Erythrocyte Membrane Model with Explicit Description of the Lipid Bilayer and the Spectrin Network

Abstract

The membrane of the red blood cell (RBC) consists of spectrin tetramers connected at actin junctional complexes, forming a two-dimensional (2D) sixfold triangular network anchored to the lipid bilayer. Better understanding of the erythrocyte mechanics in hereditary blood disorders such as spherocytosis, elliptocytosis, and especially, sickle cell disease requires the development of a detailed membrane model. In this study, we introduce a mesoscale implicit-solvent coarse-grained molecular dynamics (CGMD) model of the erythrocyte membrane that explicitly describes the phospholipid bilayer and the cytoskeleton, by extending a previously developed two-component RBC membrane model. We show that the proposed model represents RBC membrane with the appropriate bending stiffness and shear modulus. The timescale and self-consistency of the model are established by comparing our results with experimentally measured viscosity and thermal fluctuations of the RBC membrane. Furthermore, we measure the pressure exerted by the cytoskeleton on the lipid bilayer. We find that defects at the anchoring points of the cytoskeleton to the lipid bilayer (as in spherocytes) cause a reduction in the pressure compared with an intact membrane, whereas defects in the dimer-dimer association of a spectrin filament (as in elliptocytes) cause an even larger decrease in the pressure. We conjecture that this finding may explain why the experimentally measured diffusion coefficients of band-3 proteins are higher in elliptocytes than in spherocytes, and higher than in normal RBCs. Finally, we study the effects that possible attractive forces between the spectrin filaments and the lipid bilayer have on the pressure applied on the lipid bilayer by the filaments. We discover that the attractive forces cause an increase in the pressure as they diminish the effect of membrane protein defects. As this finding contradicts with experimental results, we conclude that the attractive forces are moderate and do not impose a complete attachment of the filaments to the lipid bilayer.

Introduction

The human red blood cell (RBC) repeatedly undergoes large elastic deformations when passing through narrow blood vessels. The large flexibility of the RBCs is primarily attributable to the cell membrane, as there are no organelles and filaments inside the cell. The RBC membrane is essentially a two-dimensional (2D) structure, comprised of a cytoskeleton and a lipid bilayer, tethered together. The lipid bilayer includes various types of phospholipids, sphingolipids, cholesterol, and integral membrane proteins, such as band-3 and glycophorin (see Fig. 1 A). The RBC membrane resists bending but cannot sustain in-plane static shear stress as the lipids and the proteins diffuse within the lipid bilayer at equilibrium. The RBC membrane cytoskeleton is a 2D sixfold structure consisting of spectrin tetramers, which are connected at the actin junctional complexes. The cytoskeleton is tethered to the lipid bilayer via “immobile” band-3 proteins at the spectrin-ankyrin binding sites and via glycophorin at the actin junctional complexes (see Fig. 1 A). Although the mechanical properties and biological functions of the RBC membrane have been well studied in the past decades, the interactions between the lipid bilayer and cytoskeleton as well as the interactions between the cytoskeleton and transmembrane proteins are not yet fully understood. The cytoskeleton plays a major role in the integrity of the RBC membrane, as is evident in blood disorders where defects in membrane proteins lead to membrane loss and reduced mechanical robustness of the RBC (1–3). In hereditary spherocytosis (HS), the tethering of the cytoskeleton to the lipid bilayer (vertical interaction in Fig. 1 A) is partially disrupted resulting in membrane loss and subsequently in the spherical shape of the RBCs. In hereditary elliptocytosis (HE), the cytoskeleton is disrupted at α-β spectrin linkages or at spectrin-actin-4.1R junctional complexes (horizontal interactions in Fig. 1 A) (1–4). This partial disruption of the cortex diminishes the ability of the RBC to recover its biconcave shape after undergoing large deformations. In addition, because the spectrin filaments act as barriers, restricting the lateral diffusion of the “mobile” band-3 proteins, defects in the vertical and horizontal interactions between the cytoskeleton and the lipid bilayer modify the regular diffusion of band-3 proteins (5–9).

Figure 1.

(A) Schematic of the human RBC membrane. The blue sphere represents a lipid particle and the red sphere signifies an actin junctional complex. The gray sphere represents a spectrin particle and the black sphere represents a glycophorin particle. The yellow and green circles correspond to a band-3 complex connected to the spectrin network and a mobile band-3 complex, respectively. A mesoscale detailed membrane model. (B) Top view of the initial configuration. (C) Side view of the initial configuration. (D) Side view of the equilibrium configuration. “A” type particles represent actin junctional complexes; “B” type particles represent spectrin particles; “C” type particles represent glycophorin particles; “D” type particles represent a band-3 complex that are connected to the spectrin network (“immobile” band-3); “E” type particles represent band-3 complex that are not connected to the network (“mobile” band-3); and “F” type particles represent lipid particles. To see this figure in color, go online.

Several approaches have been followed for the mathematical description and modeling of the RBC membrane. At one end of the spectrum, there are the continuum membrane models based on elasticity theory (10–18). At the other end, atomistic simulations mainly study the behavior of lipids in the lipid bilayer (19–22). However, it is challenging for continuum models to account for the detailed structure and defects in the RBC cytoskeleton, as it is not feasible for atomistic methods to simulate a representative sample of the RBC membrane including lipids, membrane proteins, and the cytoskeleton. Because of these limitations, particle-based mesoscale models were introduced to study the biomechanical behavior of the RBC membrane. These models fall into two main groups. In one group, the membrane is modeled as a 2D canonical hexagonal network of particles where the immediate neighbors are connected via a worm-like chain (WLC) potential that represents a spectrin filament. The bending rigidity induced by the lipid bilayer is represented by a bending potential applied between two triangles with a common side (23–27). In the other model group, the lipid bilayer is simulated by coarse-grained methods where each lipid molecule is coarse-grained into several connected beads (28–33). The lipids are forced to assemble the bilayer by additional solvent particles, in the case of explicit solvent models, or by an additional potential, in the case of solvent-free models. In the explicit solvent models, hydrophobic interactions are employed between the lipids and solvent particles to represent effects of the water molecules (28,34–39). In the implicit solvent models, the effect of the solvent is taken into account by employing orientation-dependent interaction potentials between the lipid particles (30,31,40–43). At a higher level of coarse-graining, a group of lipid molecules are coarse-grained into one bead (44,45). Drouffe et al. (40) simulated biological membranes by introducing a one-particle-thick, solvent-free, coarse-grained model, in which the interparticle interaction is described by a Lennard-Jones (LJ) type pair potential depending not only on the distance between the particles but also on their directionality. Noguchi and Gompper (46) developed a one-particle thick, solvent-free, lipid bilayer model by introducing a multibody potential that eliminated the need of the rotational degree of freedom. Yuan et al. (44) introduced a similar approach, but instead of the LJ potential, a soft-core potential was used to better represent the particle self-diffusion. An overview of particle-based models for the RBC membrane can be found in recent reviews (43,47–49).

Absence of explicit representation of either the lipid bilayer or the cytoskeleton in the aforementioned particle-based models limits their applications in the study of the interactions between the cytoskeleton and the lipid bilayer, and between the cytoskeleton and diffusing membrane proteins. Recently, a model consisting of two layers of 2D triangulated networks, where one layer represents the cytoskeleton and the other layer represents the lipid bilayer, was introduced (50). This approach is computationally very efficient and it can be used in simulations of blood flow. However, it is not capable of modeling the interactions between the spectrin filaments and the lipid bilayer, which is frequently essential, as exemplified in the study of the diffusion of membrane proteins such as the band-3 (5–9). Li and Lykotrafitis (45) introduced a two-component RBC membrane model where the lipid bilayer consists of CG particles of 5 nm size whereas the cytoskeleton is comprised of particles that represent actin junctional complexes and form a canonical hexagonal network. These “actin junction” particles are connected by the WLC potential representing a spectrin filament. Because of the implicit representation of the spectrin filaments, this approach does not consider the interactions between the filaments and the lipid bilayer as well as between the filaments and membrane protein during diffusion. Auth et al. (51) developed an analytical model to study the cytoskeleton and lipid bilayer interactions. This analytical model is able to estimate the pressure exerted on the lipid bilayer by the cytoskeleton because of the thermal fluctuations of the spectrin filaments. The computed pressure is then used to obtain the deformation of the lipid bilayer induced by the spectrin filaments, which was calculated to be 0.1 ∼ 1 nm. This deformation is small compared with the local deformation (∼ 15 nm) of the lipid bilayer caused by the lateral compression applied by the spectrin cytoskeleton (15,52).

In this paper, we introduce a two-component implicit-solvent CGMD RBC membrane model comprised of the lipid bilayer and the RBC cytoskeleton by extending a previously developed RBC membrane model (45). The key feature of this new model is the explicit representation of the cytoskeleton and the lipid bilayer by CG particles. We determine the parameters of the model by matching the bending stiffness and shear modulus of the membrane model with existing experimental results (53–55). Then, we extract the timescale of our simulations by measuring the viscosity of the membrane model and comparing it with the experimentally obtained values. The timescale is later confirmed by measuring the thermal fluctuation frequency of the lipid bilayer and the spectrin filaments. In addition, we measure the pressure exerted by the spectrin filaments on the lipid bilayer. These results are compared with analytically estimated pressure values (51,56). Finally, we investigate the effects of the attraction between spectrin filaments and the lipid bilayer on the pressure applied by the cytoskeleton on the lipid bilayer in the case of the normal RBC membrane and for membranes with defective proteins.

Methods

The proposed model describes the RBC membrane as a two-component system, comprised of the cytoskeleton and the lipid bilayer. We first introduce the cytoskeleton, which consists of spectrin filaments connected at the actin junctional complexes forming a hexagonal network. The actin junctional complexes are represented by red particles (see Fig. 1 B–D) that have a ∼ 15 nm diam. and are connected to the lipid bilayer via glycophorin. Spectrin is a protein tetramer formed by head-to-head association of two identical heterodimers. Each heterodimer consists of an α-chain with 22 triple-helical segments and a β-chain with 17 triple-helical segments (57). In the proposed model, the spectrin is represented by 39 spectrin particles (gray particles in Fig. 1 B–D) connected by unbreakable springs. The spring potential, $u_{\text{cy}}^{\text{s}\text{-}\text{s}}\left(r\right)=k_0{\left(r\text{-}{r}_{\text{eq}}^{\text{s}\text{-}\text{s}}\right)}^2/2$, is plotted as the green curve in Fig. 2, with equilibrium distance between the spectrin particles $r_{\text{eq}}^{\text{s}\text{-}\text{s}}=L_{\max }/39$, where L_max is the contour length of the spectrin (∼ 200 nm) and thus $r_{\text{eq}}^{\text{s}\text{-}\text{s}}\cong 5$ nm. The spectrin chain is linked to the band-3 particles (yellow particles) at the area where the α-chain and the β-chain are connected. The two ends of the spectrin chains are connected to the actin junctional complexes via the spring potential $u_{\text{cy}}^{\text{a}\text{-}\text{s}}\left(r\right)=k_0{\left(r\text{-}{r}_{\text{eq}}^{\text{a}\text{-}\text{s}}\right)}^2/2$, where the equilibrium distance between an actin and a spectrin particle is $r_{\text{eq}}^{\text{a}\text{-}\text{s}}=10$ nm. The spring constant k₀ will be determined subsequently. Spectrin particles that are not connected by the spring potential interact with each other via the repulsive part of the L-J potential as follows:

$$u_{rep}\left(r_{\text{ij}}\right)=\{\begin{array}{cc}4\epsilon \left[{\left(\frac{\sigma }{r_{\text{ij}}}\right)}^{12}-{\left(\frac{\sigma }{r_{\text{ij}}}\right)}^6\right]+\epsilon & r_{\text{ij}}<R_{\text{cut,LJ}}=r_{\text{eq}}^{\text{s}\text{-}\text{s}}\\ 0& r_{\text{ij}}>R_{\text{cut,LJ}}=r_{\text{eq}}^{\text{s}\text{-}\text{s}}\end{array},$$ (1)

where ε is the energy unit and σ is the length unit. r_ij is the distance between spectrin particles. The cutoff distance of the potential R_cut,_LJ is chosen to be the equilibrium distance $r_{\text{eq}}^{\text{s}\text{-}\text{s}}$ between two spectrin particles. The potential is plotted as the black curve in Fig. 2. The spring constant k₀ = 57 ε/σ² is chosen to be identical to the curvature of $u_{\text{LJ}}\left(r_{ij}\right)=4\epsilon \left[{\left(\sigma /r_{ij}\right)}^{12}-{\left(\sigma /r_{ij}\right)}^6\right]+\epsilon$ at the energy well bottom to reduce the number of free parameters. The cytoskeleton introduced previously is directly connected to the lipid bilayer via the band-3 particles D and the glycophorin particles C (see Fig. 1 C). In addition, we have investigated the effect of attractive interactions of various strengths between spectrin filaments and the lipid bilayer on the diffusion of band-3 particles. Previous studies on lipid–spectrin filaments interactions have suggested that spectrin binds to the negatively charged lipid surfaces with association constants of 2-10 × 10⁶ M⁻¹ (58–63), whereas the association constants of spectrin-ankyrin, ankyrin-band-3, spectrin-protein 4.1-actin are 2 × 10⁷ M⁻¹, 2 × 10⁸ M⁻¹, and 2 × 10¹² M⁻², respectively (64,65). For simplicity, we applied the attractive part LJ potential between spectrin and lipid particles as follows:

$$u_{att}\left(r_{\text{ij}}\right)=\{\begin{array}{cc}4n\epsilon \left[{\left(\frac{\sigma }{r_{\text{ij}}}\right)}^{12}-{\left(\frac{\sigma }{r_{\text{ij}}}\right)}^6\right]+n\epsilon & r_{\text{ij}}>r_{\text{eq}}^{\text{l}\text{-}\text{s}}\\ 0& r_{\text{ij}}<r_{\text{eq}}^{\text{l}\text{-}\text{s}}\end{array},$$ (2)

where n is a parameter used to tune the attractive energy between the spectrin filaments and lipid bilayer. $r_{\text{eq}}^{\text{l}\text{-}\text{s}}=5\text{nm}$ is the equilibrium distance between spectrin particles and lipid particles. Our simulation results show that the cytoskeleton is completely attached to the lipid bilayer when n ≥ 0.2. Because the cytoskeleton of the RBC membrane is a highly flexible structure with a bending rigidity of κ ∼ 0.024 – 0.24k_BT (66,67), which is about two orders of magnitude smaller than the bending rigidity of the lipid bilayer κ ∼ 10 - 20 k_BT (53,54) as well as the bending rigidity of the RBC membrane κ ∼ 10 - 50 k_BT (53), we did not consider bending rigidity for the spectrin network in this model.

Figure 2.

The interaction potentials employed in the membrane model. The blue curve represents the pairwise potential between lipid particles. The green curve represents the spring potential between spectrin particles. The red curve represents the spring potential between actin and spectrin particles. The black curve represents the repulsive L-J potential between the lipid and spectrin particles. To see this figure in color, go online.

Three types of CG particles are introduced to represent the lipid bilayer and band-3 proteins (see Fig. 1 B–D). The blue color particles denote a cluster of lipid molecules. Their 5 nm diam. is approximately equal to the thickness of the lipid bilayer. The black particles represent glycophorin proteins with the same diameter as the lipid particles. The band-3 protein consists of two domains: (1), the cytoplasmic domain of band-3 with a dimension of 7.5 × 5.5 × 4.5 nm that contains the binding sites for the cytoskeletal proteins, and (2), the membrane domain, with a dimension of 6 × 11 × 8 nm, whose main function is to mediate anion transport (68,69). We represent the membrane domain of band-3 by a spherical CG particle with a radius of 5 nm. The volume of the particle is similar to the excluded volume of the membrane domain of a band-3. However, when band-3 proteins interact with the cytoskeleton, the effect of the cytoplasmic domain has to be taken into account and thus the effective radius is ∼ 12.5 nm. One third of band-3 particles, which are connected to the spectrin network, are depicted as yellow particles (see Fig. 1 B). The rest of the band-3 particles, which are free to diffuse in the lipid bilayer, simulate the mobile band-3 proteins and they are shown as green particles (see Fig. 1 B). The CG particles, which form the lipid bilayer and transmembrane proteins, carry both translational and rotational degrees of freedom (x_i, n_i), where x_i and n_i are the position and the orientation (direction vector) of particle i, respectively. The rotational degrees of freedom obey the normality condition |n_i| = 1. Thus, each particle effectively carries 5 degrees of freedom. x_ij = x_j - x_i is defined as the distance vector between particles i and j. r_ij ≡ |x_ij| and ${\hat{\mathbf{x}}}_{\text{ij}}={\mathbf{x}}_{\text{ij}}/r_{\text{ij}}$ are the distance and the unit vector respectively. The particles, forming the lipid membrane and membrane proteins, interact with one another via the pairwise potential as follows:

$$u_{\text{mem}}\left({\text{n}}_{\text{i}},{\text{n}}_{\text{j}},{\text{x}}_{\text{ij}}\right)=u_{\text{R}}\left(r_{\text{ij}}\right)+A\left(\alpha ,a\left({\text{n}}_{\text{i}},{\text{n}}_{\text{j}},{\text{x}}_{\text{ij}}\right)\right)u_{\text{A}}\left(r_{\text{ij}}\right),$$ (3)

with

$$u_{4-8}\left(r_{\text{ij}}\right)=\{\begin{array}{cc}{u}_R\left(r_{ij}\right)=k\epsilon {\left(\left(R_{cut,mem}\text{-}{r}_{ij}\right)\text{/}\left(R_{cut,mem}\text{-}{r}_{eq}\right)\right)}^8\text{-}k\epsilon & \mathrm{for}{r}_{\text{ij}}\text{<}{R}_{\text{cut,}\text{mem}}\\ \begin{array}{l}{u}_{\text{A}}\left(r_{\text{ij}}\right)=-2k\epsilon {\left(\left(R_{\text{cut,}\text{mem}}-r_{\text{ij}}\right)\text{/}\left(R_{\text{cut,}\text{mem}}-r_{\text{eq}}\right)\right)}^4-k\epsilon \\ u_{\text{R}}\left(r_{\text{ij}}\right)=u_{\text{A}}\left(r_{\text{ij}}\right)=0,\end{array}& \begin{array}{l}\mathrm{for}{r}_{\text{ij}}<R_{\text{cut,}\text{mem}}\\ \mathrm{for}{\text{r}}_{\text{ij}}\ge {\text{R}}_{\text{cut,}\text{mem}}\end{array}\end{array},$$ (4)

where u_R(r_ij) and u_A(r_ij) are the repulsive and attractive components of the pair potential, respectively. α is a tunable linear amplification factor. The function A(α,a(n_i,n_j,x_ij))= 1+α(a(n_i,n_j,x_ij) -1) tunes the energy well of the potential, through which the fluid-like behavior of the membrane is regulated. In the simulations, α is chosen to be 1.55 and the cutoff distance of the potential R_cut,mem is chosen to be 2.6σ. The parameters α and R_cut,mem are selected to maintain the fluid phase of the lipid bilayer. Detailed information about the selection of the potential parameters can be found from our previous work (45). k is selected to be 1.2 for the interactions among the lipid particles and k = 2.8 for interactions between the lipid and the protein particles, such as glycophorin and band-3. Fig. 2 shows only the potential between lipid particles (blue curve). The interactions between the cytoskeleton and the lipid bilayer are represented by the repulsive part of the L-J potential as shown in Eq. 1. The cutoff distance of the potential R_cut,_LJ is adjusted to be the equilibrium distances between different pairs of CG particles. The equilibrium distance $r_{\text{eq}}^{\text{a}\text{-}\text{l}}$ between the actin particles and the lipid particles is 10 nm whereas $r_{\text{eq}}^{\text{a}\text{-}\text{b}}$ between the actin particles and the band-3 particles is 20 nm. The equilibrium distance $r_{\text{eq}}^{\text{l}\text{-}\text{s}}$ between the spectrin particles and the lipid particles is 5 nm, whereas the equilibrium distance $r_{\text{eq}}^{\text{b}\text{-}\text{s}}$ between the spectrin particles and the band-3 particles is 15 nm.

The equation of translational motion for all the CG particles is as follows:

$$m_{\text{i}}{\ddot{\mathbf{x}}}_{\text{i}}=-\frac{\partial \left(V\right)}{\partial {\mathbf{x}}_{\text{i}}},$$ (5)

where for the CG particles forming the lipid membrane and the transmembrane proteins $V={\sum }_{j=1}^N\left(u_{\text{mem},\text{ij}}+u_{\text{LJ},\text{ij}}\right)$, and for the CG particles comprising the cytoskeleton, $V={\sum }_{j=1}^N\left(u_{\text{LJ},\text{ij}}\right)+u_{\text{cy,}\text{i}}$. The equation of rotational motion for the CG particles forming the lipid bilayer and proteins in the lipid bilayer is as follows:

$${\tilde{m}}_{\text{i}}{\ddot{\mathbf{n}}}_{\text{i}}=-\frac{\partial \left(\sum _{j=1}^N{u}_{\text{mem,ij}}\right)}{\partial {\mathbf{n}}_{\text{i}}}+\left(\frac{\partial \left(\sum _{j=1}^N{u}_{\text{mem,ij}}\right)}{\partial {\mathbf{n}}_{\text{i}}}\cdot {\mathbf{n}}_{\text{i}}\right){\mathbf{n}}_{\text{i}}-{\tilde{m}}_{\text{i}}\left({\dot{\mathbf{n}}}_{\text{i}}\cdot {\dot{\mathbf{n}}}_{\text{i}}\right){\mathbf{n}}_{\text{i}},$$ (6)

where ${\tilde{m}}_i$ is a pseudo-mass with units of energy × time², and the right-hand side of Eq. 6. conforms to the normality constraint |n_i| = 1. The pseudomass ${\tilde{m}}_i$ is chosen to be ${\tilde{m}}_i\sim m_i^l{\sigma }^2$, where $m_i^l$ is the mass of the lipid particles.

The system used in this paper consists of N = 29,567 CG particles. The dimension of the membrane is ∼ 0.8 × 0.8 μm. The numerical integrations of the equations of motion (Eqs. 5 and 6) are performed using the Beeman algorithm (70,71). The temperature of the system is maintained at k_BT/ε = 0.22 by employing the Nose-Hoover thermostat. The model is implemented in the NAT ensemble (32,72,73). Periodic boundary conditions are applied in all three directions. Because the model is solvent-free and the membrane is a 2D structure, we controlled the projected area instead of the volume. The projected area is adjusted to result in zero tension for the entire system at the equilibrium state. Given the diameter of the lipid particles $r_{\text{eq}}^{\text{l}\text{-}\text{l}}=2^{1/6}\cdot \sigma =5\text{nm}$, the length unit σ is calculated to be σ = 4.45 nm. The timescale that guides the choice of the timestep in the molecular dynamics (MD) simulations is $t_s={\left(m_i{d}^2/\epsilon \right)}^{1/2}$. The timestep of the simulation is selected to be Δt = 0.01t_s. Since the CG particles used in the simulations do not correspond to real molecules, the employed timescale does not have an immediate correlation with the real system. The timescale in our simulation is established by measuring the viscosity of the membrane model and comparing it with the experimentally measured value for the RBC membrane. For an independent confirmation, we also obtain the timescale by measuring the thermal fluctuation frequencies of the spectrin filaments and the lipid membrane. The details of the timescale will be discussed in a subsequent section. For simplicity, we assume that the cytoskeleton possesses a perfect 2D sixfold triangular structure with a fixed connectivity and that each spectrin filament is anchored to the band-3 proteins at its midpoint. In reality, the RBC cytoskeleton contains numerous defects while the band-3-ankyrin connections and the cytoskeleton undergo dynamic remodeling (74–76).

Results and Discussion

Membrane properties

Membrane fluidity

In this section, we show that the proposed model of the RBC membrane reproduces the appropriate mechanical properties. To ensure that our system represents the lipid bilayer, it is necessary to examine whether the CG particles can diffuse in a fluidic manner. Fig. 3 A and B show the positions of the CG particles of a thermally equilibrated fluid membrane at two separate times. The tiles with different colors are used to differentiate the positions of the particles at the initial moment. After 1 × 10⁶ time steps, the colored particles are mixed due to diffusion, demonstrating the fluidic nature of the membrane. We further verify the fluidic characteristics of the membrane by first showing that the mean squared displacement (MSD), defined by $1/N\langle {\sum }_{j=1}^{j=N}{\left[{\mathbf{x}}_j\left(t\right)-{\mathbf{x}}_j\left(0\right)\right]}^2\rangle$, increases linearly with time (see Fig. 3 C), and second by showing that that the correlations between the CG lipid particles are lost beyond a few particle diameters, suggesting a typical fluidic behavior of the membrane model at $k_{\text{B}}T/\epsilon =0.22$ and R_cut,mem = 2.6σ (see Fig. 3 D).

Figure 3.

(A) Thermally equilibrated membrane of N = 29,567 particles at a reference time, representing a membrane with dimension of ∼ 0.8 × 0.8 μm. The tiles with different colors are used to differentiate the positions of the particles at a reference time. (B) After 1 × 10⁶ time steps, the particles are mixed because of diffusion, demonstrating the fluidic behavior of the membrane model. (C) Linear time dependence of the mean square displacement (MSD) of the proposed membrane model. (D) Radial distribution function of the 2D fluid membrane embedded in three dimensions. To see this figure in color, go online.

Measurements of membrane bending rigidity, shear modulus, and viscosity

In this section, we compute the bending rigidity, shear modulus, and viscosity of the membrane model. We obtain the bending rigidity of the membrane by measuring the fluctuation spectrum of the membrane at zero tension and then fitting the data to the expression of the Helfrich free energy in the Monge representation (14,16,18,53) as follows:

$$\langle {\left|\tilde{h}\left(q\right)\right|}^2\rangle =\frac{k_{B}T}{l^2\left(\gamma q^2+\kappa q^4\right)},$$ (7)

where $\tilde{h}\left(q\right)$ is the discrete Fourier transform of the out-of-plane displacement $h\left(\mathbf{r}\right)$ of the membrane, defined as follows:

$$\tilde{h}\left(q\right)=\frac{l}{L}\sum _{n}h\left(\mathbf{r}\right)e^{i\mathbf{q}\cdot \mathbf{r}},$$ (8)

where L is the lateral size of the supercell, $l=r_{\text{eq}}^{\text{l}\text{-}\text{l}}=2^{1/6}\sigma$ sets the mesh size equal to size of lipid particles, and q is the wave vector $\mathbf{q}=\left(q_x,q_y\right)$, i.e., $\left(q_x,q_y\right)=2\pi \left(n_x,n_y\right)/L$. The power spectrum in the Fig. 4 exhibits a q⁻⁴ dependence on the wave vector. The deviation at large wave vectors is because of the limitation defined by the size of the CG particles, as at wave lengths smaller than the particle size, the continuum approximation breaks down. The bending rigidity of the membrane is found to be κ = 11.3k_BT (see Fig. 4), which lies within the experimental range of (10k_BT – 20k_BT) for lipid bilayer (53,54).

Figure 4.

Vertical displacement fluctuation spectrum of membrane model as a function of the dimensionless quantity $q\sigma /\pi$, where q is the wave number and σ is the unit length corresponding to ∼ 4.45 nm. To see this figure in color, go online.

For the measurements of the shear moduli, the membrane is sheared up to a shear strain of 1 with the strain rate $\dot{\gamma }$ = 0.001σ/t_s, as shown in Fig. 5 A and B. The response of the membrane to shearing is illustrated by the blue line in Fig. 5 C. The proposed model captures the experimentally identified stiffening behavior of the RBC membrane, which is attributable to the spectrin network. The shear modulus of the membrane at small deformations is ∼ 12 μN/m while it is increased to 27 μN/m at engineering shear strain of 0.9. The initial elastic shear modulus of the model is larger than the experimentally measured values of 4 - 9 μN/m (53,55). This is most likely attributable to the implementation of a perfect network in the present model, whereas the spectrin network of the RBC membrane is not perfect (24,27,77,78) probably because of ATP-induced dissociations (15,51). Experimental measurements showed that ATP-induced dissociations played a crucial role in forming defects in the cytoskeleton. In addition, a larger shear modulus was measured in ATP-depleted RBCs compared with normal RBCs (79). At the strain rate of $\dot{\gamma }$ = 0.001σ/t_s, the shear viscosity of the lipid bilayer is not detectable and the total resistance to the shearing is caused only by the cytoskeleton. This is confirmed by the fact that the shear stress-strain response of the cytoskeleton (red curve in Fig. 5 C) follows the pattern of the shear stress-strain response of the entire membrane (blue curve in Fig. 5 C). This result is in agreement with our previous two-component membrane model where the spectrin network is represented implicitly (45). At the higher strain rates of 0.005σ/ts and to 0.01σ/ts, the viscosity of the lipid bilayer contributes to the total resistance during shearing. For example, the total value of the measured shear stress corresponding to 0.01σ/ts strain rate (green curve in Fig. 5 C) is the sum of the shear stress attributable to the cytoskeleton (purple curve in Fig. 5 C) and of the shear stress attributable to the viscosity of the lipid bilayer. By subtracting the shear stress attributable to the cytoskeleton from the total shear stress, we obtain the viscous shear stress to be τ = 0.008 ε/σ². At the strain rate of $\dot{\gamma }$ = 0.005σ/t_s, we computed the viscous shear stress to be τ = 0.004 ε/σ². We assume that during the shearing process, the homogeneous lipid bilayer behaves as a simple 2D viscous Newtonian fluid. Then, by applying the shear stress - strain rate relation $\tau =\mu \dot{\gamma }$, we calculate the shear viscosity of the lipid bilayer to be $\mu =0.8\epsilon t_{s}L/{\sigma }^3$, where L is the length of the membrane model. The shear viscosity of the lipid bilayer can also be measured from shearing the membrane without the cytoskeleton. At the same shear strain rate, the measured shear stress remains constant with respect to different shear strains. At the strain rate of $\dot{\gamma }$ = 0.001σ/t_s, no shear stress is measured from the lipid bilayer. At the strain rate of $\dot{\gamma }$ = 0.005σ/t_s, the shear stress is measured to be τ = 0.004 ε/σ². When the strain rate is further increased to $\dot{\gamma }$ = 0.01σ/t_s, the shear stress is measured to be τ = 0.008 ε/σ². Therefore, the obtained shear viscosity of the lipid bilayer is consistent with the value we measured from shearing the lipid bilayer with the cytoskeleton. The comparison between the numerical value predicted by the model and the experimental value is performed in a subsequent section where we determine the timescale $t_s$ for our simulations.

Figure 5.

(A) Thermally equilibrated fluid membrane at a temperature of k_BT/ε = 0.22. (B) Sheared thermally equilibrated membrane at engineering shear strain of 1. (C) Shear stress-strain response of the membrane at two different strain rates. The red curve represents the response of the “bare” spectrin network. The blue and green curves signify the shear stress obtained at the strain rates of 0.001σ/t_s and 0.01σ/t_s, respectively. The purple curve represents the shear stress measured from the “bare” spectrin network plus the 8μN/m attributed to viscosity. To see this figure in color, go online.

Measurements of fluctuation frequencies of the spectrin filaments and the lipid bilayer

The fluctuation frequency of the spectrin filaments in the cytoskeleton is measured by tracking the normal displacement with respect to the lipid bilayer of a single spectrin particle in the middle of the two binding sites. Then, we apply Fast Fourier Transform (FFT) on the displacements and compute the power spectral density (PSD) of the vibrations. The corresponding results in Fig. 6 A show that the dominant fluctuation frequency of the filaments is measured to be approximately f_c = 0.009/t_s. Regarding the fluctuation of the lipid bilayer, we measure the average fluctuation displacement of the lipid bilayer patch in one triangular compartment formed by the spectrin filaments. The frequency is then obtained by using the PSD plot and it is found to be f_l = 0.001/t_s, which is approximately one order of magnitude smaller than the fluctuation frequency of the filaments. Next, in a different simulation, we measure the fluctuation frequencies of the filaments in the “bare” cytoskeleton, meaning that the lipid bilayer is completely removed from the RBC membrane leaving only the cytoskeleton. Since the cytoskeleton applies compression on lipid bilayer, the lipid bilayer is crumpled at equilibrium and it exerts an extension force on the cytoskeleton. Therefore, when the lipid bilayer is removed, an effective repulsive potential is applied between two neighboring actin junctions in the cytoskeleton to represent the effect of the lipid bilayer. The effective repulsive potential $u_{A-A}\left(r\right)$ is described as follows:

$$u_{A-A}\left(r\right)=\alpha \frac{4\pi {\kappa }_{lipid}\left(R_{flat}-r\right)}{3R_{flat}}H\left(R_{flat}-r\right),$$ (9)

where R_flat is the distance between the two neighboring action junctions when the membrane becomes flat. We take R_flat = 1.2 R_eqA-A, where R_eqA-A is the equilibrium distance between the action junctions. R_eqA-A is ∼ 90 nm in our simulations. κ_lipid is the bending stiffness of the lipid bilayer. H(x) is the Heaviside step function. α is chosen to be 0.36 so that the pressure of the “bare” cytoskeleton is nearly zero at equilibrium temperature. More details about the “bare” cytoskeleton can be found in a previous work by one of the authors (75). We found that the frequency of the “bare” spectrin network is f_c = 0.001/t_s (see Fig. 6 B), which is one order of magnitude smaller than the fluctuation frequency of a spectrin filament in the cytoskeleton that is connected to the lipid bilayer. The reason for this difference is that in the “bare” spectrin network, the spectrin filaments are connected only to the actin junctional complexes, whereas in the case of the cytoskeleton attached to the lipid bilayer, the spectrin filaments are also anchored to the band-3 proteins. This means that in the “bare” network each filament is connected only at its two ends whereas in the proposed membrane model each filament has an additional binding site in the middle causing faster vibrations. The result is in agreement with the theoretical prediction that the conformation time t_c of a spectrin filament depends on the number of parts per filament divided by the band-3 binding sites, N_b, $\left(t_{\text{c}}\text{∼}1/N_b^3\right)$ (80). In the proposed membrane model, each filament comprises two parts (N_b = 2), as there is one band-3 binding site per filament. Therefore, the conformation time for the filaments in the RBC membrane is ∼ 8 times smaller than the conformation time for the “bare” network in which each filament has only one part (N_b = 1).

Figure 6.

(A) Power spectral density (PSD) corresponding to thermal fluctuations of the membrane model. The data marked as red circles are generated from the average displacement of the particles belonging to a triangular compartment of the lipid bilayer. The data marked as blue squares are generated from a particle positioned in the middle of a spectrin filament. f_c represents the thermal fluctuation frequency of the cytoskeleton; and f_l represents the thermal fluctuation frequency of the lipid bilayer. Inset: equilibrium state of the proposed RBC membrane model. (B) PSD corresponding to thermal fluctuations of a particle positioned in the middle of a spectrin filament in the “bare” cytoskeleton. Inset: equilibrium state of “bare” cytoskeleton. To see this figure in color, go online.

Timescales of the simulations

As we described previously, because the particles introduced in CGMD simulations do not correspond to real atoms or molecules, the timescale $t_s={\left(m_i{\sigma }^2/\epsilon \right)}^{1/2}$ defined in the MD simulation is not directly related to the evolution of a real physical system. Only through comparison with a physical process can such a correspondence be established. In this paper, we compare the shear viscosity obtained previously, with the experimentally measured value of 3.6 × 10⁻⁷ N·s/m (81), via which we conclude that the timescale of our simulation is approximately t_s ∼ 3 × 10⁻⁶s. The corresponding timestep of the simulations is then $\Delta t=0.01t_s\sim 3\times {10}^{-8}s$. After determining the timescale by employing the viscosity of the RBC membrane, we need to test the self-consistency of the model by confirming that its predictions for the fluctuation frequencies of the spectrin filaments and of the lipid bilayer are in agreement with the expected physical values. The thermal fluctuation frequency of the lipid membrane obtained from our simulation is f_l = 0.001/t_s ∼ 333Hz, which is comparable with the theoretical and experimentally measured frequency ∼ 1000Hz (15,79). Analytical estimation shows that the conformation time ${\tau }_{conform}$ for the spectrin filaments in the cytoskeleton is ${\tau }_{conform}={\eta }_{cytoplasm}{d}_{ee}^3/\left(N_b^3{k}_{B}T\right)\text{∼}100\mu \text{s}$, where d_ee is the end to end distance of the spectrin filaments, N_b is the number of parts per filament divided by the band-3 binding sites and η_cytoplasm is the viscosity of the cytoplasm (80). In our model, the thermal fluctuation frequency of the spectrin filament is measured to be f_c = 0.009/t_s. By using the timescale obtained above, we determine that the model predicts the conformation time for the spectrin filaments to be ${\tau }_{conform}\text{∼}330\mu \text{s}$, which is comparable with the analytically estimated conformation time ∼ 100μs (80).

Mechanical interaction between the spectrin network and the lipid bilayer

The lipid bilayer and the spectrin network along with their interactions play an essential role in the structure and biological functions of the RBC. Experimental measurement of the pressure applied on the lipid bilayer by spectrin filaments is challenging, as the length scale of the interactions between the cytoskeleton and lipid is not easily accessed by dynamic cell mechanics experiments. In this section, we compute the pressure exerted on the lipid bilayer by the spectrin filaments and investigate how it is affected by the attraction between the spectrin filament and the lipid bilayer. The pressure obtained from the simulations is compared with analytically estimated values for the case of flexible linear polymer attached to the lipid bilayer at its two ends (51,56). Lastly, we examine how defects in the anchoring of the spectrin network to the lipid bilayer and defects in the structure of the spectrin network influence the pressure applied on the lipid bilayer.

In the proposed model, the interaction between the spectrin filaments and the lipid bilayer is described by the repulsive part of the LJ-12 potential, as shown in Eq. 1. The filament-induced local pressure is calculated by using the expression proposed by Cheung and Yip (82), who consider the pressure as summation of the momentum flux and force across a planar unit area in a time interval. Because in our model the spectrin filaments do not penetrate the lipid bilayer and the local deformation of the lipid bilayer caused by the spectrin filaments is ∼ 1 nm (51), the pressure is measured by considering only the forces between the lipid bilayer and the spectrin filaments. In the RBC membrane without defects, the two ends of the spectrin filaments are connected at the actin junctional complexes, which are anchored to the lipid bilayer by glycophorin proteins. The spectrin filaments are additionally anchored to the lipid bilayer via band-3 proteins at the spectrin-ankyrin binding sites. The blue curve in Fig. 7 A shows that the pressure increases at the area close to band-3 binding site, where the spectrin particles are connected to the lipid bilayer. The pressure is large there because of the continuous interactions between the spectrin particles and the lipid bilayer. When moving away from the band-3 binding site, the pressure drops rapidly as the frequency of interactions between the spectrin particles and the lipid bilayer reduces. The pressure measured from the simulation is close to the analytical pressure distribution introduced in an earlier study (51) as follows:

$$\begin{array}{l}P\left({\rho }_1,{\rho }_2,\rho \right)=\frac{k_{B}T}{4\pi R_g^2}{e}^{{\left({\rho }_1-{\rho }_2\right)}^2\text{/}\left(4R_g^2\right)}{e}^{-{\left(|{\rho }_1-\rho |+|{\rho }_2-\rho |\right)}^2\text{/}\left(4R_g^2\right)}\frac{|{\rho }_1-\rho |+|{\rho }_2-\rho |}{|{\rho }_1-\rho {|}^3|{\rho }_2-\rho {|}^3}\\ \times \left[|{\rho }_1-\rho ||{\rho }_2-\rho |\left({\left(|{\rho }_1-\rho |+|{\rho }_2-\rho |\right)}^2-6R_g^2\right)+2R_g^2{\left(|{\rho }_1-\rho |+|{\rho }_2-\rho |\right)}^2\right],\end{array}$$ (10)

where R_g is the radius of gyration of the spectrin filaments, ρ is the pressure measurement location, ρ₁ and ρ₂ are the anchoring locations of the filament to the lipid bilayer. Assuming that the measurement points always remain on the line connecting the two anchoring locations, the pressure profile based on Eq. 10. is plotted as the purple curve in Fig. 7 A. We note that the pressures in the analytical estimation are significantly higher than the simulation measurements at the two ends of the filaments. The reason is that in the analytical calculations, the filaments are assumed to be directly attached to the lipid bilayer, whereas in our numerical simulation and in the actual RBC membrane, they are connected to the actin junctional complexes that then bind to the lipid bilayer through the glycophorin proteins.

Figure 7.

The pressure distribution exerted by the spectrin filaments on the lipid bilayer with attraction parameters (A) n = 0; (B) n = 0.05; (C) n = 0.1; and (D) n = 0.2. d_ee is the end-to-end distance of the spectrin filaments. The blue curve represents the pressure distribution measured from the membrane model. The purple curve represents the pressure distribution obtained from the analytical estimation for a normal membrane (56). The red curve represents the pressure distribution applied on the membrane with ankyrin protein defects. The black curve represents the pressure distribution obtained analytically for a membrane with ankyrin protein defects (56). The green curve represents the pressure distribution measured from the membrane with spectrin protein defects. To see this figure in color, go online.

Next, we investigate the effect that the introduced attractive forces between the spectrin filament and the lipid bilayer have on the pressure applied to the lipid bilayer by the spectrin filaments. The parameter n in the Eq. 2. is selected to be 0.05, 0.1, and 0.2, as the spectrin-lipid interaction is relatively weak compared with the two primary connections between the cytoskeleton and lipid bilayer (58–65). We chose 0.2 as the maximum value of n because at this value the spectrin filaments are completely attached to the lipid bilayer. The pressure plots corresponding to n = 0.05 and 0.1 (blue curves in the Fig. 7 B and C) show that the pressure profiles are similar to the profile obtained when n = 0, but the magnitudes of the pressure increase as n increases. The increased attractive forces reduce the average distances between the filaments and the lipid bilayer, resulting in more frequent interactions between the spectrin particles and the lipid bilayer, and subsequently in a larger pressure on the lipid bilayer. When n = 0.2, the attractive forces are large enough to cause attachment of the entire filament to the lipid bilayer, and thus generate a dramatic increase in the pressure (Fig. 7 D). We note that the pressure is lower at the two ends of the filament because these points are connected to actin junctional complexes and not directly to the membrane. Previous experimental studies (5–9,83) and analytical modeling (56,84–89) have shown that the pressure fence induced by the cytoskeleton hinders the lateral diffusion of mobile band-3 proteins and lipid molecules. Therefore, it is reasonable to predict that an increase in the attractive forces between the filament and lipid bilayer causes a reduction in the lateral diffusion of the band-3 proteins and lipid molecules.

Another question is how disruption of the connections between spectrin filaments and immobile band-3 proteins (vertical interactions in Fig. 1 A) modifies the pressure field applied on the lipid bilayer by the spectrin network. This is relevant in HS where defects in proteins such as ankyrin and band-3 proteins, which play an important role in the coupling between the lipid bilayer and the cytoskeleton, result in a partial detachment between the network and the lipid bilayer. The red curve in the Fig. 7 A shows that when only repulsive interaction between the lipids and the spectrin filaments is assumed (n = 0), the pressure is higher at the two ends of the filaments, compared with the middle portion of the filaments. This reduction is attributable to a decrease in the frequency of interactions between the middle sections of the filaments and the lipid bilayer. In particular, the pressure profile is consistent with the analytically estimated pressure distribution defined by Eq. 10 (see black curve in Fig. 7 A) with the exception that the analytical pressure is much higher at the two ends of the filament. This is because of the assumption in the theoretical analysis that the two ends are directly anchored to the lipid bilayer whereas in our simulations they are connected to the junctional complexes. Therefore, we infer that a disruption of the connection between the filaments and the lipid bilayer at the band-3 anchoring points weakens the pressure fence. This causes an increase in the probability for band-3 particles to cross the boundaries of the filament-formed compartment, thus it enhances the diffusion of band-3 proteins in HS (6,56,90,91). The introduction of small attractive forces between the lipid bilayer and the spectrin network (n = 0.05 and n = 0.1), when the connection between the filament and the lipid at the band-3 site is severed, results in a constant pressure along the filament. We note that the pressure is higher than in the case of only repulsive interactions between the lipid bilayer and the network. When the attractive force is large (n = 0.2), and the entire filament is in contact with the surface of the lipid bilayer, the pressure is similar with the pressure field developed in the normal membrane with the exception of the middle segment of the filament. In conclusion, the disruption of the anchoring of the spectrin filaments to the lipid bilayer results in a lower and nearly constant pressure along the filament. Since the spectrin filament-induced pressure acts as a pressure fence, hindering the lateral movement of mobile band-3 proteins and lipid molecules, it is predicted that the lower pressure facilitates the diffusion of lipids and band-3 proteins across the spectrin filaments. This results in a higher mobile band-3 diffusion coefficient compared to the band-3 diffusion coefficient in the normal RBC membrane.

Finally, we explore how the dissociation of a spectrin filament to two filaments affects the pressure applied to the membrane. Here, we aim to simulate defective spectrin dimer-dimer interactions that most commonly happen in HE (3). The pressure applied from both filaments on the membrane is shown as green curves in Fig. 7 A–D for different attractive forces between the filaments and the membrane. We observe that when no attractive forces (n = 0) or small attractive forces (n = 0.05) are assumed, the pressure is lower than in the normal membrane or in a membrane with defects in the vertical interactions. It is reasonable to assume that a weaker pressure fence on the lipid bilayer causes an increase in the probability of band-3 particles to cross the spectrin filaments, justifying the observed higher diffusion coefficients of band-3 proteins in HE (6,56,90,91). We also note that our simulations show that the pressure applied on the lipid bilayer in the case of dimer-dimer disruption, which corresponds to HE, is lower than the pressure measured in the case of disruption of vertical interactions, which corresponds to HS. Based on this result, we conjecture that our model predicts greater band-3 diffusion in HE than in HS, in agreement with experimental observations (91). Of course, direct study of band-3 diffusion is required to validate these predictions. When the attractive forces are large (n = 0.2), the difference between the pressure measured in the normal membrane and in the membrane with protein defects becomes negligible with the exception of the area close to the band-3 binding point in the middle where the pressure is large in the normal membrane. This means that if the attractive force between the spectrin network and the lipid bilayer are large, the diffusion coefficients of band-3 should have been similar in normal RBCs and in RBCs from patients with HS or HE. However, since the band-3 diffusion coefficients measured in normal RBCs are smaller than the ones measured in RBCs in HS and HE (91), we conjecture that the attractive forces between the spectrin filaments and the lipid bilayer should be low and the parameter n that defines the strength of the attractive forces cannot be larger than n = 0.1, which justifies our selection of n previously.

Conclusions

In this paper, we introduce a particle-based model for the erythrocyte membrane that accounts for the most important structural components of the membrane, including the lipid bilayer, the spectrin network, and the proteins that play an important role in the anchoring of the spectrin cortex to the lipid bilayer, as well as the band-3 proteins. In particular, five types of CG particles are used to represent actin junctional complexes, spectrin, glycophorin, immobile band-3 protein, mobile band-3 protein, and an aggregation of lipids. We first demonstrate that the model captures the fluidic behavior of the lipid bilayer and then that it reproduces the expected mechanical material properties of bending rigidity and shear modulus of the RBC membrane. The timescale of our simulations, which is found to be t_s ∼ 3 × 10⁻⁶s, is inferred by comparing the viscosity of the membrane model to experimentally measured values. Then, the self-consistency of the model with respect to the timescale is tested by comparing the computed vibration frequency of the spectrin filaments and lipid membrane to analytically obtained values. We confirm that vibration frequency of the spectrin filaments and lipid membrane measured from the proposed membrane model, are also in agreement with experimental values. At last, we study the interactions between the cytoskeleton and the lipid bilayer, and measure the pressure applied on the membrane by the spectrin filaments. We also investigate how disruption of the connection between the spectrin network and the lipid bilayers, which simulates defects in HS, and rupture of the dimer-dimer association, which simulates defects in HE, affect the pressure exerted on the lipid bilayer. We show that overall the introduction of defects in the spectrin network or in the vertical connection between the lipid bilayer and the cytoskeleton results in lower pressure, which is consistent with prediction in (51). In addition, we find that the defects related to HE have a stronger effect than the defects related to HS. This result implies that diffusion of band-3 proteins in RBCs from patients with HS and HE is enhanced compared to the normal RBCs. Moreover, elliptocytes exhibit more prominent diffusion of band-3 proteins than spherocytes. Both conclusions are supported by experimental results. The level of attraction forces between the lipid bilayer and the membrane cytoskeleton is another important parameter that regulates the pressure applied by a spectrin filament to the lipid bilayer. We show that as the attractive force increases, it causes an overall increase in the pressure and it diminishes the differences in the pressure generated by membrane protein defects and, consequently, the differences in the diffusion of band-3 proteins in normal and defective erythrocytes. Because this finding is not supported by experimental results, we conjecture that the attractive force between the lipid bilayer and the spectrin filaments should be low, resulting in a membrane model where the filaments are not completely attached to the lipid bilayer. A detailed study of the band-3 diffusion in this model and direct comparison with experimental results is necessary to form a more accurate picture of how the model regulates diffusion. Because of the explicit representation of the lipid bilayer and the cytoskeleton, the proposed model can be potentially used in the investigation of a variety of membrane related problems in RBCs in addition to diffusion. For example, membrane loss through vesiculation and membrane fragility in spherocytosis and elliptocytosis, interaction between hemoglobin fibers and RBC membrane in sickle cell disease, and RBC adhesion are problems where the applications of the proposed membrane model could be beneficial.