Introduction

Biomolecular condensates (BC) formed by liquid-liquid phase separation (LLPS) have been recognized to provide spatiotemporal organization of various processes inside biological cells1,2,3,4. This recognition that the cytosol is reversibly organized into heterogeneous regions to carry out specific functions recalls an earlier discovery that the cell membrane accommodates liquid-ordered nanodomains, which perform signaling and trafficking functions by condensating biomolecules5,6,7,8,9. However, in contrast to the vast attention given in recent years to BCs inside the cytosol and nucleus1,2,3,4, BCs at the cell membrane have not been studied to the same extent so far.

Recent research provides increasingly more interesting examples of BCs that form and function at the cell membrane. In particular, various transmembrane proteins together with their cytoplasmic binding partners have been found to condensate into (sub)micrometer-size fluid clusters10,11,12,13,14,15,16. LLPS has been implicated in the formation and function of dynamic clusters of diverse membrane receptors, including immune receptors, cell adhesion receptors, Wnt (Wingless/Int-1) receptors, and glycosylated receptors17. More recently, it has been shown in microscopy experiments that BCs and lipid nanodomains are coupled during T cell activation, indicating that LLPS of peripheral membrane proteins can regulate the functional organization of lipids in the cell membrane and, conversely, that local segregation of lipids in the cell membrane can facilitate protein condensation15. In addition, it has been discovered that BCs are capable of remodeling membranes in various biological processes18,19,20,21,22. Taken together, membrane BCs—possibly engaging lipid nanodomains—seem to be involved in lots of cellular processes, which are only beginning to be studied and understood. Therefore, novel approaches to control condensation of biomacromolecules at the cell membrane will be of great significance for understanding and regulating diverse cellular activities.

Here, we study condensation of membrane receptors and lipid nanodomains that is stimulated by cell adhesion to a supported lipid bilayer (SLB), see Fig. 1a. The cell membrane contains receptors (green) that can bind their ligands (violet) anchored in the SLB. The specific receptor-ligand binding mediates the adhesion of the cell membrane to the SLB. The receptor and ligand molecules are mobile and exhibit affinity for association with lipid nanodomains (orange) present both in the cell membrane and in the SLB. The SLB has a smooth profile of height h and width w. We introduce a statistical mechanical model for the system illustrated in Fig. 1a and solve this model using Monte Carlo simulations and mean-field theory. We find that the specific adhesion of the cell membrane to the SLB can cause condensation of the receptor proteins associated with lipid nanodomains even if such condensates do not form in free, non-adhered cell membranes. Further, we discover that the nanoscale topography of the SLB can facilitate the condensation process, which can be attributed to (1) an interplay between the receptor-ligand binding and membrane bending, (2) an effective attraction between the receptor-ligand complexes induced by thermal undulations of the cell membrane, and (3) a difference in bending moduli between the membrane matrix and lipid nanodomains. We quantify how these three factors influence the condensation point. Our findings indicate that SLBs with nanoscale topography can be introduced as physical stimuli for tuning condensation of membrane adhesion receptors and lipid nanodomains, and thereby directing cellular activities.

Fig. 1: Nanoscale topography of the supported lipid bilayer (SLB) affects the condensation in the adhesion system.
figure 1

a Cartoon of the system under study. The receptor and ligand molecules are shown in green and violet, respectively, and lipid nanodomains in orange. The height and width of the SLB profile are denoted by h and w, respectively. b Heat capacity per membrane patch, Cv, as a function of contact energy between lipid-nanodomain patches, U, obtained from Monte Carlo (MC) simulations of systems with planar (pSLB), corrugated (cSLB) and egg-carton (eSLB) profiles of the SLB. The vertical dashed lines indicate the heat capacity maxima. Here, the area concentration of adhesion proteins is cp = 1000/μm2. In the cases of the cSLB and eSLB, h = 15 nm and w = 200 nm. ce Snapshots from the MC simulations with the pSLB (c), cSLB (d) and eSLB (e). The color code is as in (a). The snapshots in the middle correspond to U = U* at which the heat capacity attains the maximum value, as indicated by the vertical dashed lines in (b). The snapshots on the left-hand side correspond to U < U* and show the simulation system in a homogeneous fluid phase. The snapshots on the right-had side correspond U > U* and illustrate phase separation in the simulation system, where fluid clusters of receptors associated with lipid nanodomains in the cell membrane are in register with fluid clusters of ligands associated with lipid nanodomains in the SLB.

Methods

Mesoscale model of cell membrane adhesion

Cell membrane adhesion involves multiscale phenomena, ranging from biomolecular interactions at the nanoscale all the way up to membrane deformations and phase separation at the microscale23. To deal with the resulting complexity, we adapt a mesoscale model for adhesion of multi-component membranes24,25,26,27,28 and incorporate the essential physics of the system under study, i.e.: (1) elastic deformations of the cell membrane, (2) two-dimensional diffusion of the receptors and ligands, (3) interactions of the receptors and ligands with lipid nanodomains, (4) diffusion, fusion and fission of lipid nanodomains, (5) the specific receptor-ligand binding, and (6) the nanoscale profile of the SLB. More specifically, we consider a cell membrane in contact with a SLB as illustrated in Fig. 1a. Both the membrane and the SLB are on average parallel to a horizontal reference plane. The membrane contains receptors that can bind to their ligands anchored in the SLB. The specific receptor-ligand binding mediates the adhesion of the membrane to the SLB. The receptors and ligands are mobile and have affinity for association with lipid nanodomains present in the cell membrane and in the SLB. Lipid nanodomains diffuse, change their shapes, fuse and break apart. On the one hand, the fusion of lipid nanodomains decreases the extent of the hydrophobic mismatch between the lipid nanodomains and the membrane matrix. On the other hand, the mixing entropy acts to separate lipid nanodomains.

We adapt the Helfrich model for membrane elasticity29, which implies that the cell membrane and the SLB are represented here as two-dimensional surfaces. We assume that the SLB adheres to the substrate so strongly that its shape follows precisely the profile of the substrate surface. We consider three types of the SLB profile: a planar profile with zSLB(x, y) = 0 (Fig. 1c), a corrugated profile with \({z}_{{{{{{{{\rm{SLB}}}}}}}}}(x,y)=h\cos (2\pi x/w)\) (Fig. 1d), and an egg-carton profile with \({z}_{{{{{{{{\rm{SLB}}}}}}}}}(x,y)=h\cos (2\pi x/w)\cos (2\pi y/w)\) (Fig. 1e)28. Here, zSLB(x, y) denotes the height of the SLB above the reference plane. The parameters h and w are in the range from 5 to 15 nm and from 150 to 300 nm, respectively.

To make use of statistical mechanics methods, it is convenient to impose a square lattice on the reference plane and, thus, discretize the cell membrane and the SLB into quadratic patches30. Each lattice site corresponds to one patch of the cell membrane and one patch of the SLB. Here, we take the lattice constant a = 5 nm to capture the complete spectrum of bending deformations of fluid membranes25,31. We label the lattice sites by index i = (ix, iy) which is a set of two integer numbers that specify the (x, y) coordinates in the reference plane. We denote by \({z}_{i}^{{{{{{{{\rm{o}}}}}}}}}\) the height of the cell membrane (o = +) or of the SLB (o =−) above the reference plane at lattice site i. The total bending energy of the cell membrane and SLB reads28,32,33:

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{be}}}}}}}}}={\sum}_{{{{{{{{\rm{o}}}}}}}}=+,-} {\sum}_{i}\frac{{\kappa }_{i}^{{{{{{{{\rm{o}}}}}}}}}}{2{a}^{2}}{\left({\Delta }_{{{{{{{{\rm{d}}}}}}}}}{z}_{i}^{{{{{{{{\rm{o}}}}}}}}}\right)}^{2}$$
(1)

where the sum runs over all lattice sites. \({\kappa }_{i}^{{\rm o}}\) is the membrane bending rigidity at site i, and \({\Delta }_{{{{{{{{\rm{d}}}}}}}}}{z}_{i}^{{{{{{{{\rm{o}}}}}}}}}\) is the discretized Laplacian of the height field \(\{{z}_{i}^{{{{{{{{\rm{o}}}}}}}}}\}\), which is proportional to the membrane’s local mean curvature24. In major parts of this study we assume for simplicity that κi = κ = 25 kBT25, where kB and T denote the Boltzmann constant and room temperature, respectively. However, to investigate possible effects of the larger rigidity of lipid nanodomains, we take κi = κr if the membrane at site i is within a lipid nanodomain, and κi = κm otherwise.

By analogy to lattice-gas models, we assume that any patch of the cell membrane can accommodate only one receptor, and a single patch of the SLB can be occupied by only one ligand. Thus, the distribution of receptors in the cell membrane can be described by a set of binary variables \({m}_{i}^{+}\). The value \({m}_{i}^{+}=1\) indicates that a receptor is present at site i, whereas \({m}_{i}^{+}=0\) indicates that site i contains no receptors. Likewise, the distribution of ligands in the SLB is described by a set of variables \({m}_{i}^{-}\), with \({m}_{i}^{-}=1\) and \({m}_{i}^{-}=0\) indicating the presence and absence of a ligand on lattice site i, respectively.

A receptor-ligand complex forms if (1) one receptor and one ligand occupy the same lattice site, i.e., \({m}_{i}^{+}={m}_{i}^{-}=1\), and (2) the local separation between the membrane and the SLB at this site, denoted here by li, is within the receptor-ligand binding range, i.e., \({l}_{{{{{{{{\rm{c}}}}}}}}}-\frac{{l}_{{{{{{{{\rm{b}}}}}}}}}}{2} \, < \, {l}_{i} \, < \, {l}_{{{{{{{{\rm{c}}}}}}}}}+\frac{{l}_{{{{{{{{\rm{b}}}}}}}}}}{2}\), where lc is the length of the extracellular domains of the receptor-ligand complex and lb is the width of the receptor-ligand binding potential. The receptor-ligand interaction energy takes the following form25,27,34:

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{R-L}}}}}}}}}={\sum}_{i}{m}_{i}^{+}{m}_{i}^{-}{V}_{{{{{{{{\rm{b}}}}}}}}}({l}_{i})$$
(2)

where

$${V}_{{{{{{{{\rm{b}}}}}}}}}({l}_{i})=-{U}_{{{{{{{{\rm{b}}}}}}}}}\,\theta \left(\frac{{l}_{{{{{{{{\rm{b}}}}}}}}}}{2}-| {l}_{i}-{l}_{{{{{{{{\rm{c}}}}}}}}}| \right)$$
(3)

is a square-well potential with depth Ub, width lb and centered at li = lc. The Heaviside’s step function θ(  ) is used here to describe the prerequisite for the formation of a receptor-ligand complex. In this study we use typical values of the potential parameters: Ub = 6 kBT, lc = 15 nm and lb = 1 nm26,28. Here, the binding energy Ub can be estimated from the relation \({K}_{{{{{{{{\rm{D}}}}}}}}}={a}^{2}{l}_{{{{{{{{\rm{b}}}}}}}}}{e}^{{U}_{{{{{{{{\rm{b}}}}}}}}}/{k}_{{{{{{{{\rm{B}}}}}}}}}T}\), which is in a range of about 5–10 kBT for the experimentally measured dissociation constant KD35,36,37.

Without loss of generality, we assume that the area concentration of receptors is equal to the area concentration of ligands, i.e.:

$${c}_{{{{{{{{\rm{p}}}}}}}}}=\frac{{\sum }_{i}{m}_{i}^{+}}{N{a}^{2}}=\frac{{\sum }_{i}{m}_{i}^{-}}{N{a}^{2}}$$
(4)

where N is the total number of lattice sites. In this study, cp is varied up to 2000 μm−238,39,40,41.

We describe the spatial distribution of lipid nanodomains in the membrane by a set of binary variables \({n}_{i}^{+}\). The value \({n}_{i}^{+}=1\) indicates that the membrane patch at site i is within a lipid nanodomain. The value \({n}_{i}^{+}=0\) means that the membrane patch at site i is part of the membrane matrix. Likewise, the spatial distribution of lipid nanodomains in the SLB is described by a set of variables \({n}_{i}^{-}\) with values \({n}_{i}^{-}=0\) or \({n}_{i}^{-}=1\). The propensity of lipid nanodomains to fuse can be captured by the following energy term42,43:

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{r-r}}}}}}}}}=-U{\sum}_{\langle i,j\rangle }\left({n}_{i}^{+}{n}_{j}^{+}+{n}_{i}^{-}{n}_{j}^{-}\right)$$
(5)

where the sum runs over all pairs of nearest-neighbor lattice sites, and U > 0 is a contact energy between nearest-neighbor lipid-nanodomain patches. U/a can be interpreted as energy per unit length of the interface between the membrane matrix and lipid nanodomains.

We also assume that the number of lipid-nanodomain patches in the membrane, \({\sum }_{i}{n}_{i}^{+}\), is equal to the number of lipid-nanodomain patches in the SLB, \({\sum }_{i}{n}_{i}^{-}\). Therefore, the cell membrane has the same area fraction xr of lipid nanodomains as the SLB, i.e.:

$${x}_{{{{{{{{\rm{r}}}}}}}}}=\frac{{\sum }_{i}{n}_{i}^{+}}{N}=\frac{{\sum }_{i}{n}_{i}^{-}}{N}$$
(6)

We assume here that xr takes values between 0.1 and 0.344,45.

The affinity of the receptors and ligands for association with lipid nanodomains can be described by the following energy term27,46:

$${{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{r-p}}}}}}}}}=-{U}_{{{{{{{{\rm{a}}}}}}}}} {\sum}_{i}\left({n}_{i}^{+}{m}_{i}^{+}+{n}_{i}^{-}{m}_{i}^{-}\right)$$
(7)

where Ua > 0 is the energy of association of a membrane protein (receptor or ligand) with a lipid nanodomain. To obtain typical area concentrations of membrane proteins within lipid nanodomains, ranging from around 103 to 104 μm−2 as determined experimentally47, we take Ua = 3 kBT unless specified otherwise.

Simulations and theory

The energy terms introduced above add up to the configurational energy, or Hamiltonian, \({{{{{{{\mathcal{H}}}}}}}}={{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{be}}}}}}}}}+{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{R-L}}}}}}}}}+{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{r-r}}}}}}}}}+{{{{{{{{\mathcal{H}}}}}}}}}_{{{{{{{{\rm{r-p}}}}}}}}}\). We use Monte Carlo (MC) simulations and numerical calculations within a mean field (MF) approximation to explore the phase behavior and equilibrium properties of the system under study. Details on the MC simulations and MF theory are given in the Supplementary Methods.

Results

Effect of SLB nanoscale topology

We performed MC simulations of the mesoscale model for membranes adhered to SLBs with planar (pSLB), corrugated (cSLB) and egg-carton (eSLB) profiles (for details see section “Monte Carlo simulations” in the Supplementary Methods). We set the area concentration of adhesion proteins cp = 1000/μm2. The height and width of the cSLB and eSLB profile were h = 15 nm and w = 200 nm. We performed the MC simulations with different values of the contact energy between lipid-nanodomain patches, U, and measured the heat capacity per membrane patch, Cv. We found that the shape of the SLB affects the dependence of Cv on U, see Fig. 1b. This observation suggests that the phase behavior of the simulation system is affected by the nanoscale topography of the SLB.

Snapshots from the MC simulations are shown in Fig. 1c–e, where the color code is as in Fig. 1a, i.e., the membrane patches marked in orange represent lipid nanodomains, and locations of the receptors and ligands are indicated in green and violet, respectively. The snapshots in the center correspond to U = U* at which the phase transition occurs (U* = 1.36 kBT for pSLB, 1.18 kBT for cSLB, and 1.06 kBT for eSLB, see Fig. 1b). At the phase transition, large clusters form within the system, as indicated by the distribution of cluster sizes (Supplementary Fig. 1a–c). It is evident that the heat capacity per membrane patch attains its maximum value at the phase transition point. Therefore, we identify the phase transition points from the peaks in the plots of heat capacity against contact energy between lipid-nanodomain patches. This strategy for determining phase behavior has been commonly employed in studies involving the Ising model48,49. The snapshots on the left- and right-hand side correspond to U < U* and U > U*, respectively. At U < U*, the adhesion proteins and lipid nanodomains are distributed more-or-less uniformly both in the membrane and in the SLB. The system is in a homogeneous fluid phase. At U > U*, a cluster of lipid nanodomain-associated receptors in the membrane is observed to be in register with a cluster of lipid nanodomain-associated ligands in the SLB. Although the two clusters are held together via receptor-ligand complexes, they change their shapes, sizes and location in the course of the MC simulations. Thus, at U > U* the system is fluid and phase separated, and the transition at U = U* can be regarded as condensation. Consistently, the area of the largest lipid cluster increases abruptly in the vicinity of the phase transition point (Supplementary Fig. 1d).

To quantify how the adhesion of the membrane to the SLB affects the condensation, we performed MC simulations in a broad range of parameters U and cp. We measured Cv as a function of U and cp, and determined U* from the maxima of Cv versus U curves (analogously as in Fig. 1b). Figure 2a, b shows the dependence of U* on cp for systems with cSLBs and eSLBs, respectively. The data points in blue, red and green correspond to different values of h and w. The data points in black provide a reference line as they represent the results of simulations with the pSLB.

Fig. 2: Phase diagram of the adhesion system with different profiles of the SLB.
figure 2

The lipid-nanodomain contact energy at which the condensation occurs, U*, is plotted as a function of the area concentration of adhesion proteins, cp, for systems with cSLBs and eSLBS obtained from the Monte Carlo (MC) simulations (a, b) and mean field (MF) theory (c, d). The curves in red, blue and green correspond to different heights h and widths w of the SLB, as indicated in the insets. The curves in black correspond to systems with the pSLB. The acronyms SLB, cSLB, eSLB, and pSLB have the same meaning as in Fig. 1.

At cp = 0, i.e., in the absence of the adhesion proteins in the simulation system, the membrane does not interact with the SLB. Then the condensation occurs at U* ≈ 2 kBT and results solely from the short-range attraction between lipid nanodomains. At cp > 0, i.e., in the presence of the receptors and ligands, the condensation occurs at U* < 2 kBT and, thus, is facilitated by the adhesion of the membrane to the SLB. Here, U* is found to decrease with cp, which means that the stronger the adhesion gets, the smaller the contact energy U is needed to bring about the condensation of the receptor-ligand complexes and lipid nanodomains. The decrease of U* with cp is larger in the cases of cSLBs and eSLBs than in the case of pSLB, implying that the shape of the SLB has an impact on the condensation process.

We also performed numerical calculations within the MF theory (for details see section “Mean field approximation” in the Supplementary Methods) and determined U* versus cp curves for systems with cSLBs and eSLBs, see Fig. 2c, d. The MF and MC results are in good qualitative agreement, whereas some quantitative discrepancies originate from the MF approximation, which assumes small fluctuations in the local density of lipid nanodomains.

The MF theory allows us to efficiently explore how the condensation point depends on model parameters. Examples of U* versus cp curves for different amounts of lipid nanodomains and different widths and heights of the SLB are shown in Supplementary Fig. 2. These results imply that a proper choice of the SLB geometry and adhesion protein concentration can cause a decrease in U* by up to 40% compared to the value of U* in free, non-adhered membranes.

Interplay between receptor-ligand binding and membrane bending

The influence of the SLB topography on the condensation point results primarily from an interplay between the receptor-ligand binding and membrane bending. Figure 3a, b shows average profiles of the membrane adhered to eSLBs with h = 5 and 15 nm, respectively. These profiles were generated in MC simulations with U = 0.88 kBT, cp = 1000/μm2 and w = 200 nm by simply averaging the local positions of each membrane patch. The spatial distributions of the receptors and ligands are marked in hues of green and violet, respectively. It can be clearly seen in Fig. 3a that the membrane follows the shape of the eSLB with h = 5 nm. In this case, the energy cost of membrane bending is overcome by the receptor-ligand binding. In contrast, Fig. 3b shows that the membrane does not follow the shape of the eSLB with h = 15 nm. In this case, the membrane is significantly bent only at one spot, where a cluster of receptor-ligand complexes keeps it at a distance l ≈ lc = 15 nm from the SLB. Analogous phenomena have also been observed for the adhesion system with cSLB (Supplementary Fig. 5a, b). Clearly, formation of such clusters can facilitate the condensation of receptor-ligand complexes.

Fig. 3: Interplay between the receptor-ligand binding and membrane bending.
figure 3

The effect of SLB topography on the clustering of adhesion proteins and lipid nanodomains results primarily from this interplay. ab Simulation snapshots showing average profiles of the membrane adhered to eSLBs with (a) h = 5 nm and (b) h = 15 nm. The spatial distributions of receptors and ligands are marked in hues of green and violet, respectively. c Average bending energy Ebe and root-mean-square deviation (RMSD) of the membrane adhered to the eSLB. Both Ebe and RMSD are computed with respect to the eSLB profile and shown as a function of h. If the membrane has exactly the same shape as the eSLB then RMSD = 0 and Ebe = Ebe,SLB. Here, Ebe,SLB denotes the bending energy of SLB. d Pair correlation functions of the receptor-ligand complexes, g(r), obtained for systems with the pSLB (filled circles) and with eSLBs (open circles) of heights h = 5 nm (circles in pink), h = 10 nm (circles in blue) and h = 15 nm (circles in green). ac The results were obtained in MC simulations with U = 0.88 kBT, cp = 1000/μm2, w = 200 nm and 5 × 106 MC steps for averaging. The acronyms have the same meaning as in Fig. 2.

In order to quantify to what extent the membrane bends to follow the eSLB profile, we performed MC simulations with eSLBs of different heights, ranging from 5 to 15 nm, and computed the average bending energy Ebe of the membrane and the bending energy Ebe,SLB of the SLB. We also computed the average root-mean-square deviation (RMSD) between the eSLB and membrane shapes. Figure 3c shows that the ratio of Ebe and Ebe,SLB decreases monotonically with h, and the RMSD increases monotonically with h. These results indicate that the membrane shape follows to some extent the SLB profile if h is sufficiently small, but for larger values of h, the membrane is on average almost flat compared to the eSLB. Analogous conclusions can be drawn from MC simulations of systems with cSLBs of height ranging from 5 to 15 nm, see Supplementary Fig. 5c.

The competition between the receptor-ligand binding and membrane bending impacts the spatial distribution of the adhesion proteins, see Fig. 3a, b. To quantify this phenomenon, we computed the pair correlation function g(r) of the receptor-ligand complexes, see Fig. 3d. Here, r denotes the distance between a pair of complexes. At small distances, g(r) > 1, indicating an effective attraction between the complexes. This effective attraction can be quantified by the potential of mean force, \(w(r)=-{k}_{{{{{{{{\rm{B}}}}}}}}}T\ln g(r)\), and turns out to be stronger in systems with the eSLB (open circles) than with the pSLB (filled circles). Moreover, both the magnitude and the range of this effective attraction become larger as h is increased from 5 nm (open circles in pink) to 10 nm (open circles in blue) to 15 nm (open circles in green). As w(r) becomes deeper and broader, smaller contact energy U is needed to drive the condensation of the receptor-ligand complexes associated with lipid nanodomains. Analogous conclusions can be drawn from MC simulations with cSLBs, see Supplementary Fig. 5d. Taken together, these conclusions explain why the decrease of U* with cp is steeper for systems with cSLBs and eSLBs than with the pSLB, as in the U* versus cp curves in Fig. 2.

It is worthy to mention the role of lipid nanodomains in the aforementioned competition introduced by SLBs with nanoscale topography. We computed the average receptor-ligand binding energy and the average membrane bending energy in simulations of systems with lipid nanodomains (as depicted in Fig. 1) and without lipid nanodomains (which served as control systems). The binding and bending energies rescaled by their counterparts in the control systems, \({\tilde{E}}_{{{{{{{{\rm{bi}}}}}}}}}\) and \({\tilde{E}}_{{{{{{{{\rm{be}}}}}}}}}\), are plotted against U in Fig. 4. \({\tilde{E}}_{{{{{{{{\rm{bi}}}}}}}}}\) increases with U and \({\tilde{E}}_{{{{{{{{\rm{be}}}}}}}}}\) decreases with U, implying that the clustering of lipid nanodomains facilitates the receptor-ligand binding and alleviates the membrane bending. In the case of pSLB, \({\tilde{E}}_{{{{{{{{\rm{be}}}}}}}}}=1\) independent of U, since the membrane adhered to the pSLB is on average flat. In the cases of cSLB and eSLB, \({\tilde{E}}_{{{{{{{{\rm{bi}}}}}}}}} < 1\) in a range of small U-values, meaning that the presence of lipid nanodomains actually weakens the binding of adhesion proteins, contrary to our earlier observation of enhanced binding of membrane-anchored receptors and ligands associated with “static nanodomains” of fixed shape and size27. This weakening arises from the lipid nanodomain-induced dispersion of proteins as shown in Supplementary Fig. 3, where the percentage of ligands residing in the SLB’s upper half decreases from 0.837 (left panel) to 0.578 (right panel) in Supplementary Fig. 3b and from 0.726 to 0.562 in Supplementary Fig. 3c. This is also consistent with \({\tilde{E}}_{{{{{{{{\rm{be}}}}}}}}} > 1\) since the membrane needs to follow the SLB’s profile for its adhesion receptors to bind the ligands more dispersed on cSLB or eSLB than in the control systems without lipid nanodomains. Additional simulations of the systems with different direct protein-protein attraction reveal that the condensation, in the absence of lipid nanodomains, is even harder in the case of cSLB or eSLB than in the case of pSLB (see Supplementary Fig. 4), because the local clustering of proteins disfavors the formation of protein condensates within the membrane (see the second subfigure from left to right in Supplementary Fig. 4b, c). The nanoscale topography of the SLBs thus causes a delicate interplay between receptor-ligand binding and membrane bending in the presence of lipid nanodomains.

Fig. 4: Lipid nanodomains play a role in the competition between the receptor-ligand binding and membrane bending in the systems with cSLBs or eSLBs.
figure 4

The average (a) receptor-ligand binding energy and (b) membrane bending energy for systems as in Fig. 1 with different U-values are rescaled by their counterparts in the corresponding control systems that have no lipid nanodomains. The acronyms have the same meaning as in Fig. 2.

Role of membrane thermal undulations

Thermal undulations of the cell membrane induce an effective, membrane-mediated attraction between the receptor-ligand complexes34,50,51,52. The physical picture is that the fluctuating cell membrane can gain more conformational entropy when the receptor-ligand complexes, which constrain local membrane separations, are in close proximity than when they are further apart. This attraction facilitates clustering of the receptor-ligand complexes and, consequently, can contribute to the condensation of the adhesion proteins associated with lipid nanodomains. To quantify this effect, we simulated additional control systems with rigid membranes, i.e., membranes undeformable to thermal excitations, where the fluctuation-induced, membrane-mediated attraction between the receptor-ligand complexes is absent. For a given set of model parameter values, the shape of the rigid membrane was taken to be the average profile of the flexible membrane with the bending modulus κ = 25 kBT. As in the case of Fig. 3a, b, these average profiles were obtained in MC simulations by averaging the local positions of each membrane patch. Figure 5a shows the resulting U* versus cp curves for the rigid (open symbols) and flexible (filled symbols) membranes adhered to the cSLB (circles) or eSLB (squares) with h = 10 nm and w = 200 nm. Importantly, the contact energy at which the condensation occurs, U*, is found to be clearly smaller for the flexible membranes than for the rigid membranes, independent of the SLB profile. This result confirms that the membrane-mediated attraction between the receptor-ligand complexes facilitates the condensation of the adhesion proteins associated lipid nanodomains. However, when a sufficient amount of receptors and ligands are bound at large values of cp, this attraction is weakened because membrane fluctuations are significantly suppressed. Then U* can actually gently increase with cp, as can be seen in Fig. 2.

Fig. 5: Thermal undulations and bending rigidity contrast of the cell membrane contribute to the condensation process.
figure 5

The nanoscale topography of the SLB amplifies the condensation process due to (a) an effective lateral attraction between the receptor-ligand complexes induced by thermal undulations of the membrane, and (b) a bending rigidity contrast between the membrane matrix and lipid nanodomains. a The lipid-nanodomain contact energy at which the condensation occurs, U*, as a function of the area concentration of adhesion proteins, cp, for the flexible (filled symbols) and rigid (open symbols) membranes adhered to the cSLB (circles) or eSLB (squares) with the profile height h = 10 nm and width w = 200 nm. The rigid membranes do not undergo thermally-excited shape fluctuations. b Relative shift of the condensation point \(({U}^{* }-{U}_{{{{{{{{\rm{pSLB}}}}}}}}}^{* })/{U}_{{{{{{{{\rm{pSLB}}}}}}}}}^{* }\) versus the bending rigidity ratio κr/κm. Here, κm and κr the bending moduli of the membrane matrix and lipid nanodomains, respectively, and \({U}_{{{{{{{{\rm{pSLB}}}}}}}}}^{* }\) denotes the lipid-nanodomain contact energy at which the condensation occurs in the membrane adhered to the pSLB. The acronyms have the same meaning as in Fig. 2.

If the planar substrate supporting the lipid bilayer is removed, fluctuations in the distance between the membrane and the bilayer are amplified, leading to an enhanced attraction between receptor-ligand complexes and, simultaneously, to a smaller amount of the complexes34,50,51,53. These two factors together may either promote or hinder the condensation compared to the case of pSLB. In contrast to the case of free lipid bilayer, the corrugated or egg-cartoon SLB provides a much easier way of tuning the condensation of the proteins.

Contribution from bending rigidity contrast between membrane matrix and nanodomains

Since lipid nanodomains are liquid-ordered, they are more rigid than the membrane matrix54,55,56. To examine whether this rigidity difference could further amplify the impact of the SLB topography on the condensation of the receptor-ligand complexes and lipid nanodomains, we performed simulations with different bending moduli κm and κr for the membrane matrix and lipid nanodomains. Results of these simulations with cp = 2000/μm2 and κm = 25 kBT are presented in Fig. 5b, where the shift of the condensation point, \(({U}^{* }-{U}_{{{{{{{{\rm{pSLB}}}}}}}}}^{* })/{U}_{{{{{{{{\rm{pSLB}}}}}}}}}^{* }\), is shown as a function of κr/κm for systems with cSLBs (filled circles) and eSLBs (open circles) of different heights h and widths w. Here, \({U}_{{{{{{{{\rm{pSLB}}}}}}}}}^{* }\) denotes the contact energy U at which the condensation occurs in the membrane adhered to the pSLB. We find that in all of the simulation systems \({U}^{* }-{U}_{{{{{{{{\rm{pSLB}}}}}}}}}^{* }\) is negative and decreases monotonically as κr/κm is gradually increased from 1 to 3, which means that the bending rigidity contrast between the membrane matrix and lipid nanodomains has a larger impact on the condensation process in the systems with the cSLBs and eSLBs than with the pSLB. This effect is due to an enhanced coalescence of lipid nanodomains within the cSLBs and eSLBs. Namely, if κr > κm, lipid nanodomains tend to avoid regions with large mean curvature (i.e., valleys and ridges in the cSLB or peaks and pits in the eSLB) to minimize the SLB bending energy57. Then the amount of lipid nanodomains is increased locally in the SLB regions with small mean curvature, which facilitates the coalescence of lipid nanodomains within these regions of the SLB. To support this argument we performed additional simulations of SLBs alone, i.e., systems without the membrane. Results of such simulations with κm = 25 kBT, κr = 75 kBT and different geometries of the SLB are presented in Supplementary Fig. 6. As expected, the area fraction of lipid nanodomains in the peaks and valleys of the SLB is found to decrease monotonically with U (Supplementary Fig. 6a) and the area fraction of lipid nanodomains located halfway between the peaks and valleys is found to increase monotonically with U (Supplementary Fig. 6b). Consistently, the bending energy Ebe,SLB of the SLBs decreases with U (Supplementary Fig. 6d) and this decrease is steeper for the eSLBs than for the cSLBs. Moreover, for a given value of U, the total energy of contacts between lipid nanodomains is found to be lower in the eSLBs and cSLBs than in the pSLB (Supplementary Fig. 6c), which supports our argument that the coalescence of lipid nanodomains is enhanced within the cSLBs and eSLBs. Taken together, the simulation results shown in Supplementary Fig. 6 demonstrate that if κr > κm then the periodic profiles of the cSLBs and eSLBs facilitate the coalescence of lipid nanodomains within the SLB. This curvature effect provides an additional driving force for the condensation of the adhesion proteins associated with lipid nanodomains.

Discussion

Condensation of membrane proteins and assembly of lipid nanodomains are crucial in various physiological processes, such as signal transduction5,6,7,8,9, and have been reported to be involved in many types of diseases, including neurodegeneration and cancer58. Therefore, developing feasible protocols to control reorganization of membrane proteins and lipid nanodomains is of great significance for regulating cellular activities and for opening potential avenues for disease prevention and treatment.

We put forward an effective means to control the coalescence of membrane adhesion proteins and lipid nanodomains via the specific adhesion of cell membranes to SLBs with nanoscale topography. Specifically, we demonstrate that the height h and width w of the SLB periodic profile can be used as parameters to control the coalescence of proteins and lipid nanodomains into mesoscale clusters. In the model parameter range investigated here, an appropriate choice of h and w can bring down the adhesion protein concentration at which the lipid nanodomain coalescence occurs by up to ~40%. Since our computational model is quite general and not limited to specific adhesion proteins or membrane lipid compositions, our results indicate toward generic mechanisms of control of protein and lipid nanodomain coalescence, which can be attributed to (1) the interplay between the receptor-ligand binding and membrane bending (Fig. 3), (2) the effective, fluctuation-induced, membrane-mediated attraction between the receptor-ligand complexes (Fig. 5a), and (3) the difference in the bending rigidity moduli between the membrane matrix and lipid nanodomains (Fig. 5b).

A natural extension of our current study is to further investigate the role of bilayer-substrate binding in the condensation of membrane adhesion proteins and lipid nanodomains. There is growing evidence suggesting that the substrate can influence lipid diffusion, thermal undulations, phase behavior, and mechanical properties, such as bending rigidity, of SLBs, depending on the surface structure and properties of the substrate59,60,61,62,63,64,65. For instance, substrate confinement can significantly restrict the out-of-plane undulations of SLBs59,60. Additionally, substrate patterning has proven effective in regulating the phase behavior and lipid diffusion of SLBs62,64,65. It is important to emphasize that the exact values of lipid diffusion coefficients do not affect the equilibrium phase behavior, as equilibrium quantities are independent of dynamic system properties. In our model, the suppression of SLB undulations can be incorporated by introducing an additional separation-dependent bilayer-substrate binding energy term into the Hamiltonian. The binding-induced variation in bending rigidity can be modeled by selecting appropriate values of \({\kappa }_{i}^{-}\) in Eq.(1). In our present work, we consider a symmetric system where the contact energy between nearest-neighbor lipid-nanodomain patches in both SLB and cell membrane is identical. Exploring a more general system where contact energies differ would be valuable. Our Monte Carlo simulations and mean-field theory remain applicable in these studies.

Stem cells have shown a great potential in the field of tissue engineering and regenerative medicine due to their inherent regenerative capabilities and developmental potency66,67,68,69. An ongoing challenge in clinical trials on stem-cell based therapies is to control the stem cell fate and differentiate them into the desired specific cell types66,70,71. Recent studies demonstrate that lipid nanodomain assembly plays a central role in directing the differentiation and lineage commitment of stem cells72,73,74,75,76,77, thereby indicating a promising target for controlling stem cell differentiation. For example, Jia et al. have studied the neuronal differentiation of human mesenchymal stem cells (hMSCs) at an adaptive liquid interface74. They have found that the clustering of lipid nanodomain is required for directing hMSC differentiation through a signaling mechanism involving focal adhesion kinase (FAK) pathway, and disrupting lipid nanodomain integrity by methyl-β-cyclodextrin inhibits the phosphorylated FAK expression and hMSCs neurogenesis74. Because of the substantial promoting effect of the SLB geometry on adhesion-induced coalescence of lipid nanodomains, as revealed in this study, SLBs with nanoscale topography have potential applications in controlling and directing stem cell differentiation.