Reply to Wennerström et al.: Entropic forces on a confined membrane

The comments offered by Wennerström et al. (1) on a recently published article (2) address issues that fall outside the scope of that report, to some degree. The one exception is their observation that the repulsive force cannot be analyzed by the methods used in the article for separation distances on the molecular scale, but that limitation was noted explicitly in ref. 2.

The focus of ref. 2 is on analysis of the fluctuations of an elastic bio-membrane in a thermal environment, with the membrane confined between rigid planes symmetrically positioned on either side of the membrane at a normal distance, c, from the its reference plane. The membrane is assumed to be square with each edge of length L, and periodic boundary conditions on the edges are assumed. The model problem adopted in ref. 2 is identical to that proposed originally in the ground-breaking report by Helrich (3). This choice was made to allow for a direct comparison of the results of the two analyses. As reported in ref. 2, the behavior determined by analyzing the mathematical model without approximation leads to conclusions that are qualitatively different from those in the work by Helrich (3). The determination of the partition function or free energy in ref. 2 is mathematically exact, but the analysis in ref. 3 is based on several assumptions of unknown fidelity. These assumptions are identified and discussed in ref. 2.

A principal difference between the results of the two analyses is that it was concluded in Helrich (3) that the mean entropic pressure exerted on the membrane by either confining plane varies with the distance c as c⁻³, whereas the analysis in ref. 2 leads to a c⁻¹ dependence.

A sketch of the approach adopted in ref. 2 follows. The scale invariant partition function Z is defined in terms of the elastic energy density of the membrane, and an expression for the membrane free energy A follows from A = −kT ln Z. For any value of L, the mean force f exerted on the membrane by either confining plane is the physical quantity that is work-conjugate to c with respect to A, or

Dimensionally, the right side of this expression varies with c as c⁻¹. The magnitude of the force is determined by the sensitivity of Z to changes in c. The mean pressure p = f/L² then also varies with c as c⁻¹.

A second assumption underlying the arguments of Wennerström et al. (1) is that it is characterized by a single characteristic dimension, namely, the length c. The exact solution reveals the central role of a second characteristic dimension l = L/n in the notation of ref. 2. This parameter identifies the length to which the membrane deformation is resolved in representing fluctuations in terms of a countable set of degrees of freedom.