Crowding-induced membrane remodeling: Interplay of membrane tension, polymer density, architecture

Abstract

The plasma membrane hosts a wide range of biomolecules, mainly proteins and carbohydrates, that mediate cellular interactions with its environment. The crowding of such biomolecules regulates cellular morphologies and cellular trafficking. Recent discoveries have shown that the structure and density of cell surface polymers and hence the signaling machinery change with the state of the cell, especially in cancer progression. The alterations in membrane-attached glycocalyx and glycosylation of proteins and lipids are common features of cancer cells. The overexpression of glycocalyx polymers, such as mucin and hyaluronan, strongly correlates with cancer metastasis. Here, we present a mesoscale biophysics-based model that accounts for the shape regulation of membranes by crowding of membrane-attached biopolymer-glycocalyx and actin networks. Our computational model is based on the dynamically triangulated Monte Carlo model for membranes and coarse-grained representations of polymer chains. The model allows us to investigate the crowding-induced shape transformations in cell membranes in a tension- and graft polymer density-dependent manner. Our results show that the number of membrane protrusions and their shape depend on membrane tension, with higher membrane tension inducing more tubular protrusions than the vesicular shapes formed at low tension at high surface coverage of polymers. The shape transformations occur above the threshold density predicted by the polymer brush theory, but this threshold also depends on the membrane tension. Increasing the size of the polymer, either by changing the length or by adding side chains, is shown to increase the crowding-induced curvature. The effect of crowding is more prominent for flexible polymers than for semiflexible rigid polymers. We also present an extension of the model that incorporates properties of the actin-like filament networks and demonstrate how tubular structures can be generated by biopolymer crowding on the cytosolic side of cell membranes.

Significance

The plasma membrane that defines the exterior of a cell expresses a dense layer of proteins and carbohydrates on its surface. The density and structure of these biopolymers depend on cell state and play important roles in cancer progression by rewiring cellular signaling networks. Overexpression of surface polymers drive the formation of vesicular and tubular membrane packets that mediate cell-cell communication. Here, we present a generic physics-based membrane polymer model to systematically investigate the biophysics and mechanobiology of membrane shape regulation by anchored biopolymers. We demonstrate that the density-dependent shape formation in cell membranes strongly depends on membrane tension, polymer length, degree of branching, persistence length, and ionic strength. The model can be extended to include key biochemical details, including lipid-protein compositions.

Introduction

The ability of the plasma membrane to act as an effective boundary that mediates the communication between intracellular and extracellular spaces depends on a large number of surface proteins and membrane-attached biopolymers, mainly those from the extracellular matrix and cytoskeleton. Hence, the functioning of a membrane requires a complex environment that is densely packed. Macromolecules at the membrane surface regulate membrane morphology by driving the formation of tubes and spherical vesicles, which are key to cellular response and intracellular trafficking. The known mechanisms behind complex membrane shape remodeling include lipid shape anisotropy, the interaction of different types of curvature-inducing proteins, force generation due to cytoskeletal filaments, and membrane adhesion (1, 2, 3). Crowding of macromolecules is in itself a major shape regulator (4,5). Recent studies have demonstrated the effect of protein crowding in membrane fission (6), formation of intraluminal vesicles (7), diffusion of receptors (8), and organization of ion channels (9).

On the extracellular side, the glycocalyx, a dense network of glycoproteins and proteoglycans, covers the membrane surface. Although the composition and architecture of glycocalyx depend on the type and state of the cell, from a physical perspective it can be thought of as a collection of complex biopolymers with varying stiffnesses that depend on the structure of the side chains. One of the known functions of glycocalyx is to act as a protective covering to the cell membrane and modulate cell-cell recognition and cell adhesion. Endothelial glycocalyx functions as a vascular barrier, as well as a force actuator and intracellular signaling partner by transducing external stress into the cytoskeleton (10,11). Recent studies have shown that the glycocalyx plays a vital role in many cellular processes and could be targeted for the development of novel therapeutics. For instance, it has been shown that the glycocalyx plays a significant role in immune system regulation and checkpoint inhibition (12, 13, 14). Glycosylation and glycocalyx have been shown to regulate cancer progression, cancer metabolism, and cell immunoevasion (15, 16, 17, 18, 19, 20, 21). The overexpression of the constituents of the glycocalyx, mucin, and hyaluronan are potential markers of cancer aggression. Shurer et al. have demonstrated that the density of mucin on the cell membrane can control cell membrane morphology and can drive the formation of tubules and vesicles mainly via entropic forces (22).

In cell signal transduction, membrane and its associated biopolymers, namely the glycocalyx and the cytoskeleton, function in an interdependent manner by modifying membrane shapes and polymer architecture. Hence, the high curvature deformations in membranes, which are significant in many cellular processes, are strongly correlated to the structures of the attached polymers. For decades, the deformation of polymer-decorated membranes has been of interest in biology and polymer physics. The classical polymer brush theory describes the limiting regimes of polymer-grafted membrane as a mushroom regime at low polymer densities and a brush regime at high polymer densities, as shown in Fig. 1 A. The low-density mushroom regime is characterized by the polymer separation length d being much larger than the characteristic polymer radius $R_p$. In this regime, the individual polymers adopt the maximum entropy globule conformations and are far separated to not interact with each other. Here the polymer-induced curvatures on the membrane are negligible. On the other hand, the brush regime is characterized by $d<R_p$, and the polymer adopts the low entropy linear conformations. The resulting entropic force exerts a pressure on the anchoring surface that is large enough to drive shape changes in the cell membrane. In addition to the density of polymers, there are many other factors, such as electrostatic effects, confinement, steric effects, and membrane tension, that affect the membrane remodeling by the crowding of embedded polymers.

Figure 1.

Polymer-decorated membranes. (A) An illustration of membrane in the presence of anchored biopolymers at mushroom and brush regimes. At high surface coverage of polymers, the membrane undergoes spontaneous curvature generation. (B and C) Representative membrane conformations from the simulations as a function of time for low ($A_{\text{ex}}=6\text{\%}$) and high ($A_{\text{ex}}=54\text{\%}$) excess area, respectively. (D and E) The equilibrium mean curvature distribution of the membrane in the presence of attached polymers ($N_p={10}^3$, $l_p=150$ nm) corresponding to the membranes shapes in (B) and (C). For comparison the distributions in the absence of polymers ($N_p=0$) are also shown. (F) The average mean curvature per area of the membrane as a function of $A_{\text{ex}}$ for various $N_p$ values.

Generation of lipid membrane curvature due to the crowding of peripheral membrane proteins has been shown experimentally using giant unilamellar vesicles and tagged proteins (4,5,7,23,24). Stachowiak and co-workers showed that asymmetric distribution and steric confinement of the membrane proteins to lipid domains could drive the formation of tubular structures that can extend up to micrometer length scales (4,23). Observations from Chen et al. indicate that curvature generation via a crowding mechanism requires higher protein surface density than other intrinsically curved proteins, which depends on the membrane tension (5). These findings have inspired a set of theoretical biophysics-based models that are based on the balance between bending energy of the membrane and steric repulsion between the domain confined spherical or disc-like proteins. These models have predicted some of the cell surface morphologies observed in the experiments (5,7,24,25).

Here, we present an explicit membrane model that incorporates polymer brush theory to get insight into the entropic effects of crowded polymers at the membrane surface. We employ a Monte Carlo (MC) model for membranes to explore the effect of membrane tension, fluidity, and thermal fluctuations. In this study we systematically explore the membrane conformations for varying molecular structures of polymers and membrane tension. We further extended the model to study the mechanism of membrane tubulation due to actin polymer network assembly on the cytosolic side of the plasma membrane.

Methods

We adopt a mesoscale computational approach to investigate the conformational properties of bilayer membranes with anchored polymer chains. In our simulations, the membrane surface is modeled using the dynamically triangulated MC technique developed in earlier works (26, 27, 28). In this model, the cell membrane is described as a triangular mesh constructed with $N_{tri}$ triangles that interlink $N_v$ vertices with $N_l$ links. The membrane bending energy $H_{\text{bend}}$ is computed using the Canham-Helfrich Hamiltonian (29,30), given by

$$H_{\text{bend}}=\frac{\kappa }{2}\underset{A_{\text{m}}}{\int }{\left(2H-C_0\right)}^{2}dA,$$ (1)

where κ is the bending rigidity of membrane, $A_{\text{m}}$ the membrane area, H the mean curvature, and $C_0$ the spontaneous curvature of the surface. The details of the model and MC moves to evolve the surface are provided in (28,31). Conformations accessible to the model membrane also depend on the membrane tension that controls the ratio of curvilinear area $A_{\text{m}}$ to the projected area $A_{\text{p}}$ of the surface. The dimensionless excess area, defined as $A_{\text{ex}}=\left(A_{\text{m}}-A_{\text{p}}\right)/A_{\text{p}}\times 100\text{\%}$, is used to quantify the area available in the membrane patch for characteristic deformations (31). In our studies, we systematically vary $A_{\text{ex}}$ to obtain physiologically relevant excess areas corresponding to membrane tensions (0 to –413.6 μN/m) measured in mammalian cells (31,32).

Extracellular polymers are modeled as coarse-grained linear and branched chains anchored at the membrane vertices. The interconnected polymer beads interact via a harmonic spring potential. The total bond energy of the polymers is given by

$$H_{\text{b}}=\frac{k_b}{2}\sum _{i=1}^{N_p}\sum _{j=1}^{N_b-1}{\left(r_{j,j+1}^i-r_0\right)}^2,$$ (2)

where $k_b$ is the spring constant, $N_p$ is the total number of polymers, and $N_b$ is the number of beads per polymer. The equilibrium bond length $r_0=a_0$ is the diameter of the polymer bead. The first bead of each polymer is anchored to a membrane vertex and they also interact with the spring potential given in Eq. 2, with $r_0=\left(a_0+a_v\right)/2$, where $a_v$ is the diameter of the vertex. Polymer beads are considered as self-avoiding hard spheres and they avoid membrane overlap with cutoff distance $r_{cut}=\left(a_0+a_v\right)/2$.

To study the effect of polymer rigidity, we introduce an angle potential:

$$H_{\text{ang}}=\frac{k_{ang}}{2}\sum _{i=1}^{N_p}\sum _{j=1}^{N_b-1}\left(1-\text{cos}{\theta }_j^i\right),$$ (3)

where $k_{ang}$ defines the angle stiffness and ${\theta }_j^i$ is the angle subtended by beads $j-1$, j, and $j+1$ in polymer chain i. The value of $k_{ang}$ depends on the persistence length ${\xi }_p$ of the polymer as $k_{ang}=k_{B}T\frac{{\xi }_p}{a_0}$.

To study the effect of the cortical actin network on the membrane shapes, we start with a prebuilt polymer mesh of actin filaments on the cytosolic side of the membrane. As our focus is on the effect of crowding, we ignore polymerization-depolymerization effects of the filaments and start with $N_p$ linear parent polymers with $N_b$ beads on each of them and their branches. We define a branching fraction parameter ${\nu }_{br}$ as ${\nu }_{br}=\frac{{\sum }_{i=1}^{N_{branch}}{n}_i}{N_p{N}_b}$. $n_i$ is the number of monomer beads in each branch i. A number of monomers defined by ${\nu }_{br}$ is allowed to form branches at random locations on the parent filaments and the newly formed daughter filaments. The length of the branches is restricted to be smaller than half the length of parent filaments. The branches make an angle ${\theta }_b={70}^{\circ }$ with the parent filaments (33). For this, we introduce a bending potential for the branches as

$$H_{\text{branch}}=\frac{k_{branch}}{2}\sum _{j=1}^{N_{branch}}{\left(\text{cos}{\theta }_j-\text{cos}{\theta }_b\right)}^2,$$ (4)

where $k_{branch}$ is the bending stiffness for the branches and ${\theta }_j$ is the angle made by the branch to the parent filament. ${\theta }_j$ is the minimum angle between the $r_{i,i-1}$ and $r_{i,k}$, where i corresponds to the bead on the parent filament that is connected to the branch j and k is the first monomer of the branch filament. Note that $i=1$ corresponds to the bead at the pointed end of the filament, which is attached to the membrane vertex. The pointed end of the actin filament is known to allow space for monomers to polymerize near the membrane. Hence, the bond potential (Eq. 2) for vertex attached beads are modified with $r_0=\left(3a_0+a_v\right)/2$.

To vary the range of interaction between the polymer beads in the model we modified the self-avoiding interaction between the polymer beads with Weeks-Chandler-Anderson (WCA) potential which is of the form:

$$H_{WCA}=\sum _{i,j}4{\epsilon }_{WCA}\left[{\left(\frac{r_0^b}{r_{ij}}\right)}^{12}-{\left(\frac{r_0^b}{r_{ij}}\right)}^6+\frac{1}{4}\right],r_{ij}\le r_c,$$ (5)

where ${\epsilon }_{WCA}$ is the energy parameter, $r_0^b$ the radius of beads, and $r_c=2^{1/6}{r}_0^b$ the cutoff distance for the interaction. The WCA represents a purely repulsive potential, and we vary $r_0^b$ to understand the effect of long-range interactions.

The integrated model for the membrane polymer system is evolved using the MC method. First, we equilibrate the membrane and initialize polymer chains on the equilibrated membrane patch. The polymers are then equilibrated, keeping the membrane surface frozen. The combined system is then equilibrated to obtain the final membrane conformation resulting from the polymer crowding interactions. In dynamically triangulated MC, the membrane surface is equilibrated through two independent MC moves, namely vertex moves and link flips that simulate thermal fluctuations and the fluidity of bilayer membranes (28,31). Here, we introduce an additional MC move for polymer beads in which the position $X_b$ of a bead is updated to a new random position $X_b+\delta X_b$ within a cubic box of size ${\epsilon }_b$, ${\epsilon }_b=0.05a_0$. Each MC step consists of $N_v$ attempted vertex moves, $N_p{N}_b$ attempted polymer bead move, and $N_l$ attempted link flips. All the MC moves are accepted through the Metropolis algorithm (34).

Parameters: we consider a membrane patch with $N_v=1600$ vertices. The polymer bead size is $a_0=10$ nm. We take a membrane patch with side $L=48a_0$, which sets $A_{\text{p}}=2304$ nm². The membrane bending rigidity is taken to be $20k_{B}T$. The spring constant for the bonds is taken to be $k_b=10k_{B}T/a_0^2$. The persistence length ${\xi }_p$ defines the angular stiffnesses for the semiflexible and rigid polymers. The branch stiffness for the actin filaments is taken to be $k_{branch}=1000k_{B}T$. For the long-range interactions, we take ${\epsilon }_{WCA}=1k_{B}T$. Mean curvatures presented here are ensemble averages of 10 runs where each ensemble is equilibrated for ${10}^7$ MC steps. The list of parameters are given in Table S1.

Results and discussions

Tension controls curvature of crowding-induced membrane morphologies

In cell membranes, tension is a key regulator of surface morphology irrespective of the mechanism of curvature generation. Membrane tension is also an indicator of the state of the cell in many vital cellular processes and disease progression. To investigate the role of membrane tension in the crowding-induced membrane shapes, here we consider a membrane patch with two different excess areas $A_{\text{ex}}$. A membrane surface with low $A_{\text{ex}}$ represents a tensed membrane, and high $A_{\text{ex}}$ represents a floppy/relaxed membrane (31). We also choose the number of anchored polymers and the polymer length to be $N_p={10}^3$ and $l_p=150$ nm, respectively. The simulated membrane surfaces in the presence of anchored polymers at $A_{\text{ex}}=6\text{\%}$ (Video S1) and $A_{\text{ex}}=54\text{\%}$ (Video S2) as a function of time are given in Fig. 1, B and C. Upon energy minimization, the membrane adopts conformations that are intermediates between tubular and vesicular morphologies seen in cell membranes. $A_{\text{ex}}=6\text{\%}$ shows more tubules and lower radius structures compared with $A_{\text{ex}}=54\text{\%}$. The mean curvature distributions, $p\left(H_{\text{mean}}\right)$, of the membrane at both $A_{\text{ex}}$ in comparison with a membrane patch without attached polymers are given in Fig. 1, D and E. The visible shift of $p\left(H_{\text{mean}}\right)$ along the positive curvature direction in the presence of polymers indicates the entropy-driven curvature generation observed in the membrane shapes (Fig. 1, B and C). The lower $A_{\text{ex}}$ case exhibits a second peak in $p\left(H_{\text{mean}}\right)$ due to its highly curved surface features. The average mean curvature $⟨H_{\text{mean}}⟩$ per membrane area is shown in Fig. 1 F. Clearly, $⟨H_{\text{mean}}⟩/A_m$ decreases as $A_{\text{ex}}$ increases for $N_p=1000$, but this trend changes with the surface coverage with lower $N_p$ ($N_p<700$) showing an increase in induced curvature with $A_{\text{ex}}$. The effect of surface coverage is discussed in the next section.

The total energy of the membrane polymer system is the sum of entropic and elastic energies. As we kept the number of polymers constant, the energy contributions from polymers were equal for both excess areas. To obtain the energy of the membrane patch, let us consider the energy of idealized tubular (cylindrical) and spherical vesicular morphologies. For a membrane tubule or a vesicle, the minimization of the total free energy, which is a sum of bending and tension terms yields a characteristic scaling of the radius of the feature $R\sim \sqrt{\kappa /\sigma }$, where σ is the surface tension. As we hold κ to be a constant across all simulations, a higher tension (small $A_{\text{ex}}$) is expected to generate smaller length scale features in the surface, which justifies the shape differences between the two different $A_{\text{ex}}$ values at high surface coverage of polymers ($N_p=1000$), which are above the threshold density for a crowding-induced morphological transition. In this limit, the skew in the distribution of mean curvature higher values for higher tension for the same reason as stated above, and the distribution is dominated by the curvature in the tubules and vesicles rather than the planar region (Fig. 1 F). Below the threshold (i.e., for lower $N_p$) we see the opposite effect where the skew in the distribution of mean curvature is higher for lower tension. In this limit, the surface coverage is too low for a morphological transition and the curvature distribution represents the planar region. In the absence of the polymers, the curvature distribution in the planar region is symmetric about $H=0$, but, as $N_p$ increases, the asymmetry causes a skew toward positive H, with the skew increasing with decreasing tension due to increase in membrane excess area (Fig. 1 F).

Membrane shape transition requires a threshold polymer density

As the surface density of polymers increases, they go from a mushroom to brush regime where they exert higher entropic pressure on the membrane surface to generate sufficient curvature. Hence the crowding-induced deformations on the membrane will depend on the number of surface polymers. Fig. 2, A and B show the membrane shapes and average mean curvature as $N_p$ increases for low and high excess areas. From standard polymer scaling arguments, the transition from mushroom to brush regime occurs at the threshold overlapping coverage, where $A_{\text{m}}/N_p=\pi R_p^2$. Here, $R_p$, the radius of the polymer (defined as its mean end-to-end distance) can be estimated as $R_p=a_0{N}_b^{\nu }$ with $a_0$ being the size of coarse-grained bead, $N_b$ the number of beads per polymer, and the scaling exponent $\nu =3/5$ for a polymer in good solvent. For a polymer with $N_b=15$ and $a_0=10$ nm, at threshold density, the theoretical estimate for $N_p/A_{\text{m}}\sim 123/\mu m^2$ and we expect to observe an increase in membrane curvature above this threshold. The membrane curvatures observed in our simulations with varying polymer densities and membrane excess areas are shown in Fig. 2 B. As expected, the crowding-induced curvature does increase beyond the threshold in an excess area-dependent manner. We observe the transition threshold to be lower for membranes with high $A_{\text{ex}}$ (low tension) and higher for membranes with low $A_{\text{ex}}$ (high tension). The tubule formation in membrane has been reported.

Figure 2.

Effect of polymer density. (A) Representative membrane shapes as a function of number of attached polymers for $A_{\text{ex}}=6\text{\%}$ (top panel) and $A_{\text{ex}}=54\text{\%}$ (bottom panel). (B) Average mean curvature of membrane vertices. The vertical line at $N_p/\mu m^2=123$ corresponds to the critical surface coverage of the polymers where the mushroom to brush transition occurs and the black and blue dashed lines correspond to the theoretical predictions for mushroom and brush regimes, respectively.

The induced curvature of a polymer-anchored membrane has been derived analytically using Green’s function method for the mushroom region and scaling theory for brush regime (35, 36, 37, 38). Theoretical predictions have been tested for small surface coverage of polymers and small deformations on the membrane (35,39, 40, 41, 42). In the mushroom region, the induced spontaneous curvature of the membrane depends linearly on the polymer surface density as (37,38): $C_{sp}=\frac{2b{k}_{B}T{R}_p\rho }{\kappa }$. $C_{sp}$ is the additional curvature of the membrane in the presence of attached polymers compared with the zero mean curvature membrane. The prefactor b for self-avoiding polymer chains has been reported to be $b=0.18$ in the previous studies using multiple theoretical and simulation methods (37,38,40). The factor ρ defines the surface coverage of polymers as $\rho =\frac{N_p}{A_{\text{m}}}$, where $A_{\text{m}}$ is the curvilinear area of the membrane. The expected induced curvature at the mushroom regime is minimal and is shown in Fig. 2 B as black dashed lines.

In the brush regime the energy minimization gives an implicit equation (38):

$$\frac{\partial f_c\left(x\right)}{\partial x}+\frac{4\kappa }{k_{B}T}{N}_p^{-3}{\overline{\text{Γ}}}^{-3/2\nu }x=0,$$ (6)

where $f_c\left(x\right)=\frac{{\left\{1+\left(1+\nu \right)x/\nu \right\}}^{\frac{\nu }{1+\nu }}-1}{x}-1$ and x is the reduced spontaneous curvature defined as $x=h_0{C}_{ind}/2$. Here, $h_0$ is the height of the polymer brush on the flat membrane and is given as $h_0=N_b{\overline{\text{Γ}}}^{\frac{1-\nu }{2\nu }}{a}_0$. The reduced surface coverage of the system is $\overline{\text{Γ}}=\frac{a_0^2{N}_p}{A_{\text{m}}}$. The induced curvature obtained by solving Eq. 6 is shown in Fig. 2 B as blue dashed lines. The observed curvature in simulations agrees quantitatively with the scaling theory. It can be noted that higher polymer density has not been shown to produce pearl-like structures as predicted in previous studies (22,24). This absence could be due to the additional mechanism required for pearling or the limitation of the length scales adopted in the model.

Crowding-induced curvature strongly depends on polymer structure and stiffness

The glycocalyx biopolymers anchored to cell membranes have a wide range of mechanical properties, and their expression varies among cell types. A detailed description of the physical properties of glycocalyx is given in (43). Here, we quantify the effect of length, degree of branching, and stiffness on the crowding-induced curvature remodeling of membranes. The size and length of membrane-associated polymers vary from a few nanometers to micrometers. The linear dimension of glycocalyx polymers, such as mucin ($\sim 270$ nm) and hyaluronan ($\sim 10\mu$m), are much higher than membrane-associated proteins, whose sizes are in the order of a few ten nanometers. An increase in $⟨H_{\text{mean}}⟩$ is expected with increase in polymer contour length $l_p$, as $l_p$ increases effective polymer size. Fig. 3 A shows the induced curvature of the membrane obtained as a function of $l_p$. Dashed lines in Fig. 3 A represent the theoretical predictions from Eq. 6. The longest polymer simulated here is 250 nm, which is comparable with the average length of mucin polymers.

Figure 3.

Effect of polymer structure for $A_{\text{ex}}=6\text{\%}$ and $A_{\text{ex}}=54\text{\%}$. $⟨H_{\text{mean}}⟩$ as a function of (A) contour length $l_p$ when $N_p=1000$, (B) persistence length ${\xi }_p$ when $N_p=500$, and (C) branching of the polymers. The dashed lines in (A) shows the theoretical prediction of $⟨H_{\text{mean}}⟩$. $l_p$ for (B) and (C) is 0.15 μm.

Based on their structure, glycocalyx polymers can be divided into linear and branched polymers and, based on their stiffness, they can be classified into flexible and rigid chains. We use the persistence length ${\xi }_p$ to measure the flexibility of the polymer. Polymers with $l_p\sim {\xi }_p$ are rigid rod-like, while those with $l_p\gg {\xi }_p$ are flexible random coil-like chains. Hyaluronic acid (HA) and mucins are examples of flexible polymers, their persistence lengths in normal physiological conditions are $\sim 5$ and $\sim 7.5$ nm, respectively, which can vary depending on the solvent properties and the number of side chains. Average ${\xi }_p$ for cytoskeletal polymers actin and microtubule are $7\mu$m and 5 mm, respectively. Fig. 3 B shows the effect of polymer bending rigidity on membrane curvature. $⟨H_{\text{mean}}⟩$ decreases as ${\xi }_p$ increases. When attached to a membrane surface, semiflexible and stiff polymers in a brush regime do not lose much of the conformational entropy compared with the fully flexible polymers we described in previous sections, and this justifies the reduction in $⟨H_{\text{mean}}⟩$ with the increase in ${\xi }_p$.

Previous studies have shown that the increase in density or extension of glycan side chains can increase the ${\xi }_p$ of polymers, such as mucins (44,45), and hence a reduction in $⟨H_{\text{mean}}⟩$ is expected. On the other hand, the addition of monomers can also increase the entropic effects on the surface. To check the effect of side chains on crowding-induced curvature, we introduced branches along the length of the polymers. The length of each branch was set to $bl=20$ nm. In Fig. 3 C we compare the effect of number of branches $n_{br}$ on $⟨H_{\text{mean}}⟩$ as a function of the polymer density. Increase in $n_{br}$ results in increase in $⟨H_{\text{mean}}⟩$ for all values of $A_{\text{ex}}$ and polymer density.

Explicit long-range interactions between polymers

The inherent structure of glycocalyx is complex due to the steric and electrostatic interactions between the proteins and other polymers. The presence of charged biopolymers has been observed in many metastatic cancer cells, the well known among them being HA (46,47). Another polyelectrolyte that is upregulated in cancer cells is polysialic acid (48). The long-range electrostatic interaction can alter the properties of a single polymer and the entire network of polymers on the extracellular side. This interaction will depend on the ionic strength of the solution around them. Under physiological conditions, long-range electrostatic interactions are mostly screened by oppositely charged ions in the solution. To investigate the effect of such interactions, we introduced pure repulsive interactions between the polymer beads in the form of WCA potential. This method does not include the direct charge interaction between the polymers, but the effect of the range of interaction ($r_c$) is accounted for by changing the radius of the beads ($r_0^b$). The results from the simulations are shown in Fig. 4. As the range of interaction increases, the curvature of the membrane surface increases, resulting in stable tubular structures, shown in the insets to Fig. 4.

Figure 4.

Effect of long-range interactions. Mean curvature of membrane patch as a function of range of interaction $r_0^b$ at different excess areas when $N_p=200$ and $l_p=100$ nm. Snapshots correspond to $A_{\text{ex}}=54\text{\%}$.

For a single charged polymer, such as HA, the electrostatic repulsion between similar charges tend to rigidify the polymer (49). Although an increase in stiffness is expected to reduce the induced curvature (see Fig. 3 B), we observe an increase in $⟨H_{\text{mean}}⟩$ as we increase the range of interaction in Fig. 4. This observation at the high surface coverage of polymers could be due to the direct long-range interactions between polymers being much stronger than the intrapolymer interactions.

Tubulation of the membrane due to actin-like networks

The dynamics of cytoskeletal polymers drive the membrane morphological changes in cellular processes, such as cell motility and polarization. Polymerization and depolymerization of actin filaments at the forefront of the cell determine the direction of cell migration and speed. While it is well known that the actin filaments deform the membrane by pushing it forward via polymerization reactions, some of the recent studies have shown that they are also capable of generating inward tubules in cell membranes (50). Here, we extend the membrane polymer model to study the effect of actin filament assembly and organization on the cytosolic side of the plasma membrane. Our objective here is to understand the effect of crowding due to a static actin network at the membrane surface.

The actin filaments are attached to the cell surface via actin-related proteins, such as formins and NWASP, which are also responsible for forming linear and branched actin filaments. In our model, actin filaments are modeled as membrane-anchored, semiflexible polymers with ${\xi }_p^a=10\mu m$. We study the effect of a static network by varying the number of branched polymers and ignoring the polymerization and depolymerization reactions. The number and length of the parent linear polymers are chosen to be $N_p^a=200$ and $l_p^a=100$ nm for all cases. We quantified the number and length of the branches by computing the branching fraction ${\nu }_{br}$, defined as the ratio of monomers in branched filaments to that in the linear filaments. The membrane shapes and filament-induced curvature is shown in Fig. 5. Linear polymers (${\nu }_{br}=0$) at $N_p^a=200$ does not deform membrane significantly, which is in agreement with a previous observation with changing bending stiffness (Fig. 3 B). As the branching increases, the membrane deforms and form a tubular shape at ${\nu }_{br}=3$. The membrane tubulation is only observed at higher membrane excess areas, A_ex = 33 and $53\text{\%}$, in agreement with the experimental observations (50). While our results clearly indicate that actin crowding is sufficient to induce membrane tubulation, it should be noted that the effects of actin polymerization can also enhance the membrane remodeling due to crowding.

Figure 5.

A membrane patch decorated with a branched actin network on the cytosolic side. (A) Snapshots from simulations at $A_{\text{ex}}=33\text{\%}$ as the branching fraction ${\nu }_{br}$ is varied from 0 to 3. The number of polymers was taken to $N_p^a=200$ with $l_p^a=100$ nm. (B) Mean curvature of membrane patch as a function of ${\nu }_{br}$ from three different excess areas.

Limitations of the model

The current model has been parameterized based on the macroscale physical properties of cell membranes and biopolymers, such as effective membrane bending rigidity, polymer density, and persistence length. It does not, in its current form, include any biochemical information that is reflective of the underlying chemical heterogeneity. For instance, details on the composition of the lipids and proteins in the cell membrane and the role of their dynamics in modulating the membrane bending have not been taken into account. We solely investigated the contribution of anchored polymers on the membrane structure, and have ignored any contributions to the local curvature from the lipid shape asymmetry, curvature remodeling effects of integral and peripheral membrane proteins, and lipid-protein domain-induced deformations. However, the curvature-inducing and -sensing properties of membrane proteins and lipid-proteins domains have been widely studied and can be readily incorporated to the model as described previously (51,52). Although the model can account for changes in size and persistence length of the polymers, the heterogeneity of membrane-anchored polymers in the glycocalyx and their complex chemical interactions have not been included. Extensive experimental data would be required to incorporate such features into the model. The type of anchoring proteins itself could be a factor in determining the local membrane shape (38). The model has been simplified to capture the trends in membrane shape with long-range electrostatic interactions, but the direct charge interactions between the components can have consequences, such as ion condensation and domain formation, which in turn can affect the curvature of the surface. In the case of actin-like networks, the filament polymerization-depolymerization dynamics, type of anchoring proteins, and actin network-associated molecular motors could largely determine the force applied on the membrane and promote or counteract the crowding-induced deformations.

Conclusion

We have presented an explicit computational model for a cell membrane decorated with biopolymers and determined the effect of membrane tension, polymer density, and organization in crowding-induced cell membrane remodeling. Our results show that a threshold polymer coverage is required to deform the membrane, and this threshold is dependent on the available membrane excess area. Above the threshold density the curvature and the membrane shapes generated by the crowding of biopolymers depend on polymer density and membrane tension. The curvatures of the membrane shapes generated are in the same range as the submicrometer structures observed experimentally (22). We also found that the length, addition of side chains, and flexibility of polymers can increase the crowding-induced curvature of the anchored membrane.

The model also captures the long-range interactions between the polymers. The simulation results obtained by changing the range of interaction indicate that a change in the ionic strength of solvent around a charged polymer network on the membrane surface can change the morphology of cell membranes. Finally, we extended the model to study the effect of membrane-associated actin cytoskeletal mesh. Although the curvature generated by linear actin filaments is much smaller, the formation of a branched actin network is shown to generate inward tubules on the membrane.

The current model is intended to capture the large-scale membrane deformations due to polymer crowding alone. Cell membranes, in general, are more complex in composition and dynamics. The heterogeneity in the constituent lipids and proteins itself is known to cause deviations in membrane shape. While the present model is amenable to include some of the major membrane remodeling processes, such as curvature induction by proteins and lipid-protein domains, incorporating the explicit chemical interactions between membrane constituents and anchored biopolymers would require a diverse set of quantitative information from experiments for parametrization. In addition, we expect any explicit intrapolymer and polymer-membrane interactions could enhance or oppose the entropic contributions and could aid or suppress crowding-mediated membrane morphologies.

Many different types of soft and solid tumors have now been analyzed from a mechanogenomics perspective, and how the variations in stiffness of the tumors connect to the cancer hallmarks is actively being investigated (53). Biophysical signaling is emerging as a critical area of relevance in understanding cancer progression. Recent studies have shown that the glycocalyx’s structure and organization are altered in most cancer cells, which can result in modified binding characteristics and downstream signaling (43,54). Changes in the bulk physical properties of glycocalyx can affect the cellular morphologies through multiple mechanisms that cause cell membrane remodeling, which can alter the formation of extracellular vesicles and exosomes that regulate cell-cell communications. Two relevant mechanisms of membrane shape regulation are integrin-extracellular matrix binding and cell surface polymer crowding. Our study here helps to understand the role of individual properties of glycocalyx polymers in controlling the membrane shapes.

Perturbation in tissue mechanics in solid tumors informs the mechanics of the cell by resetting the cortical tension (55). By defining a tension dependence, we believe we have described mechanistically the connection between extracellular and intracellular vesicles and trafficking and the mechanobiology of cancer. Other mechanisms of vesicle biogenesis through curvature-inducing proteins (56) and through cortical pinning of the plasma membrane-induced vesiculation (28) have both been shown to be mechanosensitive in earlier works and important in intracellular traffic, including the regulation of a class of extracellular vesicles called exosomes. This work describes the biophysics and mechanobiology of crowding-induced vesiculation that may play a major role in another class of extracellular vesicles called microvesicles. In the future, this study will be extended to understand the effect of glycosylation in mechanical signaling in cells.

Author contributions

S.K.K. and R.R. designed the research. S.K.K. carried out all simulations. S.K.K. and R.R. analyzed the data and wrote the article.

Declaration of interests

The authors declare no competing interests.

Editor: Marta Filizola.