Abstract
The phenomenon of thermal fluctuation of a biomembrane within a stack of like membranes was introduced in a pioneering paper [Helfrich W (1978) Z Naturforsch A 33(3):305–315]. Internal energy arises in a representative membrane through elastic resistance to bending deformation, and membrane motion is further restrained through steric interaction with adjacent membranes. Due to reflective symmetry within the stack, analysis of behavior can be reduced to study of a single membrane fluctuating between parallel rigid planes. The phenomenon is reexamined here from several viewpoints to quantify the dependence of system free energy on the size of the gap between membranes. This analysis is based on essentially the same formulation that was used in the original study, and it is found that analysis based on enforcement of the underlying principles can lead to an exact mathematical solution. On this basis, a self-consistent picture of behavior emerges showing a dependence of free energy on the width of the confining gap that is weaker than has been thought to prevail.
Keywords: confined fluctuations, entropic force, statistical mechanics
The physical system of interest is a nominally flat biomembrane with large surface area compared with any of its molecular dimensions; the membrane is immersed in warm water. Stimulated by Brownian motion of water molecules, the membrane undergoes random transverse fluctuations in shape. The fluctuations are sufficiently large to be observable via an optical microscope as flickering of reflected light. This motion is limited due to elastic resistance of the membrane to bending. Applied membrane tension would also limit such fluctuations, but this effect is not taken into account here. The phenomenon of interest arises when the membrane fluctuations are also limited in magnitude by parallel hard surfaces on both sides.
When fluctuating without constraint, the mean value of membrane free energy remains constant. The value of this constant is inconsequential for inferring magnitudes of forces and pressures from the free energy such that the scale for free energy is adjusted to render this constant equal to zero. When the fluctuations are confined by nearby surfaces, the resulting increase in the magnitude of the entropic free energy more than offsets the accompanying decrease in the internal energy of the system, resulting in a net increase in free energy. Further confinement caused by moving the restraining surfaces into closer proximity results in a further increase in the free energy of the membrane. This dependence of free energy on the spacing between the membrane of interest and the confining surfaces implies the existence of a restraining force with a magnitude proportional to the gradient of free energy with respect to distance. Physically, the force is an entropic force, that is, a consequence of the tendency of the membrane to increase its entropy rather than the result of some microscopic physical forces. The macroscopic force that is exerted on the confining surface is the time-averaged response to a large number of small impulses due to intermittent contact of the membrane with the confining surface. It is the dependence of this force on the mean spacing between the membrane and the confining surfaces that is of primary interest.
The confinement of a fluctuating biomembrane by its neighbors within a stack of like membranes and the entropic pressure induced as a result of that confinement were first identified by Helfrich (1). For the situation with a mean intermembrane spacing of 2c, he reasoned that the free energy of a representative membrane varies with c as 
. If so, it follows that the mean confining pressure varies with c as 
.
In general, for interaction between surfaces or lines supporting dipole distributions, the free energy of interaction will indeed vary as the inverse square of the separation distance. For example, such an interaction arises between crystallographic steps on a vicinal surface of a single crystal. The elastic field of a step is essentially a dipole field (2), and, accordingly, the steps interact through their elastic energy fields (3). Similarly, interaction between molecular surfaces that can be characterized as electrical dipole distributions underlies the definition of the van der Waals interaction between surfaces. Steric interactions between electrically neutral, nondipolar membrane surfaces appear to have a character fundamentally different from dipole-dipole behavior. Consequently, on physical grounds, an inverse square dependence of free energy on spacing that results solely from steric avoidance is curious. A recent study of the influence of a parabolic confining potential on a fluctuating membrane (4) showed a substantially weaker interaction, a result that has stimulated a reexamination of the circumstances of hard confinement.
Attention here is focused on analysis of the configuration that best represents the mechanics of mutual confinement within a stack of like membranes if that behavior is to be deduced from consideration of a single “representative” membrane. The physical model is depicted in Fig. 1. When undeformed, the membrane lies in the plane 
; the portion of the reference plane occupied by the reference configuration of the membrane is shown as cross-hatched. The uniform spacing between reference planes of adjacent membranes in the multilayer is 2c. If the total number of membranes in the stack is large, the steric interaction of adjacent membranes should exhibit reflective symmetry with respect to a plane midway between the reference planes of the two membranes. These “rigid” intermediate planes of reflective symmetry at 
are also shown in Fig. 1.
Fig. 1.
Schematic diagram of the system of interest. The cross-hatched plane is the reference plane of the membrane, and the rigid planes on either side serve to confine its fluctuations.
The transverse deflection of a material point with coordinates x, y in the reference plane is denoted by 
. The gradients 
and 
are small in magnitude compared to 1. The elastic energy per unit area of the membrane is then determined on the basis of the Helfrich (5) elastic bending energy per unit area
 |
where κ is the uniform elastic bending modulus. From the condition of reflective symmetry of deformation with respect to the intermediate planes at 
and the condition that the membranes are mutually nonpenetrating, it follows that the transverse deflection is confined to the range
 |
The case of a single confining plane was considered in an approximate analysis by Farago (6).
To set a length scale, the length λ is introduced. Most commonly, the value of λ would be set by the behavior of the membrane. For example, it could be deduced from the condition that the elastic energy of the fluctuation mode 
is equal to kT, the Boltzmann constant times the absolute temperature. The value of λ would be that wavelength for which the amplitude a is equal to the membrane thickness or some other molecular dimension. Throughout this study, values of all physical quantities are expressed in nondimensional form; lengths are normalized by λ, energies by kT, pressures by 
, and so on.
In the first of the two sections that follow, the asymptotic behavior of the free energy when 
is determined directly from the partition function and the differences in the underlying conditions from those assumed by Helfrich (1) are identified. This is followed by an analysis leading to an exact mathematical solution of the problem of an elastic membrane fluctuating within a gap of width 2c between parallel rigid surfaces; the result is valid over the full accessible range of c.
Behavior for 
An advantage of the modal representation of the deflection of a membrane in the form of a Fourier series is that the mode amplitudes can be varied independently and the mode shapes are mutually orthogonal. However, when the deflections are limited by nearby surfaces, the advantages of mode independence are lost. Consequently, an alternate representation based on strictly independent random variables is adopted for the asymptotic analysis.
For spatially periodic deflections with period L, the edge length L of the membrane is an integer multiple of λ. That integer is denoted by 
. Suppose that the deflection 
from the reference plane is described in terms of 
random deflections making up the components of the square matrix 
for 
. Each component of the matrix is the transverse deflection of a unique point on the membrane. These points might be distributed over a regular square array spanning the membrane area or according to some other prescription. Once the points are identified, the total elastic energy 
of the membrane for any choice of random variable values is determined by integration of 
over the area of the membrane 
. To do so, it is necessary to define a scheme for interpolation of deflection values between points at which the values 
are defined. Many choices that satisfy the essential requirements for continuity and differentiability are available. However, no matter which interpolation scheme is adopted, the result for small-amplitude deflections will be such that 
is a positive definite quadratic form in the components of the matrix 
. More specifically, there exists a stiffness matrix of rank four, say 
, depending only on the values of system parameters, such that
 |
where 
for any choice of 
that is not identically zero. 
only if all components of 
are zero.
In terms of nondimensional variables, the partition function in this instance is
 |
The limits of the ranges of the random variables are mutually independent, with the range being 
for every choice of i and j.
To gain some insight into the dependence of 
on gap size c, introduce a rescaled set of random variables 
. The partition function is then
 |
where the dependence on c is now explicit. For small values of c, the integrand is of the order 
; thus, to lowest order, the integral reduces to the volume integral over the interior of a hypercube in 
dimensions with edge length 2; this volume is 
. The value of the partition function to lowest order in c is then 
. The asymptotic behavior of the nondimensional free energy as 
is
 |
Its range of validity will be considered again in the next section, after a complete solution is available.
It is noted that an argument similar to that which led to the observation in Eq. 6 was proposed by Janke and Kleinert (7). However, as a result of the insertion of factors of c (or d in their notation) in the integrand of Eq. 4, a conclusion different from that shown in Eq. 6 was reached.
With a result for the dependence of free energy on c in hand, it is possible to determine the associated force exerted on the confining surface as a function of c for small values of c. If this entropic force is denoted by f, it follows that
 |
for 
.
In the analysis by Helfrich (1), it was assumed that for small values of c, the membrane could be viewed as consisting of many small squares of membrane material with each fluctuating independent of all others. Furthermore, it was assumed that the membrane patches each fluctuate in position only in the z direction. In other words, the membrane is viewed as being a one-dimensional ideal gas, a case for which the force on the confining surface due to each square varies as 
. Then, to determine a corresponding “microscale pressure” on the membrane, it was assumed that the size of each square decreases with decreasing values of c in such a way that the area of each square is 
. This assumption, which is given in equation 30 of ref. 8, leads to the conclusion that the pressure varies with c as 
.
The result reported by Helfrich (1) rests on the implicit assumption that 
, where 
is here the number of independent squares making up the membrane. Thus, it appears that the essential differences between the result reported by Helfrich (1) and the result obtained here are as follows. The point of view adopted by Helfrich (1) is that (i) the membrane can be viewed as 
noninteracting elements, (ii) these elements fluctuate independently, (iii) the area of each element scales as 
, and (iv) macroscopic pressure is the result of actual pressures on the microscale. On the other hand, the present point of view is that (i) the membrane responds as a continuous deformable elastic solid; (ii) the configurations of the membrane can be represented by means of 
degrees of freedom, where n is an intrinsic characteristic of the system that is independent of c; and (iii) the pressure arises as a result of randomly distributed impulses applied to the confining surface. In other words, only the macroscopic force due to these impulses is perceived as the entropic pressure; it is not the resultant force of pressure on the microscale. Instead of pursuing this asymptotic behavior, we turn to a method of analysis that provides the dependence of free energy on c exactly for the full range of response.
Dependence of 
on c
The transverse deflection of the membrane is represented in the modal form
 |
over the region 
, where the array of mode amplitudes 
is an 
matrix of random variables; the admissible values of the matrix elements define the accessible range of configurations. This form of the deflection 
is one of a total of four contributions to the most general form of a periodic transverse deflection with period L in both the x and y directions, with these differing only due to the combinations of trigonometric functions incorporated. Each of these contributes equally to the total free energy; consequently, it is necessary to consider only one case to determine the dependence of the free energy on c.
The elastic energy or internal energy of the membrane for any set of values of the random variables 
is found by substituting the deflection 
into the definition of membrane elastic energy density (Eq. 1), followed by integration of the result over the area of the membrane 
. The result of doing so is
 |
where κ is the nondimensional bending modulus of the membrane. With the internal energy written explicitly in terms of the random variables, attention is turned to finding the partition function 
for the system.
The partition function is defined as the result of integration of the Boltzmann factor 
over the full accessible range of each and every random variable 
. In the present instance, the range of membrane fluctuation is restricted to remain everywhere within the gap of width 2c. To satisfy this constraint, we require that there is a bound, say 
, on each mode of Eq. 8 such that
 |
The actual values of the elements of the matrix 
are determined as part of the solution process. For now, it is sufficient to note that any set of positive values for the bounds 
of individual deflection modes that satisfies
 |
will surely meet the constraint (Eq. 2) on total fluctuation amplitude. To see that this is so, Eq. 8 implies that
 |
at any point 
on the membrane surface. In view of Eq. 10, each occurrence of 
in Eq. 12 can be replaced by 
, which, given Eq. 11, ensures that the constraint condition (Eq. 2) will be satisfied.
In light of Eq. 9, the partition function for membrane fluctuation that is consistent with the constraint is
 |
where 
represents the nondimensional combination of parameters
 |
Recognizing the integral in Eq. 13 as an error function 
, it follows that
 |
The partition function provides the link between microscopic features of the system and the overall thermodynamic energy measures. The nondimensional membrane free energy 
is
 |
The task remaining is to determine the values of the parameters 
. The feature that distinguishes among choices for the components of the matrix 
is the value of 
that results from any particular choice. The point of view underlying this analysis is that the system is in thermodynamic equilibrium such that the quantities defining behavior take on time-averaged values consistent with minimum free energy. The objective, then, is to determine those values of 
that define the minimum value of 
from among all those values consistent with the constraint (Eq. 11). A mathematical technique that is well-suited for addressing such a constrained minimization task is the method of Lagrange multipliers (9).
A Lagrange multiplier, say ϕ, with value to be determined is introduced to deal with the constraint, and a modified free energy function of the 
variables 
is introduced as
 |
The 
values of 
and the value of ϕ sought are those that satisfy the 
equations
 |
in addition to 
, which is equivalent to the condition (Eq. 11). It is observed that 
for each mode. This relationship implies that 
, an observation that is incorporated in the ensuing development.
The specific equations to be solved for a prescribed value of c are Eq. 11 and the 
equations
 |
where 
is defined in Eq. 14. The unknowns are the 
components of 
and f. In fact, solving the equations is simplified considerably by prescribing a value of 
and regarding c as an unknown.
Although 
is dimensionless, it does depend on n and κ. Consequently, values of these parameters must be assigned to pursue the issue quantitatively. To see the nature of the solution, the left side of Eq. 19 is graphed in Fig. 2 for each deflection mode for 
and 
. The value of n is chosen to be small to allow for graphical visualization of relationships. Note that there are 16 modes of deformation for 
, but only 10 curves are evident in the figure. The reason is that for 
, the plots for 
and 
are identical.
Fig. 2.
Plots of 
vs. 
for 
and 
and for all values of r, s. For the fixed value of force 
, the values of the bounds on fluctuation amplitudes are those values corresponding to the intersections of the line 
, with the curve corresponding to the rs mode. Representative lines are indicated in the figure.
To solve Eq. 19 for a range of values of force f, one can draw a horizontal line on the plot with an ordinate equal to a particular value of f, say 
. Then, the abscissa values of the 
intersection points of that line with the 
curves in the figure identify the elements 
making up the full matrix of values of 
for that particular force 
. One pair of such lines is illustrated in Fig. 2. With the mathematical solution of the governing equations in hand, numerical results with a high degree of accuracy are easily extracted simply by evaluating the expressions in these equations.
Discussion
For the case with 
and 
, the dependence of the nondimensional free energy 
on normalized spacing c is illustrated quantitatively in Fig. 3. If a larger or smaller value of n would have been chosen for purposes of generating these results, the curves would be similar in shape but would have larger or smaller ordinate values accordingly. The force f representing the result of the pressure acting on each of the confining surfaces is also calculated, and a typical result is shown in Fig. 4 for the same parameter values as were chosen for Fig. 3. The form of the graph is as anticipated, with magnitude decaying to ever smaller values as c increases.
Fig. 3.
Dependence of membrane free energy 
on the degree of confinement as represented by the spacing c for the case where 
and 
. The discrete points show results obtained by numerical evaluation of the solution described in the text.
Fig. 4.
Dependence of the net force f pushing against a confining plane vs. c for the case where 
and 
, based on evaluation of the solution to determine each of the discrete points shown.
With the complete description of the dependence of the free energy 
established in the preceding section, the behavior of pressure for “small” values of c can be reexamined in a broader context. Whereas the discussion leading to Eq. 6 was based on asymptotic dependence of pressure on c as 
, it is now possible to consider the approximate dependence of pressure on c over the full range.
Consider the behavior of the left side of Eq. 19 for the case where 
for all values of 
. The result is simply
 |
Solving for 
and summing both sides of the equation over the full range of indices, it follows that
 |
a result that is identical to the asymptotic result obtained in Eq. 7. For 
, it is noted that the values of 
for the larger indices are on the order of 
. Nonetheless, the approximations invoked continue to be valid for nearly all modes under these conditions. Perhaps the best way to assess the utility of the approximation for large values of pressure or relatively small values of c is to compare it directly with the solution that is accurate over the full range of c. Such a comparison is illustrated in Fig. 5, where it is seen that the approximate solution continues to be a useful measure of entropic pressure for values of c up to quite large values, and perhaps so over the full range of c considered.
Fig. 5.
Discrete points are determined by evaluation of the terms in the exact solution, and the solid line segments connect these points. The dashed curve, on the other hand, is a graph of the approximate solution given in Eq. 7.
Finally, the task of measuring the connection between confining force and separation interval for a stack of membranes poses enormous challenges (10). Nonetheless, data resulting from such measurements have been reported in the literature (11, 12). In each case, the data show a decay in the magnitude of the confining force as the intermembrane separation distance increases, and over the limited range covered, it could be fit reasonably well by either the inverse distance cubed dependence or the inverse distance dependence.
Acknowledgments
I thank Jim Rice of Harvard University and Sam Safran of the Weizman Institute for discussions of the issues addressed here. My colleague at the University of Illinois at Urbana–Champaign, Steve Granick, provided helpful comments on a draft of this article.
Footnotes
The author declares no conflict of interest.
See Commentary on page 1976.
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