Effect of protein shape on multibody interactions between membrane inclusions

Field-mediated interactions of passive and conformation-active particles: multibody and retardation effects

Particles in soft matter interact through the deformation field they create, as in the "cheerios" effect or the curvature-mediated interactions of membrane proteins. Using a simple model for field-mediated interactions between passive particles, or active particles that switch conformation randomly or synchronously, we derive generic results concerning multibody interactions, activity driven patterns, and retardation effects.

Langmuir Monolayer Techniques for the Investigation of Model Bacterial Membranes and Antibiotic Biodegradation Mechanisms

The amounts of antibiotics of anthropogenic origin released and accumulated in the environment are known to have a negative impact on local communities of microorganisms, which leads to disturbances in the course of the biodegradation process and to growing antimicrobial resistance. This mini-review covers up-to-date information regarding problems related to the omnipresence of antibiotics and their consequences for the world of bacteria. In order to understand the interaction of antibiotics with bacterial membranes, it is necessary to explain their interaction mechanism at the molecular level. Such molecular-level interactions can be probed with Langmuir monolayers representing the cell membrane. This mini-review describes monolayer experiments undertaken to investigate the impact of selected antibiotics on components of biomembranes, with particular emphasis on the role and content of individual phospholipids and lipopolysaccharides (LPS). It is shown that the Langmuir technique may provide information about the interactions between antibiotics and lipids at the mixed film surface (π–A isotherm) and about the penetration of the active substances into the phospholipid monolayer model membranes (relaxation of the monolayer). Effects induced by antibiotics on the bacterial membrane may be correlated with their bactericidal activity, which may be vital for the selection of appropriate bacterial consortia that would ensure a high degradation efficiency of pharmaceuticals in the environment.

Membrane Remodeling by DNA Origami Nanorods: Experiments Exploring the Parameter Space for Vesicle Remodeling

Inspired by the ability of cell membranes to alter their shape in response to bound particles, we report an experimental study of long, slender nanorods binding to lipid bilayer vesicles and altering the membrane shape. Our work illuminates the role of particle concentration, adhesion strength, and membrane tension in determining the membrane morphology. We combined giant unilamellar vesicles with oppositely charged nanorods, carefully tuning the adhesion strength, membrane tension, and particle concentration. With increasing adhesion strength, the primary behaviors observed were membrane deformation, vesicle-vesicle adhesion, and vesicle rupture. These behaviors were observed in well-defined regions in the parameter space with sharp transitions between them. We observed the deformation of the membrane resulting in tubulation, textured surfaces, and small and large lipid-particle aggregates. These responses are robust and repeatable and provide a new physical understanding of the dependence on the shape, binding affinity, and particle concentration in membrane remodeling. The design principles derived from these experiments may lead to new bioinspired membrane-based materials.

Biophysics at the coffee shop: lessons learned working with George Oster

Over the past 50 years, the use of mathematical models, derived from physical reasoning, to describe molecular and cellular systems has evolved from an art of the few to a cornerstone of biological inquiry. George Oster stood out as a pioneer of this paradigm shift from descriptive to quantitative biology not only through his numerous research accomplishments, but also through the many students and postdocs he mentored over his long career. Those of us fortunate enough to have worked with George agree that his sharp intellect, physical intuition, and passion for scientific inquiry not only inspired us as scientists but also greatly influenced the way we conduct research. We would like to share a few important lessons we learned from George in honor of his memory and with the hope that they may inspire future generations of scientists.

Modeling Membrane Curvature Generation due to Membrane⁻Protein Interactions

To alter and adjust the shape of the plasma membrane, cells harness various mechanisms of curvature generation. Many of these curvature generation mechanisms rely on the interactions between peripheral membrane proteins, integral membrane proteins, and lipids in the bilayer membrane. Mathematical and computational modeling of membrane curvature generation has provided great insights into the physics underlying these processes. However, one of the challenges in modeling these processes is identifying the suitable constitutive relationships that describe the membrane free energy including protein distribution and curvature generation capability. Here, we review some of the commonly used continuum elastic membrane models that have been developed for this purpose and discuss their applications. Finally, we address some fundamental challenges that future theoretical methods need to overcome to push the boundaries of current model applications.

Curvature instability of membranes near rigid inclusions

In multicomponent membranes, internal scalar fields may couple to membrane curvature, thus renormalizing the membrane elastic constants and destabilizing the flat membranes. Here, a general elasticity theory of membranes is considered that employs a quartic curvature expansion. The shape of the membrane and its deformation energy near a long rod-like inclusion are studied analytically. In the limit where one can neglect the end effects, the nonlinear response of the membrane to such inclusions is found in exact form. Notably, exact shape solutions are found when the membrane is curvature unstable, manifested by a negative rigidity. Near the instability point (i.e., at vanishing rigidity), the membrane is stabilized by the quartic term, giving rise to a different length scale and scale exponents for the shape and its energy profile than those found for stable membranes. The contact angle induced by an applied force at the inclusion provides a method to experimentally determine the quartic curvature modulus.

Membrane-Mediated Cooperativity of Proteins

Besides direct protein-protein interactions, indirect interactions mediated by membranes play an important role for the assembly and cooperative function of proteins in membrane shaping and adhesion. The intricate shapes of biological membranes are generated by proteins that locally induce membrane curvature. Indirect curvature-mediated interactions between these proteins arise because the proteins jointly affect the bending energy of the membranes. These curvature-mediated interactions are attractive for crescent-shaped proteins and are a driving force in the assembly of the proteins during membrane tubulation. Membrane adhesion results from the binding of receptor and ligand proteins that are anchored in the apposing membranes. The binding of these proteins strongly depends on nanoscale shape fluctuations of the membranes, leading to a fluctuation-mediated binding cooperativity. A length mismatch between receptor-ligand complexes in membrane adhesion zones causes repulsive curvature-mediated interactions that are a driving force for the length-based segregation of proteins during membrane adhesion.

Curvature-Mediated Assembly of Janus Nanoparticles on Membrane Vesicles

Besides direct particle-particle interactions, nanoparticles adsorbed to biomembranes experience indirect interactions that are mediated by the membrane curvature arising from particle adsorption. In this Letter, we show that the curvature-mediated interactions of adsorbed Janus particles depend on the initial curvature of the membrane prior to adsorption, that is, on whether the membrane initially bulges toward or away from the particles in our simulations. The curvature-mediated interaction can be strongly attractive for Janus particles adsorbed to the outside of a membrane vesicle, which initially bulges away from the particles. For Janus particles adsorbed to the vesicle inside, in contrast, the curvature-mediated interactions are repulsive. We find that the area fraction of the adhesive Janus particle surface is an important control parameter for the curvature-mediated interaction and assembly of the particles, besides the initial membrane curvature.

Nano- and microparticles at fluid and biological interfaces

Systems with interfaces are abundant in both technological applications and biology. While a fluid interface separates two fluids, membranes separate the inside of vesicles from the outside, the interior of biological cells from the environment, and compartmentalize cells into organelles. The physical properties of interfaces are characterized by interface tension, those of membranes are characterized by bending and stretching elasticity. Amphiphilic molecules like surfactants that are added to a system with two immiscible fluids decrease the interface tension and induce a bending rigidity. Lipid bilayer membranes of vesicles can be stretched or compressed by osmotic pressure; in biological cells, also the presence of a cytoskeleton can induce membrane tension. If the thickness of the interface or the membrane is small compared with its lateral extension, both can be described using two-dimensional mathematical surfaces embedded in three-dimensional space. We review recent work on the interaction of particles with interfaces and membranes. This can be micrometer-sized particles at interfaces that stabilise emulsions or form colloidosomes, as well as typically nanometer-sized particles at membranes, such as viruses, parasites, and engineered drug delivery systems. In both cases, we first discuss the interaction of single particles with interfaces and membranes, e.g. particles in external fields, non-spherical particles, and particles at curved interfaces, followed by interface-mediated interaction between two particles, many-particle interactions, interface and membrane curvature-induced phenomena, and applications.

Continuum descriptions of membranes and their interaction with proteins: Towards chemically accurate models

Biological membranes deform in response to resident proteins leading to a coupling between membrane shape and protein localization. Additionally, the membrane influences the function of membrane proteins. Here we review contributions to this field from continuum elastic membrane models focusing on the class of models that couple the protein to the membrane. While it has been argued that continuum models cannot reproduce the distortions observed in fully-atomistic molecular dynamics simulations, we suggest that this failure can be overcome by using chemically accurate representations of the protein. We outline our recent advances along these lines with our hybrid continuum-atomistic model, and we show the model is in excellent agreement with fully-atomistic simulations of the nhTMEM16 lipid scramblase. We believe that the speed and accuracy of continuum-atomistic methodologies will make it possible to simulate large scale, slow biological processes, such as membrane morphological changes, that are currently beyond the scope of other computational approaches. This article is part of a Special Issue entitled: Membrane Proteins edited by J.C. Gumbart and Sergei Noskov.

Bilayer-thickness-mediated interactions between integral membrane proteins

Hydrophobic thickness mismatch between integral membrane proteins and the surrounding lipid bilayer can produce lipid bilayer thickness deformations. Experiment and theory have shown that protein-induced lipid bilayer thickness deformations can yield energetically favorable bilayer-mediated interactions between integral membrane proteins, and large-scale organization of integral membrane proteins into protein clusters in cell membranes. Within the continuum elasticity theory of membranes, the energy cost of protein-induced bilayer thickness deformations can be captured by considering compression and expansion of the bilayer hydrophobic core, membrane tension, and bilayer bending, resulting in biharmonic equilibrium equations describing the shape of lipid bilayers for a given set of bilayer-protein boundary conditions. Here we develop a combined analytic and numerical methodology for the solution of the equilibrium elastic equations associated with protein-induced lipid bilayer deformations. Our methodology allows accurate prediction of thickness-mediated protein interactions for arbitrary protein symmetries at arbitrary protein separations and relative orientations. We provide exact analytic solutions for cylindrical integral membrane proteins with constant and varying hydrophobic thickness, and develop perturbative analytic solutions for noncylindrical protein shapes. We complement these analytic solutions, and assess their accuracy, by developing both finite element and finite difference numerical solution schemes. We provide error estimates of our numerical solution schemes and systematically assess their convergence properties. Taken together, the work presented here puts into place an analytic and numerical framework which allows calculation of bilayer-mediated elastic interactions between integral membrane proteins for the complicated protein shapes suggested by structural biology and at the small protein separations most relevant for the crowded membrane environments provided by living cells.

Membrane-mediated interaction between strongly anisotropic protein scaffolds

Specialized proteins serve as scaffolds sculpting strongly curved membranes of intracellular organelles. Effective membrane shaping requires segregation of these proteins into domains and is, therefore, critically dependent on the protein-protein interaction. Interactions mediated by membrane elastic deformations have been extensively analyzed within approximations of large inter-protein distances, small extents of the protein-mediated membrane bending and small deviations of the protein shapes from isotropic spherical segments. At the same time, important classes of the realistic membrane-shaping proteins have strongly elongated shapes with large and highly anisotropic curvature. Here we investigated, computationally, the membrane mediated interaction between proteins or protein oligomers representing membrane scaffolds with strongly anisotropic curvature, and addressed, quantitatively, a specific case of the scaffold geometrical parameters characterizing BAR domains, which are crucial for membrane shaping in endocytosis. In addition to the previously analyzed contributions to the interaction, we considered a repulsive force stemming from the entropy of the scaffold orientation. We computed this interaction to be of the same order of magnitude as the well-known attractive force related to the entropy of membrane undulations. We demonstrated the scaffold shape anisotropy to cause a mutual aligning of the scaffolds and to generate a strong attractive interaction bringing the scaffolds close to each other to equilibrium distances much smaller than the scaffold size. We computed the energy of interaction between scaffolds of a realistic geometry to constitute tens of kBT, which guarantees a robust segregation of the scaffolds into domains.

The role of membrane-mediated interactions in the assembly and architecture of chemoreceptor lattices

In vivo fluorescence microscopy and electron cryo-tomography have revealed that chemoreceptors self-assemble into extended honeycomb lattices of chemoreceptor trimers with a well-defined relative orientation of trimers. The signaling response of the observed chemoreceptor lattices is remarkable for its extreme sensitivity, which relies crucially on cooperative interactions among chemoreceptor trimers. In common with other membrane proteins, chemoreceptor trimers are expected to deform the surrounding lipid bilayer, inducing membrane-mediated anisotropic interactions between neighboring trimers. Here we introduce a biophysical model of bilayer-chemoreceptor interactions, which allows us to quantify the role of membrane-mediated interactions in the assembly and architecture of chemoreceptor lattices. We find that, even in the absence of direct protein-protein interactions, membrane-mediated interactions can yield assembly of chemoreceptor lattices at very dilute trimer concentrations. The model correctly predicts the observed honeycomb architecture of chemoreceptor lattices as well as the observed relative orientation of chemoreceptor trimers, suggests a series of “gateway” states for chemoreceptor lattice assembly, and provides a simple mechanism for the localization of large chemoreceptor lattices to the cell poles. Our model of bilayer-chemoreceptor interactions also helps to explain the observed dependence of chemotactic signaling on lipid bilayer properties. Finally, we consider the possibility that membrane-mediated interactions might contribute to cooperativity among neighboring chemoreceptor trimers.

The chemotaxis system allows bacteria to respond to minute changes in chemical concentration, and serves as a paradigm for biological signal processing and the self-assembly of large protein lattices in living cells. The sensitivity of the chemotaxis system relies crucially on cooperative interactions among chemoreceptor trimers, which are organized into intricate honeycomb lattices. Chemoreceptors are membrane proteins and, hence, are expected to deform the surrounding lipid bilayer, leading to membrane-mediated interactions between chemoreceptor trimers. Using a biophysical model of bilayer-chemoreceptor interactions we show that the membrane-mediated interactions induced by chemoreceptor trimers provide a mechanism for the observed self-assembly of chemoreceptor lattices. We find that the directionality of membrane-mediated interactions between trimers complements protein-protein interactions in the stabilization of the observed honeycomb architecture of chemoreceptor lattices. Our results suggest that the symmetry of membrane protein complexes such as chemoreceptor trimers is reflected in the anisotropy of membrane-mediated interactions, yielding a general mechanism for the self-assembly of ordered protein lattices in cell membranes.

Connection between oligomeric state and gating characteristics of mechanosensitive ion channels

The mechanosensitive channel of large conductance (MscL) is capable of transducing mechanical stimuli such as membrane tension into an electrochemical response. MscL provides a widely-studied model system for mechanotransduction and, more generally, for how bilayer mechanical properties regulate protein conformational changes. Much effort has been expended on the detailed experimental characterization of the molecular structure and biological function of MscL. However, despite its central significance, even basic issues such as the physiologically relevant oligomeric states and molecular structures of MscL remain a matter of debate. In particular, tetrameric, pentameric, and hexameric oligomeric states of MscL have been proposed, together with a range of detailed molecular structures of MscL in the closed and open channel states. Previous theoretical work has shown that the basic phenomenology of MscL gating can be understood using an elastic model describing the energetic cost of the thickness deformations induced by MscL in the surrounding lipid bilayer. Here, we generalize this elastic model to account for the proposed oligomeric states and hydrophobic shapes of MscL. We find that the oligomeric state and hydrophobic shape of MscL are reflected in the energetic cost of lipid bilayer deformations. We make quantitative predictions pertaining to the gating characteristics associated with various structural models of MscL and, in particular, show that different oligomeric states and hydrophobic shapes of MscL yield distinct membrane contributions to the gating energy and gating tension. Thus, the functional properties of MscL provide a signature of the oligomeric state and hydrophobic shape of MscL. Our results suggest that, in addition to the hydrophobic mismatch between membrane proteins and the surrounding lipid bilayer, the symmetry and shape of the hydrophobic surfaces of membrane proteins play an important role in the regulation of protein function by bilayer membranes.

Directional interactions and cooperativity between mechanosensitive membrane proteins

While modern structural biology has provided us with a rich and diverse picture of membrane proteins, the biological function of membrane proteins is often influenced by the mechanical properties of the surrounding lipid bilayer. Here we explore the relation between the shape of membrane proteins and the cooperative function of membrane proteins induced by membrane-mediated elastic interactions. For the experimental model system of mechanosensitive ion channels we find that the sign and strength of elastic interactions depend on the protein shape, yielding distinct cooperative gating curves for distinct protein orientations. Our approach predicts how directional elastic interactions affect the molecular structure, organization, and biological function of proteins in crowded membranes.

Membrane-mediated protein-protein interactions and connection to elastic models: a coarse-grained simulation analysis of gramicidin A association

To further foster the connection between particle based and continuum mechanics models for membrane mediated biological processes, we carried out coarse-grained (CG) simulations of gramicidin A (gA) dimer association and analyzed the results based on the combination of potential of mean force (PMF) and stress field calculations. Similar to previous studies, we observe that the association of gA dimers depends critically on the degree of hydrophobic mismatch, with the estimated binding free energy of >10 kcal/mol in a distearoylphosphatidylcholine bilayer. Qualitative trends in the computed PMF can be understood based on the stress field distributions near a single gA dimer and between a pair of gA dimers. For example, the small PMF barrier, which is ∼1 kcal/mol independent of lipid type, can be captured nearly quantitatively by considering membrane deformation energy associated with the region confined by two gA dimers. However, the PMF well depth is reproduced poorly by a simple continuum model that only considers membrane deformation energy beyond the annular lipids. Analysis of lipid orientation, configuration entropy, and stress distribution suggests that the annular lipids make a significant contribution to the association of two gA dimers. These results highlight the importance of explicitly considering contributions from annular lipids when constructing approximate models to study processes that involve a significant reorganization of lipids near proteins, such as protein-protein association and protein insertion into biomembranes. Finally, large-scale CG simulations indicate that multiple gA dimers also form clusters, although the preferred topology depends on the protein concentration. Even at high protein concentrations, every gA dimer requires contact to lipid hydrocarbons to some degree, and at most three to four proteins are in contact with each gA dimer; this observation highlights another aspect of the importance of interactions between proteins and annular lipids.

Entropic attraction of adhesion bonds toward cell boundaries

Adhesion bonds between membranes and surfaces are attracted to each other via effective interactions whose origin is the entropy loss due to the reduction in the amplitude of the membrane thermal fluctuations in the vicinity of the adhesion bonds. These fluctuation-induced interactions are also expected to drive the adhesion bonds toward the rim of the cell as well as toward the surfaces of membrane inclusions. In this paper we analyze the attraction of adhesion bonds to the cell inner and outer boundaries. Our analysis shows that the probability distribution function of a single (diffusing) adhesion bond decays algebraically with the distance from the boundaries. Upon increasing the concentration of the adhesion bonds, the attraction to the boundaries becomes strongly self-screened.

Modeling membrane shaping by proteins: focus on EHD2 and N-BAR domains

Cellular membranes are highly dynamic, undergoing both persistent and dynamic shape changes driven by specialized proteins. The observed membrane shaping can be simple deformations of existing shapes or membrane remodeling involving fission or fusion. Here we describe several mechanistic principles by which membrane shaping proteins act. We especially consider models for membrane bending and fission by EHD2 proteins and membrane bending by N-BAR domains. There are major challenges ahead to understand the general principles by which diverse membrane bending proteins act and to understand how some proteins appear to span multiple modes of action from driving curvature to inducing membrane remodeling.

Degenerate ground-state lattices of membrane inclusions

Particles that are embedded in fluid membranes or plates can induce bending if they impose a nonzero angle of contact. This bending mediates complicated effective particle-particle interactions. In the absence of tension, these interactions are nonpairwise additive and can result in clusters of particles with specific configurations that give rise to zero total membrane bending energy. Here, we consider an infinite periodic lattice of such membrane inclusions. Upon summing the nonpairwise interactions within a regular lattice, we find an unexpected infinite number of periodic lattices that preserve zero membrane bending energy. Elliptically shaped membrane inclusions further increase the phase space of this degeneracy.

Organization of membrane-associated proteins in lipid bilayers

Lateral organization of proteins in biomembranes is vitally important to membrane functions such as signal transduction, endocytosis, and membrane trafficking. One of the major goals in current biomembrane science is to reveal the microscopic mechanism of membrane-associated protein organization in biomembranes. Here, we investigate the structural organization of membrane-associated proteins in lipid bilayers by combining self-consistent field theory with density functional theory. The present study can simultaneously take into account the entropy effect of lipids, depletion effect of membrane-associated proteins due to the presence of lipid headgroups as well as the effect of interfacial interaction. By varying the volume fraction of lipids, we examine various effects on protein organization, and reveal that a close-packed crystal structure appears at low lipid volume fractions due to interfacial energy and weak depletion effect, whereas a chain structure with branches occurs at high lipid volume fractions mainly due to strong depletion. The present results may provide some theoretical insight into further experiments on organization of membrane proteins.

Modeling 2D and 3D diffusion

Modeling obstructed diffusion is essential to the understanding of diffusion-mediated processes in the crowded cellular environment. Simple Monte Carlo techniques for modeling obstructed random walks are explained and related to Brownian dynamics and more complicated Monte Carlo methods. Random number generation is reviewed in the context of random walk simulations. Programming techniques and event-driven algorithms are discussed as ways to speed simulations.

Contributions of Gaussian curvature and nonconstant lipid volume to protein deformation of lipid bilayers

An elastic model for membrane deformations induced by integral membrane proteins is presented. An earlier theory is extended to account for nonvanishing saddle splay modulus within lipid monolayers and perturbations to lipid volume proximal to the protein. Analytical results are derived for the deformation profile surrounding a single cylindrical protein inclusion, which compare favorably to coarse-grained simulations over a range of protein sizes. Numerical results for multi-protein systems indicate that membrane-mediated interactions between inclusions are strongly affected by Gaussian curvature and display nonpairwise additivity. Implications for the aggregation of proteins are discussed.

Solvent-free simulations of fluid membrane bilayers

A molecular level model for lipid bilayers is presented. Lipids are represented by rigid, asymmetric, soft spherocylinders in implicit solvent. A simple three parameter potential between pairs of lipids gives rise to a rich assortment of phases including (but not limited to) micelles, fluid bilayers, and gel-like bilayers. Monte Carlo simulations have been carried out to verify self-assembly, characterize the phases corresponding to different potential parametrizations, and to quantify the physical properties associated with those parameter sets corresponding to fluid bilayer behavior. The studied fluid bilayers have compressibility moduli in agreement with experimental systems, but display bending moduli at least three times larger than typical biological membranes without cholesterol.

Modeling flexible amphiphilic bilayers: A solvent-free off-lattice Monte Carlo study

We present a simple, implicit-solvent model for fluid bilayer membranes. The model was designed to reproduce the elastic properties of real bilayer membranes. For this model, we observed the solid-fluid transition and studied the in-plane diffusivity of the fluid phase. As a test, we compute the elastic-bending and area-compressing moduli of fluid bilayer membranes. We find that the computed elastic properties are consistent with the available experimental data.

Dynamin recruitment by clathrin coats: a physical step?

Recent structural findings have shown that dynamin, a cytosol protein playing a key-role in clathrin-mediated endocytosis, inserts partly within the lipid bilayer and tends to self-assemble around lipid tubules. Taking into account these observations, we make the hypothesis that individual membrane-inserted dynamins imprint a local cylindrical curvature to the membrane. This imprint may give rise to long-range mechanical forces mediated by the elasticity of the membrane. Calculating the resulting many-body interaction between a collection of inserted dynamins and a membrane bud, we find a regime in which the dynamins are elastically recruited by the bud to form a collar around its neck, which is reminiscent of the actual process preempting vesicle scission. This physical mechanism might therefore be implied in the recruitment of dynamins by clathrin coats.

The many-body problem for anisotropic membrane inclusions and the self-assembly of "saddle" defects into an "egg carton"

We calculate the many-body, nonpairwise interaction between N rigid, anisotropic membrane inclusions by modeling them as point-like constraints on the membrane's curvature tensor and by minimizing the membrane's curvature energy. Because multipolar distortions of higher-order decay on very short distances, our calculation gives the correct elastic interaction energy for inclusions separated by distances of the order of several times their size. As an application, we show by thermally equilibrating the many-body elastic energy using a Monte Carlo algorithm, that inclusions shaped as "saddles" attract each other and build an "egg-carton" structure. The latter is reminiscent of some patterns observed in membranes obtained from biological extracts, the origin of which is still mysterious.

Statistical thermodynamics of membrane bending-mediated protein-protein attractions

Highly wedge-shaped integral membrane proteins, or membrane-adsorbed proteins can induce long-ranged deformations. The strain in the surrounding bilayer creates relatively long-ranged forces that contribute to interactions with nearby proteins. In contrast, to direct short-ranged interactions such as van der Waal's, hydrophobic, or electrostatic interactions, both local membrane Gaussian curvature and protein ellipticity can induce forces acting at distances of up to a few times their typical radii. These forces can be attractive or repulsive, depending on the proteins' shape, height, contact angle with the bilayer, and a pre-existing local membrane curvature. Although interaction energies are not pairwise additive, for sufficiently low protein density, thermodynamic properties depend only upon pair interactions. Here, we compute pair interaction potentials and entropic contributions to the two-dimensional osmotic pressure of a collection of noncircular proteins. For flat membranes, bending rigidities of approximately 100k(B)T, moderate ellipticities, and large contact angle proteins, we find thermally averaged attractive interactions of order k(B)T. These interactions may play an important role in the intermediate stages of protein aggregation. Numerous biological processes where membrane bending-mediated interactions may be relevant are cited, and possible experiments are discussed.