Entropic force between membranes reexamined.

L ipid bilayers serve as model surfaces that are often used to understand the behavior of actual cell membranes. Cell adhesion (1, 2), membrane fusion (1), binding-unbinding transition (3), self-assembly, and related phenomena govern an astounding array of biological functions; for example, sperm-egg fusion is the basis of mammalian reproducion (1). These phenomena are dictated by a complex interplay between the various attractive and repulsive forces that mediate between biological membranes. The key role is played by a repulsive force termed “steric hindrance,” or simply entropic pressure, the origins of which lie in the thermally excited fluctuations of membranes. Nearly four decades ago, in a landmark paper, Wolfgang Helfrich (4) proposed both the concept and the quantitative nature of this force. Freund (5) reexamines this paradigm in PNAS, and finds that the entropic force is of a remarkably different character than hitherto believed. Ubiquitous van der Waals forces provide the weak attraction between biological surfaces; these vary as 1/c for close separations and transition to 1/c at larger distances (Fig. 1 A and B) (6, 7). Here, c is the mean distance between the interacting membranes. The notable aspect of the attractive force is that it is long-ranged. A catch-all phrase, “hydration forces” (8), is used to denote the repulsive force that becomes operative when membranes are nearly touching each other; this is quite short-ranged and drops off exponentially with distance. The underlying mechanisms of hydration forces are still debated and a subject of active research (9). A notable observation is that both the van der Waals and hydration interactions would be present even if membranes were perfectly rigid. The origins of a third interaction—the so-called “entropic force”—is predicated on the fact that biomembranes are (generally) quite flexible and the energetic cost of elastic bending is low enough that at room temperature, they fluctuate and flap like flags in a strong wind. A single membrane fluctuates freely. As two membranes approach each other, they hinder or diminish each other’s outof-plane fluctuations. This hindrance decreases the entropy and the ensuing overall increase of the free-energy of the membrane system, which depends on the intermembrane distance, leads to a repulsive force that tends to push the membranes apart. Stated differently, a finite external pressure is required to maintain the mean distance between the interacting membranes. Helfrich (4), using a variety of physical arguments and approximations, postulated that the entropic force varies as 1/c. In contrast to the other known repulsive forces, this behavior is longranged and competes with the van der Waals attraction at all distances (6, 7, 10). Since Helfrich’s proposal (4), biophysicists have used the existence of this repulsive force to explain and understand a variety of phenomena related to membrane interactions. Although lipid bilayers and membranes are microscopically quite complex, their mechanical behavior (under numerous physically relevant circumstances) can be well-described by just a few continuum parameters, such as bending modulus, which sets the energy cost of bending the membrane or an out-of-plane fluctuation. Taking advantage of these parameters, Freund (5), like Helfrich and many other authors preceding him, treats membranes as idealized elastic surfaces and uses classic statistical mechanics to infer the nature of the entropic force caused by interaction Fig. 1. (A) Depicts (with some artistic license) a very small patch of interacting and thermally fluctuating lipid bilayers. The areal extent of a membrane at which a continuum analysis is valid is roughly 10–20 times larger in scale than what is shown here. Even in absence of any fluctuations, membranes feel an attractive long-range van der Waals force and a short-range exponentially decaying repulsive hydration force that is attributed to the microscopic structure of lipid-water interface, although the exact physics underlying this repulsion is still under debate (9). (B) The oft-used idealization that replaces the actual (microscopically complex) membranes, as shown in A, by a pair or stack of interacting elastic sheets.