted to c/λ ≥ 1.
Although the actual dependence of confining pressure
on c/λ is not known with certainty in any particular cases, the
form of the dominant contribution to pressure is usually assumed to be (λ/c)m where m is a positive integer. It has been
noted in the foregoing discussion that, in the present case,
the assumption that κ = 0 leads to the value m = 1 which is
the ideal gas limit. In his pioneering work on hard confinement of a membrane, Helfrich [1] concluded that m = 3 for
that case.
For the case of parabolic confinement being considered
here, the right side of Eq. (11) can easily be determined
from Eq. (10). The result of doing so is shown in Fig. 7 over
a wide range of values of n. As is evident in the figure, it
appears that m → 3 as n → 0+ . However, the value of m undergoes a dramatic transition to smaller values as n increases
and, for values of n beyond about n = 100, the value of m
approaches 2. This behavior is remarkable and it suggests
a potentially important role for the value of the parameter
n in determining the pressure of confinement. While some
connection of the value of n to the mechanism of pressure
generation can be argued, further study is required before
any conclusions can be inferred.
Fig. 7 The dependence of the exponent defining confining pressure
for small spacing according to Eq. (11) versus the number of deformation modes taken into the account for the fluctuating membrane.
Results are shown for two values of the stiffness ratio, κ/Γλ2 = 2
and 2 000
Fluctuation pressure on a bio-membrane confined within a parabolic potential well
References
1 Helfrich, W.: Steric interaction of fluid membranes in multilayer systems. Zeitschrift fur Naturforschung 33a, 305–315
(1978)
2 Boal, D.: Mechanics of the Cell. Cambrdige University Press,
Chapter 6 (2002)
3 Rädler, J.O., Feder, T.J., Strey, H.H., et al.: Fluctuation analysis
of tension-controlled undulation forces between giant vesicles
and solid substrates. Physical Review E 51, 4526–4537 (1995)
1185
4 Gov, N., Zilman, A.G., Safran, S.: Cytoskeleton confinement
and tension of red blood cell membranes. Physical Review
Letters 90, 228108 (2003)
5 Merath, R.J., Siefert, U.: Fluctuation spectra of free and supported membrane pairs. European Physics Journal E 23 103–
116 (2007)
6 Farago, O.: Membrane fluctuations near a plane rigid surface.
Physical Review E 78, 051919 (2008)
7 Helfrich, W.: Elastic properties of lipid bilayers–Theory and
possible experiments. Zeitschrift fur Naturforschung 28c, 693–
702 (1973)