Statistical Mechanics of an Elastically Pinned Membrane: Equilibrium Dynamics and Power Spectrum

Abstract

In biological settings, membranes typically interact locally with other membranes: the extracellular matrix in the exterior or internal cellular structures such as the cytoskeleton, locally pinning the membrane. Characterizing the dynamical properties of such interactions presents a difficult task. Significant progress has been achieved through simulations and experiments, yet analytical progress in modeling pinned membranes has been impeded by the complexity of governing equations. Here, we circumvent these difficulties by calculating analytically the time-dependent Green’s function of the operator governing the dynamics of an elastically pinned membrane in a hydrodynamic surrounding and subject to external forces. This enables us to calculate the equilibrium power spectral density for an overdamped membrane pinned by an elastic, permanently attached spring subject to thermal excitations. By considering the effects of the finite experimental resolution on the measured spectra, we show that the elasticity of the pinning can be extracted from the experimentally measured spectrum. Membrane fluctuations can thus be used as a tool to probe mechanical properties of the underlying structures. Such a tool may be particularly relevant in the context of cell mechanics, in which the elasticity of the membrane’s attachment to the cytoskeleton could be measured.

Significance

Cell adhesion molecules (CAMs) pin biological membranes to the surrounding scaffolds, enabling mechanical stability of cells and allowing them to respond to the local environment. We investigate the mechanics of a single pinning through its effect on the membrane shape fluctuations. Our analysis is based on analytical and numerical modeling of the membrane power spectral density, allowing us to determine the relevant timescales arising from the membrane-CAM coupling. Furthermore, we propose a method for inference of the CAMs mechanical properties in their native membrane surrounding, as opposed to most of the current methods, which require protein isolation. Most importantly, we rationalize the consequences of the pinning in a minimal model, which can be now implemented in specific biological situations.

Introduction

In ambient temperature, a phospholipid membrane can easily be deformed and exhibits appreciable fluctuations on timescales between 10⁻⁹ and 10⁻⁵ s (1, 2, 3, 4) because of its bending modulus of only a few tens of k_BT (5, 6, 7, 8, 9, 10). Although fluctuations are overdamped by the surrounding fluid (11, 12, 13, 14), mean fluctuation amplitudes of up to 100 nm were measured in different adherent cells (15). These fluctuations were used to infer mechanical properties of membranes, both in synthetic but also in cellular systems (15, 16, 17). However, in addition to the thermal driving typical for vesicles, a number of processes can actively excite membrane motion in cells (17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27). In the case of red blood cells, time and energy scales of these active contributions were recently quantified by independently measuring the fluctuations and the mechanical response of the membrane and showing that its fluctuations deviate from the fluctuation-dissipation theorem (28).

The precise mechanisms of these active fluctuations are still debated, and several propositions exist (19, 28, 29, 30), most of which are associated with the very structure of cellular membranes. Namely, in these systems, the phospholipid bilayer is locally bound to intra- and extracellular structures by flexible proteins. Typical examples of such interactions are stochastic pinnings of the membrane to the cytoskeleton (31) or the extracellular matrix (32), which occur in an ATP-dependent manner (22, 31, 33). Even in the case of permanently pinned membranes, active processes, such as the activity of pumps, can modulate the membrane dynamics (19, 29). Alternatively, the mechanical properties of the proteins pinning the membrane can be modified by the active complexation of macromolecular structures (34), hence generating deviations from thermal noise.

A powerful way to characterize the fluctuations of the membrane is by determining its time-dependent autocorrelation function and its temporal Fourier transform, the power spectral density (PSD). Both can be measured in state-of-the-art experiments (17, 28, 35). Fluctuation measurements are not limited to simple phospholipid bilayers but can characterize complexly structured membranes and their coupling to surrounding structures. Several techniques have been developed for measuring the fluctuations of membranes interacting with flat substrates (scaffold mimics). In these systems, it is now well established that not only does the interaction with the scaffold introduce nonspecific interactions (5, 7, 36, 37, 38, 39, 40, 41) but also that the proteins binding locally affect the fluctuations in a nonmonotonous manner, depending on the density of stochastic pinnings (32, 42). Actually, fluctuations of the pinned membrane were used to infer the distribution of pinning sites through the construction of so-called fluctuation maps (43, 44, 45), enabling direct comparison with theoretical approaches (4, 46).

From the modeling point of view, the presence of the pinning introduces a challenge, which is often circumvented by homogenizing the effects of interactions with the scaffold (28, 47). Alternative approaches, in which the local pinnings remain explicit, commonly involve simulations (32, 48, 49, 50). Most of this work has been relying on the Helfrich description of the membrane energetics (5, 6) combined with the Stokes fluid dynamics, an approach that was experimentally verified on artificial vesicles whose fluctuations are purely thermal (for reviews, see (1, 15) and references therein).

Meanwhile, analytic modeling of pinned membranes has mostly been focused on understanding static properties of the membrane shape and fluctuations (47, 51, 52, 53). However, the analytic treatments of the membrane dynamics, even in the context of equilibrium, remain scarce (54, 55). Specifically, the PSD of a pinned membrane has not yet been found analytically, even in the passive case. Given that the PSD measurements are becoming ever more sophisticated, this is an important step toward understanding both the passive and the active systems.

In this work, we address this issue and provide an analytical expression for the PSD of a membrane permanently pinned by a flexible protein. We describe in detail the effect of the pinning on the membrane’s equilibrium dynamics. Finally, we provide an exact analytical method for calculating the pinning stiffness from the experimentally measured PSD, accounting for the finite resolution of the setup. In addition to adhesive systems, the calculated PSD can also help understand the effects of experimental setups, such as beads attached to the membrane in optical traps. However, most importantly, this contribution enables further theoretical and experimental studies of the dynamics in active membranes.

Materials and Methods

Equation of motion

The system of our investigation consists of a tensed membrane (bending rigidity κ, tension σ) that is locally pinned to a scaffold (e.g., extracellular matrix or a cytoskeleton) by a harmonic spring of an elastic constant λ and rest length l₀, which models the flexibility of a formed ligand-receptor bond between the membrane and the scaffold. The proximity of the scaffold introduces additional nonspecific interactions with the membrane, as shown experimentally (4, 56) for the membrane-substrate interactions. The physical origin of these interactions comes from various electrostatic and steric effects (excluding specific ligand-receptor interactions) and can be approximated by a global harmonic potential of strength γ and a minimum positioned at a distance h₀ from the membrane (4) (Fig. 1). The above-described system is captured in the linearized Monge gauge (53, 57, 58) by the following Hamiltonian:

$$\mathcal{H}=\underset{A}{\int }\text{d}\mathbf{r}\left[\frac{\kappa }{2}{\left({\mathit{\nabla }}^{2}u\left(\mathbf{r}\right)\right)}^2+\frac{\sigma }{2}{\left(\mathit{\nabla }u\left(\mathbf{r}\right)\right)}^2+\frac{\gamma }{2}{\left(u\left(\mathbf{r}\right)\right)}^2+\frac{1}{2}\lambda {\left(u\left(\mathbf{r}\right)-\left(l_0-h_0\right)\right)}^2\delta \left(\mathbf{r}\right)\right],$$ (1)

where u(r) is the local displacement from the minimum of the nonspecific potential (Fig. 1). Here and throughout the work, the energy scale k_BT is set to unity, with Boltzmann constant denoted as k_B and absolute temperature T.

Figure 1.

Snapshot from the Langevin simulation of a locally pinned membrane fluctuating in a nonspecific potential (left). A sketch of the system is shown (right). The membrane is residing in a harmonic potential of strength γ at h₀ separation from a flat substrate and pinned by an elastic spring of stiffness λ and rest length l₀ positioned at r = 0. To see this figure in color, go online.

The dynamics of an overdamped membrane in a hydrodynamic surrounding is captured by the Langevin equation (32, 40, 50, 59, 60, 61, 62, 63)

$$\frac{\partial u\left(\mathbf{r},t\right)}{\partial t}=\int d{\mathbf{r}}^{\mathbf{\prime }}\mathit{\Lambda }\left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }}\right)\left(-\frac{\delta \mathcal{H}}{\delta u\left({\mathbf{r}}^{\mathbf{\prime }}\right)}+f\left({\mathbf{r}}^{\mathbf{\prime }},t\right)\right)\equiv \mathit{\Lambda }\left(\mathbf{r}\right)\text{∗}\left(-\frac{\delta \mathcal{H}}{\delta u\left(\mathbf{r}\right)}+f\left(\mathbf{r},t\right)\right),$$ (2)

which states that the velocity of the membrane profile u(r, t) is given by a convolution (denoted by the asterisk symbol) of the forces acting on the membrane with the hydrodynamic kernel Λ(r), the Oseen tensor. External forces on the system are denoted by f(r, t), whereas the internal forces acting to minimize the system’s Hamiltonian (1) are given by the first variation (53)

$$\frac{\delta \mathcal{H}}{\delta u\left(\mathbf{r}\right)}=\left(\kappa {\mathit{\nabla }}^4-\sigma {\mathit{\nabla }}^2+\gamma +\lambda \delta \left(\mathbf{r}\right)\right)u\left(\mathbf{r},t\right)-\lambda \left(l_0-h_0\right)\delta \left(\mathbf{r}\right).$$ (3)

Together with Eq. 2, this leads to the equation for the dynamics of an overdamped pinned membrane

$$\mathcal{D}\left[u\left(\mathbf{r},t\right)\right]=\mathit{\Lambda }\left(\mathbf{r}\right)\text{∗}\left[f\left(\mathbf{r},t\right)+\lambda \left(l_0-h_0\right)\delta \left(\mathbf{r}\right)\right],$$ (4)

with the operator $\mathcal{D}$ set as

$$\mathcal{D}=\frac{\partial }{\partial t}+\mathit{\Lambda }\left(\mathbf{r}\right)\text{∗}\left(\kappa {\mathit{\nabla }}^4-\sigma {\mathit{\nabla }}^2+\gamma +\lambda \delta \left(\mathbf{r}\right)\right).$$ (5)

To model the damping of the membrane close to a permeable wall by the surrounding viscous fluid (59, 61), we will use the free-membrane Oseen tensor Λ(r) defined by its Fourier transform Λ_k,

$${\mathit{\Lambda }}_k={\left(4\eta k\right)}^{-1},$$ (6)

where η is the viscosity of the surrounding fluid. Although Eq. 6 is the free-membrane Oseen tensor, it is a good approximation in the vicinity of a permeable wall. For a longer pinning molecule, the average distance between the membrane and the impermeable wall increases, and consequently, the free-membrane approximation becomes even better. In the presence of an impermeable wall, the damping coefficients are modified (64, 65), which, in the case of protein-mediated adhesion, typically only affects the amplitude of the first few membrane modes (62).

Furthermore, if the membrane is surrounded by two different fluids with viscosities η₁ and η₂, the viscosity η in the damping coefficients is replaced by the arithmetic mean η = (η₁ + η₂)/2 (15). We have assumed here that the hydrodynamic surrounding is viscous; however, the same formalism can be generalized to the case of a viscoelastic surrounding by the introduction of a viscoelastic kernel (66, 67).

The system described by Eqs. 1, 2, 3, 4, 5, and 6 with σ = γ = 0 was analytically investigated in (55). However, the introduction of the γ term in the Hamiltonian has, until now, impeded the analytical investigation of the pinned-membrane dynamics. We traverse this problem in the following sections.

Dynamics of thermal fluctuations

In thermal equilibrium, forces f(r, t) in the Langevin Eq. 4 are associated with the stochastic thermal noise, characterized by a vanishing mean

$$\langle f\left(\mathbf{r},t\right)\rangle =0$$ (7)

and spatiotemporal correlations obeying the fluctuation-dissipation theorem

$$\langle f\left(\mathbf{r},t\right)f\left({\mathbf{r}}^{\mathbf{\prime }},t^{\prime }\right)\rangle =2{\mathit{\Lambda }}^{-1}\left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }}\right)\delta \left(t-t^{\prime }\right).$$ (8)

Here, Λ⁻¹(r) is defined by

$$\mathit{\Lambda }\left(\mathbf{r}\right)\text{∗}{\mathit{\Lambda }}^{-1}\left(\mathbf{r}\right)=\delta \left(\mathbf{r}\right).$$ (9)

Simulation methods

Equation 4 for the membrane dynamics, subject to thermal noise defined with Eqs. 6, 7, and 9, is the foundation of our Langevin dynamics simulations of the membrane, described previously in full detail (50). In this case, one pinning site is placed in the middle of a simulation box (periodic boundary conditions) of a size 640 × 640 nm for a tensionless membrane and a size 5120 × 5120 nm at finite tensions. The simulations are performed with a temporal and lateral resolution of 10⁻⁹ s and 10 nm, respectively. The membrane height profile is recorded as a function of time and analyzed to extract the membrane shape and correlation functions.

Results

Time-dependent Green’s function

The solution of Eq. 4 provides the evolution of the membrane profile u(r, t). It is obtained by the integration of forces acting on the membrane, the latter accounted for by the dynamic Green’s function g(r, t|r′, t′):

$$u\left(\mathbf{r},t\right)=\underset{{\mathbb{R}}^2}{\int }\text{d}{\mathbf{r}}^{\mathbf{\prime }}\underset{\mathbb{R}}{\int }\text{d}{\text{t}}^{\prime }g\left(\mathbf{r},t|{\mathbf{r}}^{\mathbf{\prime }},t^{\prime }\right)\left(f\left({\mathbf{r}}^{\mathbf{\prime }},t^{\prime }\right)+\lambda \left(l_0-h_0\right)\delta \left({\mathbf{r}}^{\mathbf{\prime }}\right)\right)$$ (10)

Here, the Green’s function is defined by

$$\mathcal{D}\left[g\left(\mathbf{r},t|{\mathbf{r}}^{\mathbf{\prime }},t^{\prime }\right)\right]=\mathit{\Lambda }\left(\mathbf{r}\right)\text{∗}\left[\delta \left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }}\right)\delta \left(t-t^{\prime }\right)\right].$$ (11)

Besides imposing causality, this equation is subject to homogeneous spatial boundary conditions, forcing the membrane in the minimum of the nonspecific potential far from the pinning.

Free membrane, λ = 0

Recognizing the spatiotemporal translational invariance of the free-membrane system, the corresponding Green’s function g_f(r, t|r′, t′) can be written in terms of variables $\tilde{t}=t-t^{\prime }$ and $\tilde{\mathbf{r}}=\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }}$ as

$$g_f\left(\mathbf{r},t|{\mathbf{r}}^{\mathbf{\prime }},t^{\prime }\right)=g_f\left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }},t-t^{\prime }\right)\equiv g_f\left(\tilde{\mathbf{r}},\tilde{t}\right).$$ (12)

Consequently, for the free membrane, Eq. 11 becomes

$$\left[\frac{\partial }{\partial \tilde{t}}+\mathit{\Lambda }\left(\tilde{\mathbf{r}}\right)\text{∗}\left(\kappa {\mathit{\nabla }}_{\tilde{\mathbf{r}}}^4-\sigma {\mathit{\nabla }}_{\tilde{\mathbf{r}}}^2+\gamma \right)\right]g_f\left(\tilde{\mathbf{r}},\tilde{t}\right)=\mathit{\Lambda }\left(\tilde{\mathbf{r}}\right)\text{∗}\left[\delta \left(\tilde{\mathbf{r}}\right)\delta \left(\tilde{t}\right)\right].$$ (13)

The Fourier transform of Eq. 13 ($\tilde{\mathbf{r}}\to \mathbf{k}$ and $\tilde{t}\to \omega$), upon rearranging, yields

$$g_f\left(\mathbf{k},\omega \right)=\frac{1}{E_{k,\omega }},$$ (14)

with

$$E_{k,\omega }=i\omega /{\mathit{\Lambda }}_k+\kappa k^4+\sigma k^2+\gamma ,$$ (15)

where Λ_k is the spatial Fourier transform of $\mathit{\Lambda }\left(\tilde{\mathbf{r}}\right)$ and k = |k|. Transforming back to the spatial domain $\left(\mathbf{k}\to \tilde{\mathbf{r}}\right)$ provides the frequency-domain Green’s function for the free membrane

$$g_f\left(\tilde{\mathbf{r}},\omega \right)=\underset{{\mathbb{R}}^2}{\int }\frac{d\mathbf{k}}{{\left(2\pi \right)}^2}\frac{e^{i\mathbf{k}\tilde{\mathbf{r}}}}{E_{k,\omega }}.$$ (16)

For ω = 0, the Green’s function $g_f\left(\tilde{\mathbf{r}},\omega \right)$ reduces to the static correlation function (53). Consequently,

$$g_f\left(\tilde{\mathbf{r}}=0,\omega =0\right)\equiv \frac{1}{{\lambda }_m}=\frac{\text{arctan}\left(\sqrt{{\left(\frac{{\lambda }_m^0}{4\sigma }\right)}^2-1}\right)}{2\pi \sigma \sqrt{{\left(\frac{{\lambda }_m^0}{4\sigma }\right)}^2-1}}$$ (17)

represents the fluctuation amplitude, which in the tensionless case reduces to

$$g_f\left(\tilde{\mathbf{r}}=0,\omega =0;\sigma =0\right)\equiv \frac{1}{{\lambda }_m^0}=\frac{1}{8\sqrt{\kappa \gamma }}.$$ (18)

Finally, Fourier transforming $g_f\left(\tilde{\mathbf{r}},\omega \right)$ back to the temporal domain $\left(\omega \to \tilde{t}\right)$ and integrating over frequencies provides the spatiotemporal Green’s function for the free membrane

$$g_f\left(\tilde{\mathbf{r}},\tilde{t}\right)=\int \frac{d\mathbf{k}}{{\left(2\pi \right)}^2}{\mathit{\Lambda }}_k{e}^{i\mathbf{k}\tilde{\mathbf{r}}}{e}^{-{\mathit{\Lambda }}_k{E}_k\tilde{t}}\mathit{\Theta }\left(\tilde{t}\right)=\underset{0}{\overset{\infty }{\int }}\frac{dk}{2\pi }{\mathit{\Lambda }}_{k}k{J}_0\left(k\left|\tilde{\mathbf{r}}\right|\right)e^{-{\mathit{\Lambda }}_k{E}_k\tilde{t}}\mathit{\Theta }\left(\tilde{t}\right).$$ (19)

where Θ is the Heaviside step function, appearing as a consequence of causality. Moreover, $g_f\left(\tilde{\mathbf{r}},\tilde{t}\right)$ depends only on the absolute value of $\tilde{\mathbf{r}}$, as expected. For $\tilde{t}=0$, $g_f\left(\tilde{\mathbf{r}},0\right)=\mathit{\Lambda }\left(\tilde{\mathbf{r}}\right)$ naturally reduces to the Oseen tensor.

Pinned membrane

Permanent pinning breaks the spatial but keeps the temporal translational invariance. Therefore, the Green’s function of the pinned membrane must be described by two spatial variables, r and r′, and one temporal variable, $\tilde{t}=t-t^{\prime }$,

$$g\left(\mathbf{r},t|{\mathbf{r}}^{\mathbf{\prime }},t^{\prime }\right)=g\left(\mathbf{r},t-t^{\prime }|{\mathbf{r}}^{\mathbf{\prime }}\right)\equiv g\left(\mathbf{r},\tilde{t}|{\mathbf{r}}^{\mathbf{\prime }}\right).$$ (20)

In this notation, Eq. 11 for the pinned-membrane Green’s function becomes

$$\left[\frac{\partial }{\partial \tilde{t}}+\mathit{\Lambda }\left(\mathbf{r}\right)\text{∗}\left(\kappa {\mathit{\nabla }}^4-\sigma {\mathit{\nabla }}^2+\gamma +\lambda \delta \left(\mathbf{r}\right)\right)\right]g\left(\mathbf{r},\tilde{t}|{\mathbf{r}}^{\mathbf{\prime }}\right)=\mathit{\Lambda }\left(\mathbf{r}\right)\text{∗}\left[\delta \left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }}\right)\delta \left(\tilde{t}\right)\right].$$ (21)

Fourier transforming (r → k and $\tilde{t}\to \omega$) and rearranging Eq. 21 gives

$$g\left(\mathbf{k},\omega |{\mathbf{r}}^{\mathbf{\prime }}\right)=\frac{e^{-i\mathbf{k}\mathbf{r}\text{'}}}{i\omega /{\mathit{\Lambda }}_k+E_k}-\lambda g\left(\mathbf{r}=0,\omega |{\mathbf{r}}^{\mathbf{\prime }}\right)\frac{1}{i\omega /{\mathit{\Lambda }}_k+E_k}.$$ (22)

Transforming back to the spatial domain (k → r) gives

$$g\left(\mathbf{r},\omega |{\mathbf{r}}^{\mathbf{\prime }}\right)=g_f\left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }},\omega \right)-\lambda g\left(\mathbf{r}=0,\omega |{\mathbf{r}}^{\mathbf{\prime }}\right)g_f\left(\mathbf{r},\omega \right).$$ (23)

For r = 0 in Eq. 23, we find

$$g\left(\mathbf{r}=0,\omega |{\mathbf{r}}^{\mathbf{\prime }}\right)=\frac{g_f\left({\mathbf{r}}^{\mathbf{\prime }},\omega \right)}{1+\lambda g_f\left(\mathbf{r}=0,\omega \right)},$$ (24)

which, upon insertion into Eq. 23, yields the Green’s function for the pinned membrane in the spatiofrequency domain

$$g\left(\mathbf{r},\omega |{\mathbf{r}}^{\mathbf{\prime }}\right)=g_f\left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }},\omega \right)-\lambda \frac{g_f\left(\mathbf{r},\omega \right)g_f\left({\mathbf{r}}^{\mathbf{\prime }},\omega \right)}{1+\lambda g_f\left(\mathbf{r}=0,\omega \right)}.$$ (25)

For ω = 0, the Green’s function g(r, ω|r′) reduces to the static correlation function of the pinned membrane (53). Fourier transforming Eq. 25 (r → k and r′ → k′) results in

$$g\left(\mathbf{k},\omega |{\mathbf{k}}^{\mathbf{\prime }}\right)={\left(2\pi \right)}^2\delta \left(\mathbf{k}+{\mathbf{k}}^{\mathbf{\prime }}\right)g_f\left(\mathbf{k},\omega \right)-\lambda \frac{g_f\left(\mathbf{k},\omega \right)g_f\left({\mathbf{k}}^{\mathbf{\prime }},\omega \right)}{1+\lambda g_f\left(\mathbf{r}=0,\omega \right)},$$ (26)

which is the representation of the Green’s function in the Fourier space.

Note that we can write Eq. 25 in the free-membrane form (compare to Eq. 16)

$$g\left(\mathbf{r},\omega |{\mathbf{r}}^{\mathbf{\prime }}\right)=\underset{{\mathbb{R}}^2}{\int }\frac{d\mathbf{k}}{{\left(2\pi \right)}^2}\frac{e^{i\mathbf{k}\left(\mathbf{r}-{\mathbf{r}}^{\mathbf{\prime }}\right)}}{\tilde{E}\left(\mathbf{k},\omega ,\mathbf{r}\right)},$$ (27)

where

$$\tilde{E}\left(\mathbf{k},\omega ,\mathbf{r}\right)=E_{k,\omega }{\left[1-\frac{\lambda g_f\left(\mathbf{r},\omega \right)}{1+\lambda g_f\left(\mathbf{r}=0,\omega \right)}{e}^{-i\mathbf{k}\mathbf{r}}\right]}^{-1}.$$ (28)

Hence, the effect of a local pinning is described by the factor multiplying E_k,ω in Eq. 28, which introduces k and r dependence as a consequence of the translational symmetry breaking by the pinning. It is therefore impossible to interpret the effect of the pinning as a simple rescaling of the parameters κ, σ, γ because this would obviously give a translationally invariant solution. However, for the special case r = 0, we find $\tilde{E}\left(\mathbf{k},\omega ,\mathbf{r}=0\right)=E_{k,\omega }\left(1+\lambda /{\lambda }_m\right)$, and the correlation from the pinning g(r = 0, ω|r′) can be interpreted as an effective free-membrane correlation,

$$g\left({\mathbf{r}}^{\mathbf{\prime }}|\mathbf{r}=0,\kappa ,\sigma ,\gamma ,\eta \right)=\frac{g_f\left({\mathbf{r}}^{\mathbf{\prime }},\kappa ,\sigma ,\gamma ,\eta \right)}{1+\lambda /{\lambda }_m}=g_f\left({\mathbf{r}}^{\mathbf{\prime }},{\kappa }^{\text{eff}},{\sigma }^{\text{eff}},{\gamma }^{\text{eff}},{\eta }^{\text{eff}}\right)\text{,}$$ (29)

where κ^eff = (1 + λ/λ_m)κ, σ^eff = (1 + λ/λ_m)σ, γ^eff = (1 + λ/λ_m)γ, and η^eff = (1 + λ/λ_m)η. Note that even in this special case, λ enters all effective parameters equally. Moreover, all effective parameters depend on all free-membrane parameters through λ_m, which is given by Eq. 17.

Power Spectral Density

We complement the simulations of thermally fluctuating membrane with the analytic calculations based on the Green’s function approach (Eq. 10). We start with rewriting Eq. 10 as

$$u\left(\mathbf{r},t\right)=\langle u\left(\mathbf{r}\right)\rangle +v\left(\mathbf{r},t\right),$$ (30)

where

$$v\left(\mathbf{r},t\right)=\underset{{\mathbb{R}}^2}{\int }\text{d}{\mathbf{r}}^{\mathbf{\prime }}\underset{\mathbb{R}}{\int }\text{d}{\text{t}}^{\prime }g\left(\mathbf{r},t-t^{\prime }|{\mathbf{r}}^{\mathbf{\prime }}\right)f\left({\mathbf{r}}^{\mathbf{\prime }},t^{\prime }\right)$$ (31)

are the fluctuations around the ensemble averaged static profile (53):

$$\langle u\left(\mathbf{r}\right)\rangle =\underset{{\mathbb{R}}^2}{\int }\text{d}{\mathbf{r}}^{\mathbf{\prime }}\underset{\mathbb{R}}{\int }\text{d}{\text{t}}^{\prime }g\left(\mathbf{r},t-t^{\prime }|{\mathbf{r}}^{\mathbf{\prime }}\right)\lambda \left(l_0-h_0\right)\delta \left({\mathbf{r}}^{\mathbf{\prime }}\right).$$ (32)

Transforming (t → ω) Eq. 31 gives

$$v\left(\mathbf{r},\omega \right)=\underset{{\mathbb{R}}^2}{\int }\text{d}{\mathbf{r}}^{\mathbf{\prime }}g\left(\mathbf{r},\omega |{\mathbf{r}}^{\mathbf{\prime }}\right)f\left({\mathbf{r}}^{\mathbf{\prime }},\omega \right),$$ (33)

from which the PSD $\langle {\left|v\left(\mathbf{r},\omega \right)\right|}^2\rangle$ can be calculated as (see Supporting Materials and Methods, Section SI)

$$\langle {\left|v\left(\mathbf{r},\omega \right)\right|}^2\rangle =\frac{1}{{\left(2\pi \right)}^2}\underset{{\mathbb{R}}^2}{\int }\text{d}\mathbf{k}\frac{2{\mathit{\Lambda }}_k^{-1}}{{\left|E^{\text{eff}}\left(\mathbf{k},\omega ,\mathbf{r}\right)\right|}^2}=\frac{1}{{\left(2\pi \right)}^2}\underset{{\mathbb{R}}^2}{\int }\text{d}\mathbf{k}\frac{2{\mathit{\Lambda }}_k^{-1}}{{\left|E_{k,\omega }\right|}^2}{\left|1-\frac{\lambda g_f\left(\mathbf{r},\omega \right)}{1+\lambda g_f\left(\mathbf{r}=0,\omega \right)}{e}^{-i\mathbf{k}\mathbf{r}}\right|}^2.$$ (34)

With the hydrodynamic coefficients specified as in Eq. 6, Eq. 34 becomes

$$\langle {\left|v\left(\mathbf{r},\omega \right)\right|}^2\rangle =\frac{4\eta }{\pi }\underset{0}{\overset{\infty }{\int }}\text{d}k\frac{k^2}{{\left(4\eta k\omega \right)}^2+{\left(\kappa k^4+\sigma k^2+\gamma \right)}^2}{\left|1-\frac{\lambda g_f\left(\mathbf{r},\omega \right)}{1+\lambda g_f\left(\mathbf{r}=0,\omega \right)}{e}^{-i\mathbf{k}\mathbf{r}}\right|}^2.$$ (35)

For infinitely soft pinnings (λ = 0), Eq. 35 becomes homogeneous in space and reduces to the well-known result

$$\langle {\left|v_f\left(\omega \right)\right|}^2\rangle =\frac{4\eta }{\pi }\underset{0}{\overset{\infty }{\int }}\text{d}k\frac{k^2}{{\left(4\eta k\omega \right)}^2+{\left(\kappa k^4+\sigma k^2+\gamma \right)}^2},$$ (36)

which, for small and large ω, has the limiting behavior (3, 11, 24, 68)

$$\langle {\left|v_f\left(\omega \right)\right|}^2\rangle =\{\begin{array}{cc}\frac{\eta }{\sqrt{\gamma {\left({\lambda }_m^0/4+\sigma \right)}^3}},& \omega \ll {\omega }_0\\ \frac{1}{6\sqrt[3]{2{\eta }^2\kappa }}{\omega }^{-5/3},& \omega \gg {\omega }_0\end{array}.$$ (37)

Here, ${\omega }_0\equiv \sqrt[4]{\kappa {\gamma }^3}/\eta$ is a crossover frequency, defined as the intersection of the lines fitting the low- and high-frequency limits of the spectrum. The high-frequency decay ∼ω^−1.667 is related by Fourier transform to the short-time mean-square displacement behavior ∼t^0.667 (55, 60, 69). The low-frequency limit decays with σ^−1.5 for tensions $\sigma \gg {\lambda }_m^0$ (Fig. 2 b).

Figure 2.

Dynamical properties of a membrane at the pinning site. Comparison of modeling (lines) and simulations (symbols) shows excellent agreement across the entire parameter range. (a) PSD of a free membrane (Eq. 36) (full line) and a pinned tensionless membrane (Eq. 38) is shown (dashed lines for $\lambda /{\lambda }_m^0$ = 1, 3, 10, increasing in the direction of the arrows). The high-frequency regime of the PSD is unaffected by the pinning, and the free-membrane behavior (ω^−1.667) is recovered. (b) The low-frequency limit of the PSD (Eq. 39) is shown as a function of the membrane tension σ for different pinning strengths ($\lambda /{\lambda }_m^0$ = 1, 3, 10). For large tensions, a σ^−1.5 dependence is recovered irrespective of λ. (c) The low-frequency limit is shown as a function of the pinning strength for different membrane tensions ($8\sigma /{\lambda }_m^0$ = 1, 3, 10). For large bond stiffness, a λ⁻² dependence is displayed. All curves are plotted for κ = 20 k_BT, γ = 3 × 10⁻⁷k_BT/nm⁴, and η = 1 mPas. To see this figure in color, go online.

It is clear from Eq. 35 that the PSD at the pinning site r = 0 can be recast into

$$\langle {\left|v\left(\mathbf{r}=0,\omega \right)\right|}^2\rangle =\frac{\langle {\left|v_f\left(\omega \right)\right|}^2\rangle }{{\left(1+\lambda g_f\left(\mathbf{r}=0,\omega \right)\right)}^2}=\langle {\left|v_f\left(\omega ,{\kappa }^{\text{eff}},{\sigma }^{\text{eff}},{\gamma }^{\text{eff}},{\eta }^{\text{eff}}\right)\right|}^2\rangle ,$$ (38)

with effective parameters defined in Eq. 29.

In agreement with simulations, Eq. 38 shows that only the pinning stiffness, and not its length, has an effect on the PSD and that the pinning affects only the low-frequency regime (Fig. 2 a). The low-frequency behavior can be obtained upon combining Eq. 38 for ω = 0 with Eq. 37 to yield

$$\langle v{\left|\left(\mathbf{r}=0,\omega =0\right)\right|}^2\rangle ={\left(\frac{1}{1+\lambda /{\lambda }_m}\right)}^2\frac{\eta }{\sqrt{\gamma {\left({\lambda }_m^0/4+\sigma \right)}^3}}=\{\begin{array}{cc}{\left(\frac{1}{1+\lambda /{\lambda }_m^0}\right)}^2\frac{\eta }{\sqrt{\gamma {\left({\lambda }_m^0/4\right)}^3}},& 4\sigma /{\lambda }_m^0\ll 1\\ {\left(\frac{1}{1+\lambda \text{ln}\left(\sigma \right)/\left(2\pi \sigma \right)}\right)}^2\frac{\eta }{\sqrt{\gamma {\sigma }^3}},& 4\sigma /{\lambda }_m^0\gg 1.\end{array}$$ (39)

Equation 39 shows that the low-frequency spectrum is independent of membrane tension for $4\sigma /{\lambda }_m^0\ll 1$, and it decays with σ^−1.5 for membrane tensions large enough to diminish the effect of the pinning $\left(\lambda \text{ln}\left(\sigma \right)/\left(2\pi \sigma \right)\ll 1\right)$ (Fig. 2 b). For stiff pinnings $\left(\lambda /{\lambda }_m\gg 1\right)$, the low-frequency limit falls off as λ⁻² (Fig. 2 c). On the other hand, for $\lambda /{\lambda }_m\ll 1$, pinning effects vanish even in the low-frequency limit. Interestingly, the crossover frequency ${\tilde{\omega }}_0$ for the pinned membrane,

$${\tilde{\omega }}_0={\left[6\sqrt[3]{2{\eta }^2\kappa }\langle {\left|v\left(\mathbf{r}=0,\omega =0\right)\right|}^2\rangle \right]}^{-0.6},$$ (40)

defined analogously to the one for the free membrane, becomes sensitive to the elastic properties of the pinning as ${\tilde{\omega }}_0\sim {\lambda }^{1.2}$ and, as such, increases with the pinning stiffness.

The spatial range of the pinning effect on the PSD is independent of the pinning stiffness λ and is determined only by the free-membrane correlation length, as is discussed in detail in (53). For a tensionless membrane, the correlation length is given by ξ₀ = (κ/γ)^0.25. For κ = 20 k_BT and typical values of γ measured in experiments, the correlation length ξ₀ is on the order of 100 nm. The pinning effect spans several correlation lengths ξ₀ from the pinning (53). Although the pinning stiffness λ does not affect the spatial range of the pinning, it affects its amplitude. It can be seen in Fig. 2 that the low-frequency limit of the PSD at the pinning position can be modified by orders of magnitude for typical biological values of the parameters. From the above considerations, we expect that the pinning effect should be experimentally measurable.

Effect of the finite experimental resolution on the fluctuation spectrum

To compare with experiments, it is necessary to account for the finite temporal and spatial resolutions of the setup (15, 16). Averaging the true membrane profile u(r, t) over a spatial domain A and a time interval τ gives rise to the so-called apparent membrane profile $u_{\tau }^A\left(\mathbf{r},t\right)$:

$$u_{\tau }^A\left(\mathbf{r},t\right)=\underset{0}{\overset{\tau }{\int }}\frac{\text{d}{t}^{\prime }}{\tau }\underset{A}{\int }\frac{\text{d}{\mathbf{r}}^{\mathbf{\prime }}}{A}u\left(\mathbf{r}+{\mathbf{r}}^{\mathbf{\prime }},t+t^{\prime }\right),$$ (41)

from which it is straightforward to derive the apparent PSD (Supporting Materials and Methods, Section SI.A):

$$\langle {\left|v_{\tau }^A\left(\mathbf{r},\omega \right)\right|}^2\rangle ={\left(\frac{\text{sin}\left(\omega \tau /2\right)}{\omega \tau /2}\right)}^2\frac{2}{{\left(2\pi \right)}^2}\underset{{\mathbb{R}}^2}{\int }\text{d}\mathbf{k}\frac{{\mathit{\Lambda }}_k^{-1}}{{\left(\omega /{\mathit{\Lambda }}_k\right)}^2+E_k^2}{\left|\underset{A}{\int }\frac{\text{d}{\mathbf{r}}^{\mathbf{\prime }}}{A}{e}^{i\mathbf{k}\left(\mathbf{r}+{\mathbf{r}}^{\mathbf{\prime }}\right)}\left(1-\frac{\lambda g_f\left(\mathbf{r}+{\mathbf{r}}^{\mathbf{\prime }},\omega \right)}{1+\lambda g_f\left(0,\omega \right)}{e}^{-i\mathbf{k}\left(\mathbf{r}+{\mathbf{r}}^{\mathbf{\prime }}\right)}\right)\right|}^2.$$ (42)

The PSD measured around the pinning placed centrally in a circle of radius R is (Supporting Materials and Methods, Section SI.A.1)

$$\langle {\left|v_{\tau }^{R^2\pi }\left(\mathbf{r}=0,\omega \right)\right|}^2\rangle ={\left(\frac{\text{sin}\left(\omega \tau /2\right)}{\omega \tau /2}\right)}^2\frac{1}{\pi }\underset{0}{\overset{\infty }{\int }}\text{dk}\frac{k{\mathit{\Lambda }}_k^{-1}}{{\left(\omega /{\mathit{\Lambda }}_k\right)}^2+E_k^2}\frac{4}{R^2}{\left|\frac{J_1\left(kR\right)}{k}-\frac{\lambda }{1+\lambda g_f\left(0,\omega \right)}\frac{1}{2\pi }\underset{0}{\overset{\infty }{\int }}\text{d}{\text{k}}^{\prime }\frac{J_1\left(k^{\prime }R\right)}{i\omega /{\mathit{\Lambda }}_{k^{\prime }}+E_{k^{\prime }}}\right|}^2.$$ (43)

Equation 43 reduces to Eq. 38 in the limit τ, R → 0.

The high-frequency regime of the averaged PSD recovers the averaging behavior of the free-membrane—spatial averaging changes the decay from ω^−1.667 to ω⁻² as previously reported (3), whereas finite temporal resolution induces an additional attenuation of ω⁻². Hence, the PSD, which is subject to both temporal and spatial averaging, decays as ω⁻⁴.

In the low-frequency regime, temporal averaging plays no role for ω < 1/τ, whereas the finite spatial resolution has a more complex effect. Because of the interplay with the effects of the pinning, the low-frequency amplitude is not a monotone function of the averaging area (Fig. 3, b and c). Increasing the averaging area up to a critical size on the order of the correlation length of the membrane amplifies the low-frequency components. However, further increases of the averaging area begin to attenuate them. This can be understood as a competition of two effects: spatial averaging has the effect of attenuating the PSD amplitude, as can be seen for the free membrane (Fig. 3 c), but at the same time, averaging effectively reduces the impact of the pinning on the PSD, which amplifies the low-frequency components. Obviously, the amplifying effect is stronger up to the scale of the membrane correlation length, after which the first attenuating effect dominates. The free and the pinned membrane can therefore be clearly distinguished by the effect of the measurement averaging area on the measured single-point spectrum. This is especially useful in the context of experiments that do not directly supply the spatial structure of the PSD but instead provide a spatially averaged single-point spectrum with the ability to systematically vary the averaging area of the measurement. This is, for example, the case with the dynamic optical displacement spectroscopy (17). If the averaging area is too large, we enter the regime in which the measured spectra of the free and the pinned membrane are indistinguishable (Fig. 3 c).

Figure 3.

Effect of the finite resolution (averaging over a circle of radius R and a time interval τ) on the PSD at the pinning (Eq. 43). Lines represent analytical results, and symbols represent the simulation data. Note that the numerical averaging of the simulation data was done on a rectangular grid, which introduces a miniature underestimation of the averaged PSD amplitude in comparison to the analytical results, which assume averaging over a perfect circle (for more detail, see Supporting Materials and Methods, Section II.B). (a) Averaging decreases the difference between the free and the pinned PSD, therefore reducing the effect of the pinning on the PSD. Pinning has no effect on the high-frequency part of the spectrum for both the averaged and the unaveraged spectrum. (b) The low-frequency part of the PSD is a nonmonotone function of the spatial averaging area. On the other hand, increasing the averaging area always attenuates the high-frequency components, which, in this case, fall as ∼ω⁻² instead of ∼ω^−1.667. (c) The low-frequency part of the PSD for three different pinning stiffness values is shown as a function of the averaging area. The low-frequency part of the free-membrane PSD monotonically decreases, whereas the pinned-membrane PSD shows nonmonotonic behavior. The position of the peak is related to the pinning correlation length. (d) Time averaging introduces oscillatory behavior of the PSD, for which only the envelope of the PSD is shown. The combined effect of the spatial and temporal averaging gives an ∼ω⁻⁴ behavior of the high-frequency regime. Parameters: κ = 20 k_BT, σ = 10⁻²⁰k_BT/nm², γ = 3 × 10⁻⁷k_BT/nm⁴, η = 1 mPas, and $\lambda /{\lambda }_m^0$ = 10 (except in c, where the values of $\lambda /{\lambda }_m^0$ are noted in the legend). To see this figure in color, go online.

In the low-frequency regime (ω = 0), we obtain (Supporting Materials and Methods, Section SI.A.1)

$$\langle {\left|v_{\tau }^{R^2\pi }\left(\mathbf{r}=\mathrm{0,0}\right)\right|}^2\rangle =\frac{4\eta }{\pi }\underset{0}{\overset{\infty }{\int }}\text{dk}\frac{k^2}{E_k^2}{\left(\frac{J_1\left(kR\right)}{kR/2}-\frac{\lambda {\lambda }_m}{\lambda +{\lambda }_m}s\left(R\right)\right)}^2,$$ (44)

where, for brevity purposes, we introduce a reduced coefficient s(R)

$$s\left(R\right)=\frac{1}{R^2\pi \sqrt{\sigma -4\kappa \gamma }}\left(\frac{1-a_{-}R{K}_1\left(a_{-}R\right)}{a_{-}^2}-\frac{1-a_{+}R{K}_1\left(a_{+}R\right)}{a_{+}^2}\right),$$ (45)

with a_±

$$a_{\pm }={\left[\frac{\sigma }{2\kappa }\left(1\pm \sqrt{1-{\left(\frac{{\lambda }_m^0}{4\sigma }\right)}^2}\right)\right]}^{1/2}.$$ (46)

To calculate the pinning stiffness λ from the PSD, Eq. 44 can be inverted, which, upon introduction of coefficients L, a, b, and c, yields

$$\lambda ={\left(\frac{1}{L}-\frac{1}{{\lambda }_m}\right)}^{-1},$$ (47)

with

$$L=\frac{b}{2a}-\sqrt{{\left(\frac{b}{2a}\right)}^2-\frac{c}{a}}$$ (48)

and

$$a=\frac{\pi s^2\left(R\right)}{4\sqrt{\gamma {\left({\lambda }_m^0/4+\sigma \right)}^3}},b=\frac{4s\left(R\right)}{R}\left(\underset{0}{\overset{\infty }{\int }}\text{dk}\frac{k{J}_1\left(kR\right)}{E_k^2}\right),c={\left(\frac{2}{R}\right)}^2\underset{0}{\overset{\infty }{\int }}\text{dk}\frac{J_1^2\left(kR\right)}{E_k^2}-\frac{\pi }{4\eta }\langle {\left|v_{\tau }^{R^2\pi }\left(\mathbf{r}=\mathrm{0,0}\right)\right|}^2\rangle .$$ (49)

The averaged spectrum is contained in the coefficient c. When implemented numerically, Eqs. 47, 48, and 49 represent a fast and exact method for obtaining information about the pinning stiffness from the experimentally measured PSD. Here, we note that the deconvolution of the noise associated with the experimental setting should be performed before the extraction of the pinning stiffness.

Discussion

Our calculation of the Green’s function (Eq. 25) fully resolves the dynamics of an overdamped, permanently pinned membrane in a hydrodynamic surrounding (Eq. 2). The solution is general in the sense that it works for any forces acting on the membrane and enables one to study the membrane dynamics in the presence of both nonthermal and thermal perturbations. The latter case is resolved in this work by the calculation of the thermal equilibrium PSD (Eq. 34). For the specific case of hydrodynamic damping close to a permeable wall, our analytical calculation (Eq. 35) is verified with Langevin simulations in a broad range of parameters such as the pinning stiffness, membrane tension, and strength of the nonspecific potential, which were allowed to independently vary for several orders of magnitude (Fig. 2). It is shown that the pinning decreases the low-frequency amplitudes of the spectrum and pushes the crossover frequency (Eq. 40) to higher values while the high-frequency amplitudes remain unaffected (Fig. 2).

Interestingly, the pinned-membrane PSD at the pinning site is given by a product of a free-membrane PSD and a λ-dependent prefactor (Eq. 38). Assuming knowledge of the pinning stiffness λ, this enables inference of the pinned-membrane PSD at the pinning site directly from the free-membrane PSD. This approach has a clear advantage over a direct measurement of the pinned PSD because it replaces the pinned-membrane measurement with a well-established free-membrane measurement. On the other hand, if λ is not known, it can be easily determined by comparing the pinned- and free-membrane PSDs. The low-frequency limit of the PSD at the pinning site (Eq. 39) is particularly useful for getting a better understanding of the interplay of the system parameters and shows a λ⁻² decay.

These relationships, however, may not be observed experimentally because of the finite resolution of measurements. Specifically, although the effects of the temporal averaging are simple, significant spatial averaging introduces nontrivial modulations of the spectrum and breaks the relation between the free- and the pinned-membrane PSD given by Eq. 38. In this regime, the pinning stiffness can be inferred from the measured PSD with the use of the spatially averaged spectrum (Eqs. 44, 45, 46, 47, 48, and 49). It is assumed here that the parameters κ, σ, and γ can be determined independently from λ. An example of an experimental protocol that can give independent measurements of κ, σ, and γ on a pinning-free giant unilamellar vesicle is described in (17). It uses the fact that the nonspecific interaction γ contributes to the membrane dynamics only at the side of the vesicle closest to the substrate. The measured free-membrane parameter values, averaged over a large number of similar vesicles, can then be used with Eqs. 44, 45, 46, 47, 48, and 49 to estimate λ. However, we would ideally want to have independent measurements of κ, σ, γ, and λ on the same vesicle. Our proposition is to extend the experimental protocol described in (17) by using the locality of the pinning effect. First of all, if the main goal of the experiment is to measure λ, one could simplify the problem by working with deflated vesicles, making the σ contribution negligible, although the following experimental protocol can be generalized for tensed membranes if needed. In the case of tensionless vesicles, one needs three PSD measurements done on three different parts of the vesicle, each giving a one-parameter fit for κ, γ, and λ, respectively. The first measurement should be done on the part of the vesicle for which both the nonspecific interaction and the pinning interaction are absent, which gives a one-parameter fit for κ. The second measurement should be done on the vesicle part close to the substrate but outside the range of the pinning effect, which, together with the value of κ estimated from the first measurement, gives a one-parameter fit for γ. Finally, the third PSD measurement should be done around the pinning position, which—together with the values of κ and γ estimated from the first two measurements and Eqs. 44, 45, 46, 47, 48, and 49—gives a one-parameter fit for λ. Of course, the reliability of this method depends on the spatiotemporal experimental resolution. For a review about the available experimental methods for fluctuation measurements and their corresponding resolutions, see (1).

Our model serves as a starting point for the construction of more complex models. It shows that the effect of the pinning is more complicated than a simple rescaling of the membrane parameters and reveals the true nature of the parameter mixing. As shown in the study, this enables estimation of the system parameters from the experimental data. An interesting and straightforward extension of this model would be to consider the viscoelasticity of the surrounding medium, such as the cell cytoplasm, by writing the Langevin equation with a viscoelastic kernel, as described in (66). Furthermore, by generalizing calculations presented in this work to a multipinning model, one could find the PSD for an arbitrary distribution of pinnings. This can reveal the effect of pinning clustering and gives PSD maps of arbitrarily shaped pinned domains. The multipinning model can be further extended by introducing stochastic pinnings that can attach and deattach from the membrane, as is the case with most biological ligand-receptor pairs. The importance of membrane fluctuations and their relationship to rates has been emphasized in (46). Generalizing the calculation of the PSD to a stochastically pinned membrane would enable experimental determination of the (de)attachment rates. As such, one could experimentally infer both the mechanical properties of the ligand-receptor pair and the properties of the ligand-receptor chemical bond and give feedback needed to improve the rate model.

The importance of studies of isolated pinnings are further emphasized by the fact that this allows for measurements in the natural surroundings of ligand-receptor pairs, avoiding the need for their extraction and isolation, in contrast to most current single-molecule experiments.

A further challenge is to extend the equilibrium theory by including active contributions present in the biological systems. One source of active fluctuations comes from the ATP-dependent stochastic (de)attachment of the pinning to (from) the membrane, which can modulate the (de)attachment of the cell membrane to (from) the cytoskeleton or the adhesion substrate. It would enable us to explore the clustering and adhesion dynamics in the presence of the active contribution, which can inspire new experimental approaches with model systems and increase our understanding of the active processes in live cells. As an example, there is a growing body of evidence that the membrane affects the affinity (70) and the kinetic rates for protein binding (46, 50), which, in turn, affect the fluctuations and the early stage signaling in developing junctions between cells (71).

All data needed to evaluate the conclusions in the manuscript are present in the manuscript and/or the Supporting Materials and Methods. All data and computer code for this study are available on request from the authors.

Editor: Markus Deserno.

Author Contributions

A.-S.S. was in charge of overall direction and supervision. U.S. provided critical feedback and helped shape the research. J.A.J. did the analytical calculations. D.S. and R.B. did the simulations. A.-S.S. and J.A.J. wrote the manuscript with contributions from all authors.