Entropic force between membranes reexamined

Pradeep Sharma1 Department of Mechanical Engineering and Department of Physics, University of Houston, Houston, TX 77204

ipid bilayers serve as model sur- faces that are often used to un- L derstand the behavior of actual cell membranes. Cell adhesion (1, 2), membrane fusion (1), binding-un- binding 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 mem- branes. 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 quan- titative nature of this force. Freund (5) reexamines this paradigm in PNAS, and finds that the entropic force is of a re- markably different character than hith- erto believed. Ubiquitous van der Waals forces pro- vide the weak attraction between bi- ological surfaces; these vary as 1/c3 for close separations and transition to 1/c6 at larger distances (Fig. 1 A and B) (6, 7). Here, c is the mean distance between the interacting membranes. The notable as- pect of the attractive force is that it is long-ranged. A catch-all phrase, “hydra- tion forces” (8), is used to denote the re- pulsive force that becomes operative when Fig. 1. (A) Depicts (with some artistic license) a very small patch of interacting and thermally fluctuating membranes are nearly touching each other; lipid bilayers. The areal extent of a membrane at which a continuum analysis is valid is roughly 10–20 times this is quite short-ranged and drops off larger in scale than what is shown here. Even in absence of any fluctuations, membranes feel an attractive exponentially with distance. The un- long-range van der Waals force and a short-range exponentially decaying repulsive hydration force that is derlying mechanisms of hydration forces attributed to the microscopic structure of lipid-water interface, although the exact physics underlying are still debated and a subject of active this repulsion is still under debate (9). (B) The oft-used idealization that replaces the actual (micro- research (9). A notable observation is that scopically complex) membranes, as shown in A, by a pair or stack of interacting elastic sheets. both the van der Waals and hydration in- teractions would be present even if mem- pressure is required to maintain the mean physically relevant circumstances) can be branes were perfectly rigid. The origins of distance between the interacting mem- well-described by just a few continuum a third interaction—the so-called “entropic parameters, such as bending modulus, force”—is predicated on the fact that bi- branes. Helfrich (4), using a variety of fl physical arguments and approximations, which sets the energy cost of bending the omembranes are (generally) quite exible fl and the energetic cost of elastic bending is postulated that the entropic force varies as membrane or an out-of-plane uctuation. c3 low enough that at room temperature, they 1/ . In contrast to the other known re- Taking advantage of these parameters, fluctuate and flap like flags in a strong wind. pulsive forces, this behavior is long- Freund (5), like Helfrich and many other A single membrane fluctuates freely. As ranged and competes with the van der authors preceding him, treats membranes two membranes approach each other, Waals attraction at all distances (6, 7, as idealized elastic surfaces and uses clas- they hinder or diminish each other’sout- 10). Since Helfrich’s proposal (4), bio- sic statistical mechanics to infer the nature of-plane fluctuations. This hindrance de- physicists have used the existence of this of the entropic force caused by interaction creases the entropy and the ensuing overall repulsive force to explain and understand a increase of the free-energy of the mem- variety of phenomena related to membrane brane system, which depends on the in- interactions. Author contributions: P.S. wrote the paper. termembrane distance, leads to a repulsive Although lipid bilayers and membranes The author declares no conflict of interest. force that tends to push the membranes are microscopically quite complex, their See companion article on page 2047. apart. Stated differently, a finite external mechanical behavior (under numerous 1E-mail: [email protected].

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with nearby surfaces. Freund’s analysis of elements making up the membrane are a prior work (11). Parabolic confinement is concludes that across the range of in- considered to be completely unrelated to easier to handle analytically, although care termembrane separations, the entropic intermembrane distance. In his analysis, must be taken to interpret what exactly is force varies as 1/c, a significant departure the elements that constitute the mem- meant by intermembrane distance. Freund’s ’ fl from Helfrich s conclusions (4) and those brane do not uctuate independently; analysis of the softer confinement corrob- ’ of others (9). Freund s work thus appears rather, the elastic deformable membrane orates his present calculations (5) and to be unique in completing a mathematical responds in its entirety to the external provides some additional insights: in the analysis of this phenomenon. disturbances. Finally, rather than limiting analysis of fluctuating and interacting Despite four decades of work (9) that the root mean square fluctuation ampli- more or less have found agreement with tudes to lie below |c| in an average sense, membranes, it is tempting to consider only the results of Helfrich, what leads this constraint is imposed at every single the lowest modes because shorter wave- Freund (5) to this different conclusion? point of the membrane. lengths are far less probable as a result of Helfrich (4, 6) views the membrane as In Freund’s (5) analysis, each small their higher energetic cost. Freund (11) consisting of many small elements that piece of the membrane can fluctuate up to finds that nature of the confinement fluctuate independently, not unlike how c, but it need not. The actual statistical pressure strongly depends on inclusion atoms in an ideal gas behave. Drawing distribution of the fluctuation amplitudes of higher modes. furtheronthisanalogyandtreatingthe is determined by minimizing the free en- Although a clichéd statement (which membrane as an ideal gas of valleys and ergy. The final outcome is a set of equa- does not make it any less true), the test of 3 bumps, a pressure ∼1/c against the tions that must be numerically evaluated any new theory comes from careful ex- fi con ning medium can be extracted. There but yield a rather simple asymptotic re- periments. However, experiments that c → are other assumptions buried in the various lation (for 0), but that is seen to be purport to isolate the entropic force are arguments presented by Helfrich; for ex- valid across a wide range of intersepara- easier said than done. Early on, some el- ample, (i) each independent fluctuating tion distance: f ∼ n2/c. Here, f is the re- n2 egant synchrotron X-ray studies were done element is considered to have an area that sultant entropic force and can be fi is proportional to c2;(ii) recognizing that it identified with the number of degrees-of- by Sa nya et al. (12, 13), who concluded ’ is difficult to handle the statistical me- freedom representing the system. If a rep- that Helfrich s theory is qualitatively cor- chanics of the point-wise hindrance condi- resentative periodic membrane patch of rect. Another notable experimental work is tion, |h(x, y)| ≤ c, a weaker constraint is edge length L is considered, then n = L/λ that of Richetti et al. (14). Freund suggests used: 〈h2〉 ≤ c2. In other words, the hin- (i.e., it scales with membrane size). λ is that in the range of intermembrane dis- drance is imposed on the average rather a normalizing length scale but with a tances examined in some of the aforemen- than point-wise. Although the latter argu- meaningful physical interpretation: this tioned experiments, the distinction ment is a simplification designed to extract is a length scale that is large compared between 1/c and 1/c3 lies within the error analytical insights (a time-honored tradi- with molecular size, and therefore the use bars. Freund’s thought-provoking re- fl tion), the notion that each element uc- of both a continuum analysis and classic examination of the widely studied entropic fi tuates independently is rooted in the idea statistical mechanics is justi ed. A nota- force between membranes will hopefully that wavelengths below a certain value ble feature of this asymptotic result is that encourage design of further experimental are energetically very costly, and therefore the mean pressure f/n2 ∼1/c is in- work that will probe his specific assertions. hardly affect the membrane interaction. dependent of membrane size, which is The assumption that the area of each in- what we expect for periodic fluctuations. What are the physical implications of the dependent fluctuating element is related to As was initially pointed by Helfrich (4), unique force-law proposed by Freund? the intermembrane distance is harder to because of reflective symmetry, the prob- I expect the quest for an answer to this justify, something that Freund (5) aban- lem of a pair of interacting membranes question to be an interesting avenue for dons from the outset. may be replaced by a single membrane future research in membrane physics. It is precisely with the assumptions confined between two rigid walls. One mentioned above (and a few that I did not may, however, consider a softer constraint, ACKNOWLEDGMENTS. I thank Prof. Liping Liu articulate) where Freund (5) departs for example, parabolic confinement instead and Gemunu Gunaratne for our ongoing collab- fi oration on biomembranes, and Ms. Fatemeh from Helfrich (4). I have already in- of the rigid walls (square con nement). Ahmadpoor and Xin Yan for their assistance dicated that in Freund’s work the number Freund has addressed this problem in with Fig. 1 A and B.

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