Implicit solvent simulation models for biomembranes
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DOI: 10.1007/s00249-005-0013-y
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- Brannigan, G., Lin, L.C. & Brown, F.L.H. Eur Biophys J (2006) 35: 104. doi:10.1007/s00249-005-0013-y
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Abstract
Fully atomic simulation strategies are infeasible for the study of many processes of interest to membrane biology, biophysics and biochemistry. We review various coarse-grained simulation methodologies with special emphasis on methods and models that do not require the explicit simulation of water. Examples from our own research demonstrate that such models have potential for simulating a variety of biologically relevant phenomena at the membrane surface.
Introduction
Biomembranes play a central role in most processes in cellular biology (Lodish et al. 1995; Gennis 1989), and lipid bilayer membranes are becoming increasingly important for technological applications and medicine (Tien and Ottova-Leitmannova 2003). From a purely physical standpoint, lipid bilayers are a fascinating system for study because of their complex properties over a range of different sizes and time regimes. The clear importance of membranes across a range of disciplines has led to extensive study of lipids and bilayers from both experimental and theoretical perspectives (Bloom et al. 1991; Evans and Skalak 1980; Lipowsky and Sackmann 1995; Nagle and Tristram-Nagle 2000).
Computationally, atomistic simulations of lipid bilayers have provided insight into the microscopic structure within the membrane sheet (of lipids, cholesterol and water) and allowed for realistic simulation of transmembrane proteins in their native environment (Chiu et al. 2002; Feller 2000; Marrink et al. 1993; Pastor 1994; Pitman et al. 2005; Smondyrev and Berkowitz 1999; Tobias et al. 1997). However, the time scale and size limitations of fully atomic molecular dynamics (MD) simulation currently preclude detailed studies involving mixing of multiple lipid species, interactions between multiple proteins in a membrane environment, verification of “macroscopic” elastic properties (without finite size artifacts) and essentially any phenomena occurring over lengths larger than tens of nanometers and/or times longer than tens of nanoseconds. Since much biological behavior relies on complex inhomogeneous membrane environments and interactions spanning a range of length and time scales, such limitations are severe.
It is worth examining just how severe these limitations really are. A recent paper by Feller and co-workers reported MD simulation of a single rhodopsin protein in a membrane environment consisting of water, two lipid species and cholesterol for a total time of 118 ns (Pitman et al. 2005). The simulation box employed had dimensions of 5.5×7.7×10 nm^{3} (43,222 atoms). We do not claim this to be the largest MD simulation on record, but the authors themselves note that the simulation was only possible because of “recent improvements in computer power and simulation techniques.” We take this study as a benchmark for what is feasible with current MD and computer technologies. Let us imagine that MD computation scales linearly with system size regardless of how massive the simulation. Then, a reasonable estimate of “computing power” necessary to run a given simulation is given by the size of the simulation volume (i.e., number of atoms) times the duration of the simulation. By this definition, the benchmark simulation above required about 50,000 ns nm^{3} of MD computing power.
Suppose we were interested in simulating a small fraction of the surface of a cell, a 1 μm×1 μm patch (for reference, a red blood cell is approximately 7 μm in diameter) for a time of 1 ms. To allow for out-of-plane membrane undulations (see below), we set the dimension of the box normal to the membrane surface at 100 nm. The computing power required for this simulation would be 10^{14}ns nm^{3}, two billion times that required for the “state-of-the-art” calculation mentioned above! Although improvement of MD simulation capability over time is impossible to predict, it seems highly unlikely that improvement will surpass Moore’s (1985) law for the predicted exponential growth rate of CPU speed. Assuming (a la Moore) that MD capabilities will double every 1.5 years, our hypothetical simulation would only become feasible 46 years from now.
The above estimate is not expected to be quantitatively accurate. However, we emphasize that the simplifications introduced (linear scaling, Moore’s law) are likely to fail in the direction of predicting a future that is overly optimistic. A more refined treatment would almost certainly predict that it will be much more than 46 years in the future when it becomes possible to run MD on micrometer scales for millisecond durations. From a biological point of view, these scales are actually quite conservative (see Protein mobility on the red blood cell) although it is clear that simulation of such processes with atomic scale resolution will continue to be computationally prohibitive for many years to come.
The practical limitations of fully atomic simulations have led to interest in developing model systems that preserve the physical properties of biomembranes, but with reduced computational cost. Various levels of coarse-graining have been introduced (Shelley and Shelley 2000) as will be summarized below. Explicit simulation of water in biomolecular simulations typically increases the simulation size by an order of magnitude or more relative to the size of the solute(s) (Feig and Brooks 2004). Clearly, eliminating the direct simulation of solvent is a huge computational advantage if it is possible to do so while preserving physical reality.
The evolution of membrane simulation methodologies that do not require explicit simulation of water is the primary focus of this review. We realize that the adjective “implicit” has come to mean a variety of different things to various researchers. Our own definition is borrowed straight from the dictionary, which defines implicit to mean “involved in the nature or essence of something though not revealed, expressed or developed” (Gove 1970). Any form of solvent, no matter how coarse grained, removes a model from consideration as an “implicit solvent model” in our opinion. We further narrow the focus of this review to cover two specific flavors of the model, each appropriate to a different size regime. First, we consider dynamic elastic models where the influence of water is incorporated through implicit treatment of the hydrodynamics for the solvent surrounding the bilayer. Second, we consider coarse-grained molecular level models where the stability of the bilayer (due to hydrophobic interactions in the real system) is enforced by effective lipid–lipid interactions. These interactions, though not typically derived from microscopic considerations, should be viewed as potentials of mean force, reflecting the implicit role of solvent in these simulations. Prior literature has referred to such models as “solvent-free” (Brannigan and Brown 2004) and “water-free” (Farago 2003) in addition to the moniker “implicit solvent.” All such titles mean the same thing: solvent is not simulated on the computer, even though all these systems are intended to model lipid bilayers in aqueous solution. At present, these models are used primarily in Monte Carlo simulations, so the role of implicit solvent here is solely to maintain membrane integrity and reproduce thermal behavior. Hydrodynamics are not currently incorporated into this second class of models.
We note that the above definition of implicit could technically include models beyond those reviewed here. Our focus is on models that allow for general (non-planar) membrane configurations and is driven by an interest in the behavior and properties of the three-dimensional membrane itself. Another class of implicit solvent models regard the bilayer solely as a planar environment whose dielectric properties influence protein stability (Im et al. 2003). Such models are interesting in the context of protein behavior, but fall outside the scope of this review. Similarly, models that treat the membrane as a flat two-dimensional fluid (for instance, Murtola et al. 2004) hold great promise for addressing certain questions, such as domain formation in perfectly flat geometries, but also fall outside the purview of this review.
Overview of coarse-grained simulation models
Continuum models
For physical questions involving sufficiently long length and time scales, continuum models provide the only feasible simulation schemes. By coarse-graining over the lipid degrees of freedom, fluid membranes have been successfully described by infinitely thin, continuous sheets with curvature elastic energy. The solvent never appears in such a picture, although its effects are implicitly present in the elastic properties that specify the model. Canham (1970) first used such a model to show that curvature energy minimization leads to the various observed shapes of red blood cells under different conditions. Subsequently, models developed by Helfrich included spontaneous curvature of the membrane due to differences in the composition of the two leaflets (Helfrich 1973) leading to explanations of other observed shapes in vesicles (Deuling and Helfrich 1976). Further extensions have provided detailed phase diagrams for red blood cells (Lim et al. 2002; Mukhopadhyay et al. 2002), vesicles (Lipowsky 1991; Seifert et al. 1991), as well as predictions for budding and vesiculation (Miao et al. 1991, 1994).
The Helfrich model has been most widely applied to the study of bilayers with small thermal height fluctuations away from a flat reference configuration (Safran 1994). Analytically, the model contains only two experimentally measured parameters, the bending rigidity and surface tension of the sheet, and allows for exact solutions that have been verified in a variety of studies. Comparisons between theory and experiment for the mean square height fluctuations in red blood cells have been successfully made (Zilker et al. 1992). Predictions of undulation-induced, steric repulsive interactions between proximal membranes (Helfrich 1978) have been verified in stacks of lipid bilayers (Safinya et al. 1989). Helfrich models with additional harmonic interactions can also be handled analytically. Equilibrium properties can be calculated for various forms of harmonic potentials (Gov and Safran 2004; Gov et al. 2003; Lin and Brown 2004b) including localized pinning and uniform confinement.
Non-harmonic potentials within the Helfrich picture have been studied through the use of Monte Carlo. Average height fluctuations and pressures have been calculated and compared to experiment by simulating stacks of lipid bilayers interacting through van der Waals attraction and repulsive hydration potentials (Gouliaev and Nagle 1998a, b). Various other non-harmonic interactions have been studied such as unbinding transitions for membranes interacting through a finite square well potential (Lipowsky and Zielenska 1989) and membranes interacting with a surface through approximate entropic, repulsive potentials (Bar-Ziv et al. 1995; Weikl and Lipowsky 2000).
Dynamics for Helfrich membranes depend upon the viscous fluid environment surrounding the bilayer. Overdamped equations of motion for the thin sheet can be derived from the Navier–Stokes equations in the same way that one derives dynamics for a collection of interacting Brownian particles (Doi and Edwards 1986). The surrounding fluid is assumed to be incompressible and inertial effects are generally negligible, leading to low Reynolds number (Stokesian) dynamics. Small fluctuations away from a flat bilayer are also assumed. Boundary conditions dictate the form of the solutions, but the fluid is generally assumed to be infinite (Doi and Edwards 1986; Granek 1997) or bounded by a wall (Brochard and Lennon 1975; Gov et al. 2004; Seifert 1994) depending upon the physical situation. Thermal fluctuations are included by adding a Langevin-like noise term to satisfy the fluctuation–dissipation relationship (Granek 1997).
Various dynamic properties of lipid bilayer systems have been successfully studied analytically using these methods. The flicker effect of red blood cells was explained on the basis of thermal undulations of the membrane surface (Brochard and Lennon 1975). Furthermore, the structure factor of stacks of membranes as a function of wave vector and frequency has been calculated and compared favorably with experiment (Zilman and Granek 1996). Harmonic interactions can generally be handled analytically for the dynamics of Helfrich sheets. The effects of the coupling between the red blood cell bilayer and spectrin cytoskeleton have been modeled through an effective surface tension and harmonic confinement, as well as a wall boundary condition for the hydrodynamics (Gov et al. 2003). This theory has provided improved fits to experimental measurements (Zilker et al. 1987) of the frequency dependence of the mean square height amplitude. Analytical results have also been found for the time correlation functions for a membrane in a localized harmonic potential, and experiments for optical tweezers have been suggested (Granek and Klafter 2001).
Few dynamical studies on systems containing non-harmonic interactions have been performed. Studies of the formation of the immunological synapse have included Helfrich elasticity (Qi et al. 2001), but without the need for hydrodynamics or thermal fluctuations. For polymers adsorbed on surfaces, dynamics simulations (Laradji 1999), numerical solutions (Laradji 2002) and Monte Carlo simulations (Laradji 2004) have predicted the scaling of excess rigidity with the grafted polymer density. Recently, our work has focused on a simulation algorithm we have termed Fourier space Brownian dynamics (FSBD; Lin and Brown 2004a). Our method is a prescription for the time evolution of the lipid bilayer in the presence of arbitrary forces and includes hydrodynamics as well as thermal fluctuations. The FSBD algorithm is sufficiently general to find application to many biophysical processes. We discuss the algorithm and various applications in Fourier space Brownian dynamics.
Particle-based models
For many questions regarding lipid bilayers, a continuum formulation is inappropriate. The models described above are suitable for studying long-wavelength undulation behavior in specific geometries with elastic properties provided as inputs to the model. For simulations at smaller scales, in more general geometries or aimed at predicting elastic properties, particle-based methods are more versatile. In practice, simulation particles may represent single atoms, groups of atoms, single lipids or multiple lipids. Stability of the bilayer structure within such models may be accomplished by constraining such a geometry (tethered models) or as a natural consequence of interparticle interactions. Models that include solvent particles are able to enforce bilayer stability in the same general method adopted by nature (hydrophobic attractions). In the absence of explicit solvent, the hydrophobic effect must be incorporated through effective potentials between membrane particles.
Tethered models
A wide class of solvent-free particle-based models maintain membrane integrity by “tethering”: particles are either bound to each other or to the same continuous sheet. In the former case, models resemble the elastic picture discussed previously in the sense that Helfrich energetics are imposed with elastic properties assigned as input to the model. The advantage is that one may simulate geometries with a tethered model that might be impossible to treat analytically. For a review of earlier applications, see Gompper and Kroll (1997); recently such “tethered membranes” have been used to study vesicle budding (Kohyama et al. 2003; Kumar et al. 2001; Kumar and Rao 1998), inclusions (Holzöhner and Schoen 1999), vesicle adhesion (Gruhn and Lipowsky 2005), coupling between curvature and composition (McWhirter et al. 2004) and vesicle behavior in shear flow (Noguchi and Gompper 2004). The simulations of Ayton and Voth (2002) constitute a special case; while dissipative particle dynamics (DPD) is employed, the model “bonds [DPD] particles in a two-dimensional (2D) network.” The compressibility modulus k_{A} for the DPD membrane particles is matched to simulations of an atomistic solvated system; however, the bending modulus k_{c} is not an input to the model. In other words, Helfrich elasticity is not imposed, but falls out of this particular tethered system.
Another class of tethered models constrain the heads of flexible lipids to one or two common sheets, rather than tethering point particles to each other. These systems have been primarily used to investigate chain conformations (Harries and Ben-Shaul 1997), depletion forces (Sintes and Baumgärtner 1998a, b) and related properties (Whitehead et al. 2001) within a flat bilayer geometry. Recently Laradji (2004) has allowed the sheet to fluctuate according to the Helfrich elastic model.
Non-tethered solvated models
The field of coarse-grained lipid models is large and growing. The majority of approaches adopted in the literature are not based on implicit solvent, but rather include water in various approximate, particle-based forms. A comprehensive review of all models falling into this category would merit a stand-alone review in its own right, but this subject matter is perpendicular to our focus here. We include the few brief words below only to point out that implicit solvent models are certainly not the only route to adopt in lipid simulations. Indeed, most workers would argue that explicit solvent models are the most realistic and general coarse-grained models developed to date.
Solvated coarse-grained models for lipids can be divided into two general categories. Some models are developed from bottom-up, starting with atomic resolution and simplifying the description while attempting to maintain certain microscopic/mesoscopic properties of the lipids, water and other molecules. Other models have been developed from top-down in order to reproduce continuum and mesoscopic properties of the membrane without microscopic details ever being explicitly considered. Models in the former category (Izvekov and Voth 2005; Marrink et al. 2004; Shelley et al. 2001) typically use a hierarchical algorithm for integrating atomic degrees of freedom. (To our knowledge, this top-down approach has never been applied to implicit solvent models.) Models in the latter category (Ayton et al. 2001; Goetz and Lipowsky 1998; Groot and Rabone 2001; Laradji and Kumar 2004; Rekvig et al. 2003; Shillcock and Lipowsky 2002; Smit et al. 1991; Soddemann et al. 2001; Stevens 2004; Yamamoto et al. 2002;) tend to rely on ad hoc potentials, sacrificing microscopic detail for the sake of simplicity and speed. Both schemes have their own advantages, and application of one model over another is mostly dependent upon the particular applications one is interested in. From the standpoint of biology and biochemistry, the bottom-up approach has a lot of potential. It may eventually become possible to run realistic simulations of bilayers incorporating multiple lipid species, cholesterol, proteins, etc., without the computational cost inherent to fully atomic molecular dynamics. From the standpoint of physics and biophysics, the top-down approaches are appealing in their ability to study long-wavelength behavior and fundamental questions about membrane biophysics. It is these top-down models that implicit solvent models have the most in common with at present. Argument can be made that if ad hoc potentials are already present in a solvated model, then it makes sense to try to incorporate all solvent effects within effective interactions between lipids. This is the primary motivation behind the implicit solvent models discussed in Applications of molecular models.
Detailing the numerous approaches to coarse-graining is outside the scope of this review, and here we simply outline the premise behind the solvated top-down models. Most have extended the hydrophobic/polar (HP) scheme used in polymer simulations (Dill et al. 1995; Kolinski and Skolnick 2004 and references therein). “Polar” beads make up the solvent, while one or two short “hydrophobic” chains attached to a polar head bead represents a lipid. Polar beads prefer other polar beads over hydrophobic beads, inducing membrane formation. The form of the potentials vary from model to model, as do the molecular complexity and the intramolecular constraints. In these coarse-grained models, explicit solvent provides a straightforward and physically motivated, but computationally expensive, mechanism for inducing membrane aggregation.
Non-tethered implicit solvent models
Effective attractions between lipids are necessary to induce membrane aggregation in implicit solvent models. A common obstacle (Drouffe et al. 1991; Noguchi and Takasu 2001b) encountered by developers has been the tendency (Cooke et al. 2005) of standard attractive forms (i.e., van der Waals) to crystallize the membrane, i.e., the phase diagram for an assembly of such lipids does include a bilayer structure, but these structures are not fluid at any temperature. At sufficiently high temperatures the bilayer sublimes without ever passing through a fluid phase. Consequently, successful interactions have leaned towards the exotic, and early efforts (Drouffe et al. 1991; Noguchi and Takasu 2001b) implied that many-body potentials were necessary to access a fluid regime.
A successor to the Drouffe, Maggs and Leibler model, the Noguchi and Takasu model (Noguchi and Takasu 2001b) uses many-body potentials among molecules with a slightly more complex (and conventional) architecture. Each lipid is represented by a rigid chain of three beads (Fig. 1b); the many-body potential is applied to the two hydrophobic beads. This model has been used to study self-assembly of vesicles (Noguchi and Takasu 2001b), vesicle fusion (Noguchi 2002; Noguchi and Takasu 2001a), adhesion of nanoparticles to vesicles (Noguchi and Takasu 2002a), pulled vesicles (Noguchi and Takasu 2002b) and crystalline vesicles (Noguchi 2003). Very recently, Wang and Frenkel (2005) have extended this model to flexible molecules.
Later successes (Brannigan and Brown 2004; Brannigan et al. 2005; Cooke et al. 2005; Farago 2003) with pair potentials demonstrated that a many-body potential was not essential for stable fluid behavior. From a practical standpoint, a strong argument can be made for the development of pair-potential models. With pair potentials, Monte Carlo algorithms are more efficient and standard packages (such as GROMACS) can be used for dynamics. Still, difficulties in accessing a fluid phase impeded development of simple implicit solvent models. While some early implicit solvent models (Edwards et al. 1998; Morikawa and Saito 1994) for surfactants used only pair potentials, these models did not demonstrate fluid behavior.
Like the Noguchi and Takasu model (Noguchi and Takasu 2001b) [and the Soddemann, Dunweg and Kremer solvated model (Soddemann et al. 2001)] the Farago model (Farago 2003) treats each lipid as a rigid chain of three beads, one polar and two hydrophobic (Fig. 1b). Unlike earlier models, the Farago model relies only upon pair interactions between lipids. This model has been used to investigate pores (Farago 2003) and a microscopic method for calculating the bending rigidity (Farago 2004). With some similarity to the Drouffe, Maggs and Leibler model (Drouffe et al. 1991), a rigid model developed by two of us (Brannigan and Brown 2004) treats each lipid as a soft amphiphilic rod (Fig. 1c) and will be detailed in Applications of molecular models.
This first generation of solvent-free pair interaction models did not reproduce quantitative elastic behavior. Both Farago’s (Farago 2003) model and our rigid (Brannigan and Brown 2004) model produce membranes with bending rigidities k_{c} more similar to bilayers containing cholesterol [k_{c}∼ 30–100k_{B}T (Seifert and Lipowsky 1995)] than single component model membranes [k_{c}∼ 5–25k_{B}T (Seifert and Lipowsky 1995)]. As shown in Fig. 10a, the bending rigidity of our rigid model can be tuned via the rod aspect ratio, but interpolation places the region of biological relevance at thicknesses for which no stable fluid regime was observed. Furthermore, while the bending rigidity was found to decrease with the addition of a second species (Fig. 10b), we found it impossible to generate a membrane that is thermodynamically stable in the fluid phase with elastic constants as low as those mentioned above (k_{c}∼ 5–25k_{B}T).
The flexible model developed by two of us (Brannigan et al. 2005) and the Cooke, Kremer and Deserno model (Cooke et al. 2005) comprise a very recent second generation of pair-additive implicit solvent models. These models allow for molecular flexibility and have demonstrated bending elasticity in the range of biological relevance. Both are made up of flexible chains of spheres. Like the Wang and Frenkel model, the Cooke, Kremer and Deserno model uses one polar and two hydrophobic beads (Fig. 1e). Our model includes a third species: the interface bead (Fig. 1d) with a total of five beads (one polar, one interface, three hydrophobic).
A unique property of our flexible model is that it treats hydrocarbon groups at the “water”–hydrophobic interface distinctly from hydrocarbon groups in the bilayer core. In contrast, other models employ identical hydrophobic attraction among all hydrophobic beads. Although empirical in form, our interfacial attraction effectively mimics the essence of the hydrophobic effect within a bilayer geometry. This claim is supported by the range of physical behaviors effectively reproduced with this model as detailed in Thickness and heterogeneity.
Fourier space Brownian dynamics
Model and simulation algorithm
Often, the random fluctuations ζ(r,t) are purely thermal in nature. In this case, we have Gaussian white noise with the moments of ζ(r,t) specified by the fluctuation–dissipation theorem (Doi and Edwards 1986). Whether the random forces are thermal or not, once their moments are specified, the above equations completely determine the time evolution of the system.
Although it is possible to handle Helfrich theory analytically in certain circumstances as mentioned in the introduction, quantitative studies of lipid bilayer dynamics in the presence of general potentials in complex geometries are sometimes necessary. To achieve this goal, we require simulations of the membrane that are computationally efficient. Working in Fourier space is the solution and it has various other advantages that we discuss below.
The membrane is evolved forward in time (Lin and Brown 2004a) based on the principles of standard Brownian dynamics (Ermak and McCammon 1978). Although the basic idea is the same, we choose to evolve the Fourier modes since the convolution in Eq. 4 is more computationally expensive than performing a Fourier transform and using Eq. 10. In addition, it becomes straightforward to coarse grain over microscopic details by employing a cutoff for short wavelengths or, equivalently, large wave vectors.
- 1.
Compute h_{q} from the current configuration h(r) to calculate F_{q}^{free}.
- 2.
Using the current configuration h(r), evaluate in position space the force due to the interactions F_{int}(r)=− δH_{int}/δh(r) and compute the Fourier transform to obtain F_{q}^{int}.
- 3.
Pick R_{q}(Δt) from the appropriate distributions determined by the moments of the random force ζ_{q}(t).
- 4.
Compute h_{q}(t+Δt) using Eq. 11 and compute the inverse Fourier transform to obtain h(r) for use in the next iteration.
The above algorithm outlines the FSBD method for simulating lipid bilayers with Helfrich elastic energy and arbitrary interactions. In the following sections, we explore several applications of FSBD that include protein mobility on the red blood cell, membranes adhered to supported bilayers and non-thermal forces due to the pumping action of proteins embedded in the bilayer.
Applications
Protein mobility on the red blood cell
Since the protein does manage to explore the entire surface of the red blood cell, various theories have been proposed to explain how band 3 escapes confinement by the spectrin filaments as illustrated in Fig. 2. Previous theoretical studies have used simple, non-specific potential barriers at the corral boundary (Saxton 1995) or have assumed that rearrangements of the spectrin network are necessary for a protein to escape confinement (Brown et al. 2000; Leitner et al. 2000; Saxton 1989, 1990a, b). Although these mechanisms likely do play a role, we focus on thermal undulations of the bilayer as a means for promoting the passage of the protein over the cytoskeletal barrier (Brown 2003; Lin and Brown 2004a, b, 2005).
In our model, the height of the bilayer is influenced by both random thermal fluctuations and forces imposed by the attached spectrin filaments. We assume that the cytoskeleton lies in the h(r)=0 plane and only allows the protein to escape if the height of the bilayer at the corral edge exceeds the distance h_{0}∼ 6 nm (Zhang et al. 2000) that the cytoplasmic domain of band 3 protrudes beneath the bilayer into the cell. In addition, the gap must persist for a sufficiently long period of time for the protein to diffuse over the barrier. The probability that h(r,t)>h_{0} for times 0 and t can be approximated using the time correlation 〈h(r,0)h(r,t)〉. These statistical quantities allow us to estimate the macroscopic diffusion due to transverse bilayer undulations (Brown 2003; Lin and Brown 2004a, b, 2005).
Results for the calculated macroscopic diffusion constant, D_{macro} (μm^{2} s^{−1}), for various models compared to the experimentally measured value (Tomishige et al. 1998)
| Square | Triangular |
---|---|---|
Pinned | 7.0×10^{−2} | 6.6×10^{−2} |
Pinned and repulsive | 3.4×10^{−2} | 2.1×10^{−2} |
Free | 9.2×10^{−4} | |
Experiment | 6.6×10^{−3} |
Adding the cytoskeletal repulsion decreases the macroscopic diffusion constant relative to the pinned-only model. Since the membrane is forced to be above the plane at the corral edges, the modes of wavelength 2L_{c} or larger are quenched in comparison to the system without repulsion. The suppression of these modes leads to a lower value of the diffusion constant by a factor of ∼2 or 3.
Although D_{macro} computed from our most detailed theory is ∼3 times larger than the experimental value, we cannot expect better agreement due to the simplified nature of our modeling. Our intention is not to quantitatively predict D_{macro}, but rather to ascertain whether thermal bilayer fluctuations are a feasible mechanism to allow the protein to escape from the corral. Our results conclusively show this assertion to be the case; thermal membrane undulations likely play a role in the mobility of band 3 on the surface of the red blood cell.
Supported lipid bilayers
Lipid bilayers supported by solid substrates have become important tools for studying various membrane-related biophysical processes (Sackmann 1996). Supported membranes can be made to retain their fluidic properties (Sackmann and Tanaka 2000), making them useful for studying processes that involve the diffusion of components in the bilayer such as the formation of the immunological synapse (Grakoui et al. 1999).
Here, we focus on a recent study of supported intermembrane junctions (Kaizuka and Groves 2004). In these experiments, large unilamellar vesicles are ruptured over the supported bilayer and spontaneously adhere to the substrate at a random series of localized sites. The ruptured membrane is strongly pinned to the planar membrane on the support, although the nature of these adhesions is not well understood. The height of the upper membrane can subsequently be measured over a period of time using fluorescence interference contrast microscopy (Lambacher and Fromhertz 1996; Parthasarathy and Groves 2004). The resulting data are used to obtain the average height and the size of the fluctuations.
Active membranes
Thus far, we have only considered stochastic forces resulting from thermal fluctuations. Clearly, many biological processes of interest involve forces on the cell membrane of non-thermal origin. One example is the influence of membrane-bound proteins that turn energy from light or ATP into directed work. An elegant series of experiments have examined just this sort of process, but in a well-controlled environment. Prost and coworkers have studied membrane fluctuations induced by bacteriorhodopsin embedded in artificially constructed vesicles (Manneville et al. 1999, 2001). Bacteriorhodopsin exerts a force normal to the bilayer by pumping protons when activated by light. The experiments indicate a clear increase in the size of the height fluctuations when the pumps are activated. In addition, the size of the fluctuations decreases with increasing viscosity of solvent. Equilibrium averages in thermal systems can only depend on quantities specified in the Hamiltonian and not on the dynamical constants like the viscosity. These results clearly indicate that non-thermal effects in the system are being observed.
A number of theories have been proposed to explain various aspects of these experiments (Manneville et al. 2001; Prost and Bruinsma 1996; Prost et al. 1998; Ramaswamy et al. 1999, 2000). Recently, the viscosity dependence has been captured in a model by Gov (2004) where a set of uniformly distributed pumps change their state with a characteristic time constant τ. We have extended this model in several ways and used FSBD to verify our analytical results (L.C.-L. Lin et al., submitted). This application illustrates the utility of the FSBD method for explicitly time-dependent forces acting on the membrane surface.
In our model, the equations of motion are the same as in Eq. 8 except with ζ_{q}(t) representing the fluctuating force exerted by the pumps. The moments and time-correlations of this force determine the size of the height fluctuations of the membrane. We have thus reduced bacteriorhodopsin to a point-like particle in our continuum simulation. While this level of description would be laughable for questions related to detailed protein dynamics, we find that we are able to reproduce various aspects of experiment with this picture. This has important implications for our understanding of the nature of coupling between protein and membrane and, more generally, the gross effects of non-thermal energetics in the cellular environment.
There are several advantages to this model. It is easy to extend, as we will show below, and is straightforward to simulate using FSBD. In addition, this model allows for lattice sites on the membrane to be left empty thereby allowing for a non-uniform density. Consequently, we denote an average over all possible configurations of the proteins with a bar. The scaling behavior for this model is \(\overline{{{\left\langle {h^{2} } \right\rangle }}} _{{\mathbf{r}}} \sim F^{2} n\tau L^{3} /k_{\rm c} \eta \) for γ=0 and \(\overline{{{\left\langle {h^{2} } \right\rangle }}} _{{\mathbf{r}}} \sim F^{2} n\tau L/\gamma \eta \) for γ ≠ 0. Since the vesicles are of the order of tens of microns in diameter, we have taken the limit of a large system size L. Note that the fluctuations decrease with increasing viscosity as seen in the experiments.
Summary of the results for the scaling of the mean square height displacement \(\overline{{{\left\langle {h^{2} } \right\rangle }}} _{{\mathbf{r}}}\) for static and diffusing proteins in the limit of large L
| 3 state (γ=0) | 3 state (γ≠0) | 2 state (γ=0) | 2 state (γ≠0) |
---|---|---|---|---|
Static | F^{2}nτ L^{3}/κη | F^{2}nτ L/γη | F^{2}nL^{6}/κ^{2} | F^{2}nL^{2}/γ^{2} |
Diffusing | F^{2}nτ L^{3}/κη | F^{2}nτ L/γη | F^{2}nL^{5}/Dκη | F^{2}nL^{2}/γ^{2} |
Interestingly, the dependence on both the viscosity and τ disappear. Since the small q modes are the most important, we define a timescale for membrane relaxation \(\tau _{\rm m} = 1/\omega _{{{\bf q}}0}\) with q_{0} representing the longest wavelength mode. In the large L limit, τ_{m} becomes very large in comparison to τ and the pumps are switching rapidly between the on and off states relative to the membrane relaxation time. As a result, each protein essentially exerts a constant average force on the membrane and the resulting configuration is simply a static balance between this constant force and the elastic bending energy. Therefore, neither τ_{m}, which contains η, nor τ affects the mean square height. This observation is contrasted with the model containing three-state proteins where the pump can switch between positive and negative values with zero average force. In that case, there is no static balance of forces and the membrane can relax. Both η and τ play a role since the rate at which the force switches signs compared to the rate of membrane relaxation determines the size of the height fluctuations.
Applications of molecular models
There are a number of questions for which continuum models are inappropriate. For instance, elastic constants such as the bending rigidity are simply inputs to a continuum model. Understanding what determines the bending rigidity requires an atomic or molecular model. Because atomic MD calculations of mesoscale membranes are computationally demanding, running the multiple simulations necessary to quantify trends is often computationally prohibitive. Coarse-grained models are useful for questions that require not only large systems, but also many systems or extensive sampling. We provide some examples of such questions, addressed using Monte Carlo simulations and our rigid and flexible models (see Fig. 1).
Model details
The single component version of our rigid model (Fig. 1c) is presented in detail in Brannigan and Brown (2004) and Brannigan et al. (2004), while the extension to multicomponent systems is presented in Brannigan and Brown (2005). Any two molecules interact through three potentials: a soft excluded volume repulsion between the distance of closest approach, a Lennard-Jones attraction between the points on the tail ends and an attractive alignment potential that depends on the relative orientations and the distance between the centers of mass. The model is constructed such that for flat homogeneous membranes at constant area, the average potential energy does not depend on the molecular length λ_{i}. For the heterogeneous systems, the same energetic parameters are used both for pairs of like and unlike molecules; all differences between species are reflected in rod length alone.
The five-bead version of our flexible model (Fig. 1d) is presented in Brannigan et al. 2005. All beads interact through the repulsive component of a standard Lennard-Jones potential. Hydrophobic–hydrophobic and hydrophobic–interface pairs interact through the attractive component of a Lennard-Jones potential as well. Interface–interface pairs interact through a truncated 1/r^{2} potential. All bond lengths are constrained and there are no dihedral potentials. There is a bond angle potential that is minimized when the chains are straight.
Phase behavior
Like thin films and smectic liquid crystals (Cheng et al. 1988; Chou et al. 1998; deGennes and Prost 1993; Pindak et al. 1981), melting in lipid bilayers (Luzzati 1968; Luzzati and Husson 1962) proceeds by a multistep process. Although the gel mesophase of lipid bilayers has been identified with the hexatic mesophase of two-dimensional melting (Smith et al. 1988, 1990), it is unclear to what degree the phase behavior in such systems is governed by the same physics. While phase diagrams (nematic, smectic, isotropic, etc.) have been measured for numerous liquid crystal models (Allen 1993) including models with dipole moments (Weis et al. 1992) and hence qualitatively similar to our own rigid model, fewer simulation studies have investigated melting within the smectic phase. Furthermore, the existing studies (Aoki and Yonezawa 1992; Zangi and Rice 2003) typically involve stacks of single layers, rather than one bilayer. Since experiments (Chao et al. 1996; Cheng et al. 1988; Geer et al. 1992) suggest that location within the lamellar stack affects phase behavior, one cannot simply map the behavior of smectics onto that of a single bilayer, even if the bilayer is composed of rigid molecules. As a liquid crystal like model for bilayers, our rigid model is well suited to studying the relationship between generic quasi two-dimensional systems and lipid bilayers. We summarize a study investigating the effect of molecular elongation on bilayer phase behavior, first presented in Brannigan et al. (2004).
Main melting transition
Hopping regime
While merely obeying Eq. 22 does not indicate that a system is fluid, quantitative comparisons of diffusion constants do seem to be reasonable indicators of a transition. Using our rigid model we observed a jump in diffusion at the main transition temperature (Fig. 7b). For long molecules, we also observed a low temperature jump, which corresponds to a dramatic increase in the percentage of hopping molecules. As described in Brannigan et al. (2004) the diffusion constant (and the bond order) were used to designate a plausible solid–hexatic transition.
Composition dependence of bending elasticity
Measurements of the fluctuation spectrum (〈|h_{q}|^{2}〉) as a means to extract elastic constants from bilayer simulation have now become commonplace. Numerous simulation models (Brannigan and Brown 2004; Brannigan et al. 2005; Chiu et al. 2003; Cooke et al. 2005; Drouffe et al. 1991; Farago 2003; Goetz et al. 1999; Lindahl and Edholm 2000a; Loison et al. 2003; Marrink and Mark 2001; Stevens 2004) have verified that the Helfrich Hamiltonian describes fluctuations at wavelengths longer than the bilayer thickness. Although all have reported bending rigidities, the models and simulation conditions vary so much from study to study that it is difficult to infer precisely why, for instance, sphingomyelin (Chiu et al. 2003) is so much stiffer than glycerolmonoolein (Marrink and Mark 2001) and dipalmitoylphosphatidylcholine (Lindahl and Edholm 2000a). Relatively few studies (Brannigan and Brown 2005; Brannigan et al. 2004, 2005; Illya et al. 2005; Imparato et al. 2005) have quantified how controlled changes in molecular architecture affect bilayer bending elasticity. Simplified molecular models are especially appropriate for understanding the influence of gross molecular properties, such as shape, chain length and flexibility.
Thickness and heterogeneity
On dimensional grounds alone one can relate the bending rigidity k_{c} to the compressibility modulus k_{A} and the membrane thickness d, according to k_{c}=k_{A}d^{2}/b, where b is a dimensionless constant. Various theories have been put forth regarding the calculation of b. Continuum elastic theory suggests that b ranges from 4 to 48, depending on the stress profile (Bloom et al. 1991; Brannigan and Brown 2005; Evans 1974) and whether the leaflets are bound or unbound. A calculation that treats each leaflet as a polymer brush results in b=24 (Rawicz et al. 2000).
Tail flexibility
Inclusions
While coarse-grained models cannot necessarily be used to make quantitative predictions regarding the deformation profile, they are well suited for testing the multitude of available analytical theories. For instance, one persistent question in the theory of inclusion-induced deformations is whether the elastic constants that determine the deformation profile are the same as those that determine thermal fluctuations. We tested our simulations against an analytical theory that consistently predicts both thickness fluctuations in an unperturbed bilayer and the thickness deformation profile in a perturbed bilayer (G. Brannigan and F.L. Brown, submitted). The analytical theory represents an extension of the works of Lipowsky and Grotehans (1994) and Aranda-Espinoza et al. (1996).
In principle, this test could be conducted using any model; in practice a sufficiently large box is needed with extensive sampling. Coarse-grained models are the only strategy presently feasible. The signature of the analytical model proposed by Aranda-Espinoza et al. (1996) is a non-monotonic deformation profile, but this is only expected for low protein concentrations. Venturoli et al. (2005), Nielsen et al. (2005) and ourselves (G. Brannigan and F.L. Brown, submitted) all observed non-monotonic behavior using coarse-grained models. The studies of Venturoli et al. and Nielsen et al., however, were more focused on protein tilting within the bilayer than testing analytical theories for the deformation profile, and the thickness fluctuations for an unperturbed bilayer were not reported. [Conversely, thickness fluctuations in atomistic bilayers (Chiu et al. 2003; Lindahl and Edholm 2000a; Marrink and Mark 2001) have been reported, but these models have not been used to study inclusions. However, as reported in Brannigan and Brown (submitted), the results of these atomistic simulations are consistent with our analytical theory for thickness fluctuations.]
Our simulation model for inclusions (Fig. 13) is similar to those used by Nielsen et al. (2005) and Venturoli et al. (2005). Inclusions are constructed from rigid chains with one head bead, one interface bead and three tail beads; if the surrounding lipids were completely straight, there would be no hydrophobic mismatch. The same interactions are used for, as an example, interface beads regardless of whether they are in lipid or inclusion molecules. Four concentric rings comprise the inclusion, and all intermolecular and intramolecular distances are held fixed.
Figure 14b depicts the thickness deformation profile δ(r)=(h(r) − h_{0})/h_{0} at a distance r from the center of a cylindrical inclusion, where h(r) is defined in Fig. 14a and h_{0} is the average thickness of an unperturbed membrane. The profile is clearly non-monotonic, as originally predicted by Huang (1986). The curve is a fit to the theory of Aranda-Espinoza et al. (1996), as described in Brannigan and Brown (submitted), involving the following fit constants: the compressibility modulus k_{A}, the spontaneous curvature of a monolayer c_{0} and the area derivative c′_{0} of the spontaneous curvature evaluated at zero tension. These values were found to compare favorably (i.e., within error bars) to measurements via the thickness fluctuations and the stress profile.
There are two remarkable points to be made about this work. First, it is surprising that the continuum elastic theory of Aranda-Espinoza et al. (1996) is so successful given that the length scales considered are only several lipid diameters. Secondly, it is very encouraging to see that the deformations predicted by the solvent-free model employed by us agree not only with the analytical theory, but also with more complex simulations requiring the explicit simulation of solvent (Venturoli et al. 2005). The fact that our flexible model is able to capture not only thermal fluctuations correctly, but also more severe deformations of the bilayer indicates that it has quite general utility in modeling mesoscopic membrane behavior.
Conclusion
Relative to analytical theory, fully atomic simulation and solvated coarse-grained simulation, solvent-free simulation models for biomembranes are in their infancy. Still, these methods have clear practical appeal and have already begun to show compelling evidence of their suitability for use in mesoscale simulations. The models considered in detail here demonstrate that (1) it is possible and useful to extend dynamic Helfrich models to anharmonic potentials and complex geometries via Brownian dynamics simulation implemented in Fourier space and (2) fluid lipid bilayers may be stabilized by effective pair potentials that obviate the need to explicitly simulate solvent in molecular level simulations.
Acknowledgments
This work was supported in part by the NSF (MCB-0203221, CHE-0349196, CHE-0321368) and the donors of the American Chemical Society Petroleum Research Fund (PRF 42447-G7). F. B. is an Alfred P. Sloan Research Fellow. F. B. thanks the NSF for travel funds to participate in the “Biophysical Chemistry Meets Molecular Medicine” workshop.