Simple differences in the protein-membrane attachment mechanism have functional consequences for surface mechanics

Abstract

This paper describes two methods for propagating coupled membrane and embedded particle dynamics with ensembles that are valid to second order in the deformation of the membrane. Proteins and functional lipids associate with cellular membranes, and their attachments influence membrane physical and dynamical properties. Therefore, it is necessary to accurately model the coupled dynamics of the membrane and any associated material of interest. We have developed two methods for coupling membrane and particle dynamics that differ in the binding mechanism of the particle to the surface. The “on-surface” mechanism should be used for particles that slide along the membrane; this description leads to an effective reduction in the membrane surface tension. The “in-surface” mechanism treats the particles as tightly bound to the lipidic binding sites; the method avoids double counting lateral entropy of implicitly modeled lipids. We emphasize the differences between these two mechanisms, when it is appropriate to use them, and how the methods differ from previously used dynamic methods.

I. INTRODUCTION

The cell membrane is vital in many biological processes. It provides both structure and protection while mediating processes that occur within the cell. It is a complex and dynamic fluid system, made of many different types of lipid molecules and proteins. The spatial organization of the molecular elements of the plasma membrane and organelles are currently of intense interest both experimentally^1–4 and theoretically.^5–8

Protein and lipid species can be integral parts of the membrane or peripherally bound to the surface. They will move with the surface and influence its dynamics. An efficient mathematical representation of a membrane surface introduces a standard complication of these dynamics: an auxiliary coordinate system is necessary to locate proteins and lipids on the membrane surface. Usually, this coordinate system is constructed by a projection onto the xy plane. In the standard methodology, the projected diffusion of particles is slowed on curved surfaces to reflect the longer path traveled.^9–11 However, coupling the motion of particles to the membrane deformation without also coupling the dynamics of membranes to the embedded particles breaks detailed balance and does not generate the desired equilibrium distribution. This work fixes that flaw.

Experimental studies are now able to track how fluorescently tagged proteins and lipids move on curved surfaces such as clathrin-coated pits and sites of exocytosis.^12–15 Polarized total internal-reflection fluorescence (TIRF) microscopy correlates the membrane curvature and fluorophore location on oriented bilayers. To differentiate driven and diffusive transport at these sites, it is necessary to model how proteins and lipids move on these surfaces and thus to have an accurate method for propagating their dynamics. The problem is further complicated in that surface embedded biomolecules deform the membranes around them, either by enforcing their own shape or by modifying membrane curvature energetics.^6,16–22 Thus, it is essential to correctly propagate the dynamics of surface biomolecules and the membrane simultaneously.

There are multiple simulation techniques available for studying the cell membrane.^{6,17,20,23,24} In order to reach the length and time scales necessary to study the dynamics of the spatial organization, coarse-graining methods must typically be leveraged. On large length scales, it is sufficient to model the membrane as a continuous elastic sheet. It has been shown by Canham and Helfrich that membrane fluctuations can be described by the membrane’s bending modulus and surface tension.^25,26

Membrane fluctuations and diffusivity are intricately coupled. For example, we have previously shown that membrane protein attachments that suppress membrane fluctuations undergo membrane mediated interactions and have slowed projected diffusion.^8,27 In that example, we investigated diffusion in-plane using Brownian dynamics, that is, not strictly bound to the membrane, which is fluctuating out-of-plane. However, most surface biomolecules, especially lipids, are more tightly bound to the membrane itself than, for example, to the cytoskeleton.

The mechanism of how a molecule is attached to the membrane matters. Shown in Fig. 1 are illustrations of two different binding mechanisms to a surface. The first mechanism [Fig. 1(a)], in which the bound molecule is a fundamentally different unit from the underlying surface, is termed “on-surface.” A simple example of this mechanism would be the adsorption of an amphiphilic molecule to an oil–water interface. The bound molecule stabilizes the surface and decreases its tension, in part due to its lateral entropy. In this case, the number of binding sites is proportional to the area of the surface. The second mechanism [Fig. 1(b)], in which the bound molecule is directly attached to a lipid in the surface, is termed “in-surface.” Due to the strong interaction with the bound lipid, the bound molecule does not have a lateral entropy distinct from that lipid. The number of binding sites is proportional to the area times the density of binding sites, $\mathcal{P}$. For this mechanism, the binding site density decreases as the bilayer expands, consistent with binding to individual lipids rather than the surface. Examples of this mechanism would be a protein binding to a charged phosphatidylinositide-phosphate (PIP) lipid or a His-tagged protein binding to a lipid-nitrilotriacetic molecule. Intermediate cases are less clear; an amphipathic helix inserted into a bilayer appears to be a lipid in its own right and thus would appear to contribute its own lateral entropy. However, if it binds some lipids strongly, it may display behavior in between the two modes. It may be necessary to explicitly model, as particles, some specific lipid species (e.g., PIP) that bind proteins. No additional entropic contribution for these lipids should be applied; the continuum model would already account for the lateral entropy of all lipids.

FIG. 1.

(a) With on-surface attachment, a molecule (represented as a rectangle) binds nonspecifically to the membrane surface so that an increase in the bilayer area yields more possible attachment positions and therefore an increase in entropy. (b) With the in-surface attachment, the molecule binds to a specific lipid. In this case, the lateral entropy does not change upon binding.

Diffusion on curved surfaces is a well-studied problem that is applied in fields beyond biophysics. To investigate the diffusion along a curved surface requires a curvilinear coordinate system with modified expressions for gradients and related operators. Specifically, one must replace the Laplace operator in the flat diffusion equation with the Laplace-Beltrami (LB) operator,¹⁰ which takes into consideration the conversion between the surface coordinates and the real position in three dimensional space, accomplished using the metric tensor.

In one dimension, the metric tensor reduces to a scalar metric factor. Figure 2 shows a simple scheme relating the area available for a particle to the one dimensional metric factor, 1 + (dh/dx)², where the derivative is denoted ∂_xh in this work. This can be applied to a particle diffusing along a static curved membrane.

FIG. 2.

The local Jacobian factor determines the area a membrane-bound particle can explore relative to its projected (xy) area. An embedded particle is more likely to be found in the projected Cartesian space dx if the metric factor is higher. The Jacobian factor is the square root of the metric factor employed in this work. The Laplace operator can be modified to correct diffusion for the metric factor. See, for example, Ref. 10.

Previous works have studied how the membrane alters particle diffusion using the LB operator to describe particle motion and Fourier Space Brownian Dynamics (FSBD) to describe membrane fluctuations.^9–11 This method will be referred to as the Laplace-Beltrami with Fourier Space Brownian Dynamics (LB + FSBD) method throughout this article. However, as pointed out by Sigurdsson and co-workers,²⁸ including the Laplace-Beltrami operator to describe particle motion but not including the particles’ effect on membrane fluctuations results in inconsistent probability distributions for the membrane and the particles. We will derive a model that instead couples particle and membrane motions from a Fokker-Planck equation chosen to produce the correct probability distribution, denoted the on-surface method. The coupling is due to the additional area available to a particle as a result of membrane fluctuations. This raises a key question: Do membrane-bound particles induce membrane fluctuations to increase their entropy (lateral area)? In contrast, in the in-surface method, particles are embedded in the membrane, equivalently to a lipid. Here, the particle positions do change with the membrane undulations, decoupling them at second order and achieving the same projected density of particles. An example of how the lipids flow is shown in Fig. 3.

FIG. 3.

For in-surface diffusion, the attached particle moves with the flow of lipids. This yields the correct projected density distribution of particles without energetic coupling to surface fluctuations. Lateral gray arrows indicate the particle flow field as a consequence of the membrane moving from the old conformation (gray) to the new one (black).

Below is the theoretical development of the in- and on-surface methods, including the ramifications for diffusional dynamics, deformation-driven fluctuations in the projected particle distributions, and the tension that emerges from the on-surface model. The results are summarized in Table I of the Conclusions.

TABLE I.

A side-by-side comparison of different ramifications to using the on-surface and in-surface models presented here along with the LB + FSBD method described in Appendix B and a planar diffusion model.

	On-surface	In-surface	LB + FSBD	Planar
Effect on surface	Δσ = −ρkBT	Δσ = 0	Δσ = 0	Δσ = 0
Tension
Projected density	$\propto \sqrt{g\left(\mathit{r}\right)}$	$\propto \sqrt{g\left(\mathit{r}\right)}$	$\propto \sqrt{g\left(\mathit{r}\right)}$	Uniform
Distribution
Projected particle	D/D0 < 1	D/D0 = 1	D/D0 < 1	D/D0 = 1
Diffusion
Notes	Alters lateral	Computationally	Inconsistent	Diffusion
	Entropy	Expensive	Distributions	In-plane

II. MODEL DEVELOPMENT

A. Surface parameterization and elasticity

Given a flat patch of membrane, typically modeled with periodic boundary conditions, the so-called Monge gauge is a convenient choice for parameterizing deformations of the surface. Rather than using specialized surface coordinates, the position of the surface is determined by the planar Cartesian coordinates. A simple scalar function $h\left(\mathit{r}\right)$ describes the height of the surface. The position on the surface is then

$${\mathit{r}}_{\mathrm{o}\mathrm{n}}\left(x,y;\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)=\left\{\begin{array}{c}x\hfill \\ y\hfill \\ h\left(x,y;\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)\hfill \end{array}\right\}\left(\mathrm{M}\mathrm{o}\mathrm{n}\mathrm{g}\mathrm{e}\right).$$ (1)

Equation (1) is the representation of the surface in terms of the frequently employed Monge gauge. The Monge gauge is most effective when combined with the Fourier representation of $h\left(\mathit{r}\right)$,

$$\begin{array}{cc}\hfill h\left(\mathit{r}\right)& =L^{-2}\sum _{\mathit{k}}{\tilde{h}}_{\mathit{k}}{\mathrm{e}}^{ı\mathit{k}\cdot \mathit{r}}\hfill \\ \hfill {\tilde{h}}_{\mathit{k}}& ={\iint }_{L^2}\mathrm{d}\mathit{r}h\left(\mathit{r}\right){\mathrm{e}}^{-ı\mathit{k}\cdot \mathit{r}},\hfill \end{array}$$ (2)

where L is the length of the membrane and L² = A is the area of the membrane. The Monge gauge is also convenient for linking membrane bound particles to off-membrane structures because the x coordinate of the particle, r_x, is the same as its surface parameter, x. The area of a patch on the Monge surface, which in the planar configuration had area dxdy, now has area $\sqrt{\left(1+{\left(\nabla h\right)}^2\right)}\mathrm{d}x\mathrm{d}y$. The spatial variation of this area element is the source of the inconsistent probability distributions that are remedied in this work.

In contrast, the formula developed in this work for in-surface embedding as opposed to on-surface embedding applies a coordinate change,

$$\begin{array}{cc}\hfill x& =x^{\prime }+u_x\left(x^{\prime },y^{\prime };\left\{{\tilde{h}}_k\right\}\right),\hfill \\ \hfill y& =y^{\prime }+u_y\left(x^{\prime },y^{\prime };\left\{{\tilde{h}}_k\right\}\right),\hfill \end{array}$$ (3)

where x′ and y′ are the Cartesian coordinates of a point on the membrane in the flat reference state, but which change as the membrane fluctuates. The priming of coordinates is unnecessary for the on-surface mechanism, for which the x and y coordinates do not change with $\left\{{\tilde{h}}_k\right\}$. The surface is

$${\mathit{r}}_{\mathrm{i}\mathrm{n}}\left(x^{\prime },y^{\prime };\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)=\left\{\begin{array}{c}{x}^{\prime }+u_x\left(x^{\prime },y^{\prime };\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)\hfill \\ y^{\prime }+u_y\left(x^{\prime },y^{\prime };\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)\hfill \\ h\left(x^{\prime }+u_x\left(x^{\prime },y^{\prime };\left\{{\tilde{h}}_{\mathit{k}}\right\}\right),y^{\prime }+u_y\left(x^{\prime },y^{\prime };\left\{{\tilde{h}}_{\mathit{k}}\right\}\right);\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)\hfill \end{array}\right\}\left(\mathrm{i}\mathrm{n}-\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\right).$$ (4)

Conceptually, we imagine the transformation u(x′, y′) to be a lateral flow field that describes how lipids move to maintain homogeneous lipid spacing. The transformation u(x, y) developed in Appendix A, Eq. (A4), makes the metric factor spatially invariant up to second order in the height deformation. With this modification, the lateral position of the particle depends on the set $\left\{{\tilde{h}}_{\mathit{k}}\right\}$. The flow field u(x, y) itself is second order in the height. The curvature, a result of out of plane deformation h(x, y) proportional to ${\tilde{h}}_{\mathit{k}}$, is thus only affected beyond second order in the magnitude of the height deformation. The energetics of membrane modes are decoupled at second order using both the Monge and in-surface gauges.

Note that for the in-surface representation, and unlike the Monge gauge, the x and y variables do not directly specify the spatial position of the surface. For both the in-surface and on-surface binding mechanisms, a particle’s dynamics are computed using the coordinates x and y, with the final position given by either r_on(x, y) or r_in(x, y). In both cases, the surface is parameterized by the Fourier coefficients ${\tilde{h}}_{\mathit{k}}$.

1. Surface metrics

In previous works, we modeled particles that diffuse in the xy plane,^8,27 for example, modeling strong attachment of a curved membrane to an effectively rigid body, such as the cytoskeleton. In that work, the particles are not assumed to be on the membrane surface and instead diffuse in the xy plane. In this case, the Laplace operator in the Fokker-Planck equation should be replaced by the Laplace-Beltrami operator to take into account the membrane shape. We treat the particles as diffusing on a surface whose motion itself varies with time. This means that the motion of the particle depends on the current membrane shape. The slope of the membrane increases the particle density when projected onto the flat xy reference plane. This mathematically comes from the fact that the area of the curved surface is larger than that of the projected. The conversion factor that accounts for this excess area is the square root of the determinant of the metric tensor. The metric tensor, inverse metric tensor, and the determinant of the metric tensor (the metric factor) for the on-surface binding mechanism are

$$\begin{array}{ll}\hfill \mathbf{O}\mathbf{n}-\mathbf{s}\mathbf{u}\mathit{r}\mathbf{f}\mathbf{a}\mathbf{c}\mathbf{e}& \left(\mathbf{M}\mathbf{o}\mathbf{n}\mathbf{g}\mathbf{e}\right)\mathbf{m}\mathbf{e}\mathbf{t}\mathit{r}\mathbf{i}\mathbf{c}\mathbf{s}\hfill \\ \hfill g_{\alpha \beta }& ={\delta }_{\alpha \beta }+{\partial }_{\alpha }h{\partial }_{\beta }h,\hfill \end{array}$$ (5)

$${\left(g^{-1}\right)}_{\alpha \beta }={\delta }_{\alpha \beta }-{\partial }_{\alpha }h{\partial }_{\beta }h/g\left(\mathit{r}\right),$$ (6)

$$g\left(\mathit{r}\right)=1+{\left(\nabla h\left(\mathit{r}\right)\right)}^2.$$ (7)

For the in-surface binding mechanism, the quantities are

$$\begin{array}{l}\hfill \mathbf{I}\mathbf{n}-\mathbf{s}\mathbf{u}\mathit{r}\mathbf{f}\mathbf{a}\mathbf{c}\mathbf{e}\mathbf{m}\mathbf{e}\mathbf{t}\mathit{r}\mathbf{i}\mathbf{c}\mathbf{s}\\ \hfill g_{\alpha \alpha }& =1+g_{\mathrm{i}\mathrm{n}}\left(\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)+\mathcal{O}\left[{\tilde{h}}_k^4\right],\hfill \end{array}$$ (8)

$$g_{\alpha \beta ,\alpha \ne \beta }={\partial }_{\alpha }h{\partial }_{\beta }h-{\partial }_{\beta }{u}_{\alpha }-{\partial }_{\alpha }{u}_{\beta }+\mathcal{O}\left[{\tilde{h}}_k^4\right],$$ (9)

$${\left(g^{-1}\right)}_{\alpha \alpha }=1-g_{\mathrm{i}\mathrm{n}}\left(\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)+\mathcal{O}\left[{\tilde{h}}_k^4\right],$$ (10)

$${\left(g^{-1}\right)}_{\alpha \beta ,\alpha \ne \beta }=-{\partial }_{\alpha }h{\partial }_{\beta }h-{\partial }_{\alpha }{u}_{\beta }-{\partial }_{\beta }{u}_{\alpha }+\mathcal{O}\left[{\tilde{h}}_k^4\right],$$ (11)

$$g\left(\mathit{r}\right)=g_{\mathrm{i}\mathrm{n}}\left(\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)+\mathcal{O}\left[{\tilde{h}}_k^4\right],$$ (12)

where $g_{\mathrm{i}\mathrm{n}}\left(\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)$ is spatially independent at this order of ${\tilde{h}}_k$ [see Eq. (A5), Appendix A].

2. Elasticity

The energy expression developed by Canham and Helfrich^25,26 describes membrane shape deformations according to the bending rigidity (κ) and bare surface tension (σ),

$$E_{\mathrm{m}}=\int \mathrm{d}\mathbf{S}\left[\sigma +\frac{\kappa }{2}{\left(K-c_0\right)}^2+{\kappa }_g{K}_G\right],$$ (13)

where K is the total curvature, c₀ is the spontaneous curvature, K_G is the Gaussian curvature, and κ_g is the saddle splay modulus. Parameterizing the surface in the linearized Monge gauge, this expression is simplified,

$$E_{\mathrm{m}}\left[h\left(\mathit{r}\right)\right]={\int }_{L^2}\mathrm{d}\mathit{r}\left[\frac{\kappa }{2}{\left[{\nabla }^{2}h\left(\mathit{r}\right)\right]}^2+\frac{\sigma }{2}{\left[\nabla h\left(\mathit{r}\right)\right]}^2\right].$$ (14)

The first term describes the energy cost to bend the membrane away from a flat sheet, and the second term is the energy associated with changes in the membrane area. The energy is truncated to quadratic order in the height of the membrane, $h\left(\mathit{r}\right)$, which is valid if fluctuations in the height are small. The Gaussian curvature term is a constant over a fixed membrane topology and therefore is omitted, while the spontaneous curvature only contributes a constant in the linearized Monge gauge. With the Fourier representation of Eq. (2), Eq. (14) becomes

$$E_{\mathrm{m}}\left(\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)=\frac{1}{L^2}\sum _{\mathit{k}}\left(\frac{\sigma }{2}{k}^2+\frac{\kappa }{2}{k}^4\right)|{\tilde{h}}_{\mathit{k}}{|}^2,$$ (15)

where the energies of each mode are now decoupled. Since $h\left(\mathit{r}\right)$ is real-valued, the condition ${\tilde{h}}_{\mathit{k}}={\tilde{h}}_{-\mathit{k}}^{*}$ must be satisfied. The vectors r and k are both two-dimensional vectors. We follow the formalism described in Ref. 8 for the Fourier space implementation and rewrite this expression in terms of k > 0, which is defined as the set of wave vectors $\left\{\left(2\pi p/L,2\pi q/L\right):\left(1\le p\le P,q=0\right)\mathrm{o}\mathrm{r}\left(-P\le p\le P,1\le q\le Q\right)\right\}$, where P and Q determine the magnitude of the largest wave vector,

$$E_{\mathrm{m}}\left(\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)=\frac{1}{L^2}\sum _{\mathit{k}>0}\left(\sigma k^2+\kappa k^4\right)|{\tilde{h}}_{\mathit{k}}{|}^2.$$ (16)

This distinction allows for the treatment for the real and complex parts of the coefficients independently.

When implementing this scheme on a computer, one should allow all Fourier amplitudes ${\tilde{h}}_{\mathit{k}}$ with nonzero wavevector to be complex in order to maintain translational invariance. This requires special care if one wants to go back and forth between a real-space and Fourier-space description of the membrane using the discrete Fourier transform, which imposes constraints on the Fourier amplitudes at the edge of the Brillouin zone that in turn might lead to unphysical artifacts.⁸

B. Statistical mechanics of the surface-particle coupling

The partition function Z that describes a system of N particles diffusing along a membrane is

$$\begin{array}{ccc}\hfill Z& =\hfill & \frac{1}{N!}{\int }_{-\infty }^{\infty }{\mathrm{d}}^M\left\{{\tilde{h}}_{\mathit{k}}\right\}\mathcal{P}{\left(\left\{{\tilde{h}}_k\right\}\right)}^N{\int }_{L^2}{\mathrm{d}}^2{\mathit{r}}_1\hfill \\ \hfill & \hfill & \times \sqrt{g\left({\mathit{r}}_1\right)}\dots {\int }_{L^2}{\mathrm{d}}^2{\mathit{r}}_N\sqrt{g\left({\mathit{r}}_N\right)}{\mathrm{e}}^{-E\left(\left\{{\tilde{h}}_{\mathit{k}}\right\},\left\{{\mathit{r}}_{\mathbf{i}}\right\}\right)/k_{\mathrm{B}}T},\hfill \end{array}$$ (17)

where M is the number of Fourier modes that describe the membrane shape and $\mathcal{P}\left(\left\{{\tilde{h}}_k\right\}\right)$ is the density of binding sites. The energy $E\left(\left\{{\tilde{h}}_{\mathit{k}}\right\},\left\{\mathit{r}\right\}\right)$ includes $E_m\left(\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)$ as well as any particle-dependent energetics. For the on-surface mode, the value of $\sqrt{g\left(\mathit{r}\right)}$ varies with space at second order so that particles when projected onto the xy-plane are more likely to be found where $\sqrt{g\left(\mathit{r}\right)}$ is large. For this model, $\mathcal{P}$ is unity. For the in-surface mechanism, the value of $\sqrt{g\left(\mathit{r}\right)}$ is a constant with respect to particle position [$1+{\sum }_{\mathit{k}}|\mathit{k}{|}^2{L}^{-4}{\tilde{h}}_{\mathit{k}}{\tilde{h}}_{-\mathit{k}}$, Eq. (8)] and can be pulled out of the spatial integrals. For this model, $\mathcal{P}$ is inversely proportional to the area of the surface,

$$\mathcal{P}=1\left(\mathrm{o}\mathrm{n}-\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\right),$$ (18)

$$\mathcal{P}\left(\left\{{\tilde{h}}_k\right\}\right)={\left(1+\sum _{\mathit{k}}|\mathit{k}{|}^2{L}^{-4}{\tilde{h}}_{\mathit{k}}{\tilde{h}}_{-\mathit{k}}\right)}^{-1}\left(\mathrm{i}\mathrm{n}-\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}\right).$$ (19)

For the in-surface mode, $\mathcal{P}{\left(\left\{{\tilde{h}}_k\right\}\right)}^N$ cancels the contribution of the metric factors precisely. The following analysis of the coupling of particle position to membrane fluctuations thus applies only to the on-surface mechanism.

The probability distribution function that describes the system of particles diffusing along a fluctuating membrane is

$$\rho \left(\left\{{\tilde{h}}_{\mathit{k}}\right\},\left\{{\mathit{r}}_i\right\}\right)=\frac{1}{Z}\sqrt{\prod _{i}g\left({\mathit{r}}_i\right)}{\mathrm{e}}^{-E/k_{\mathrm{B}}T},$$ (20)

where Z is the partition function and $\sqrt{{\prod }_{i}g\left({\mathit{r}}_i\right)}$ accounts for the increase in density of the particles in the projected plane due to the excess area. g(r_i) is the metric factor at the position of the ith particle. For the on-surface mechanism, the metric-factor couples particles and surface fluctuations. We can absorb its effect by introducing an effective energy

$$E_{\mathrm{eff}}=E-\frac{1}{2}{k}_{\mathrm{B}}T\sum _i\ln g\left({\mathit{r}}_i\right).$$ (21)

It is clear from this expression that the particle coordinates in the plane are drawn to regions of the membrane with a high gradient. Since we have chosen to represent the energies up to this point in quadratic order in the derivatives of the membrane height, we will do the same with this new energy term as well,

$$E_{\mathrm{eff}}=E-\frac{k_{\mathrm{B}}T}{2}\sum _i{\left(\nabla h\left({\mathit{r}}_i\right)\right)}^2.$$ (22)

1. Definition of tension

There are a few quantities that function as tensionlike factors in surface dynamics. The frame tension, $-\frac{\mathrm{d}F}{\mathrm{d}{A}_p}$, which multiplies the projected area, is so-called because it is an external force applied to the frame of the surface. The bare tension, σ, is a model parameter multiplying the surface’s area in the energy defined in Eq. (15). The fluctuation tension is the coefficient, proportional to q², recorded from the fluctuation spectrum of the surface. The internal tension is $\frac{\mathrm{d}F}{\mathrm{d}A}$ and is not directly observable in the same way as the frame tension. However, considerable theoretical effort over the last three decades has yielded equivalence of each tension within a sign consistent with whether the force is external or internal, in the linearized Monge gauge, to the appropriate first order in the area strain consistent with that linearized Monge Hamiltonian.^29–34 This includes the internal and fluctuation tensions.

In this work, it is shown that with the on-surface attachment mechanism, the bare tension σ is reduced. This affects the fluctuation spectrum, which due to their equivalence changes the apparent fluctuation tension. We find this is true regardless of if the membrane is fluctuating at constant area-per-lipid in equilibrium with a lipid reservoir, if the membrane’s area is fluctuating, or even if only the projected area fluctuates (constant area, constant particle number). However, with these different ensembles, the mechanism of the change in bare surface tension differs.

First, consider how the area changes are manifest in the different ensembles. The area strain α_A of a quantity A is

$${\alpha }_A=\frac{A-A_0}{A_0},$$ (23)

where A₀ is the resting area.

Consider the following difference between an incompressible and compressible bilayer, in the linearized Monge gauge. For a compressible surface fluctuating on a fixed projected area, the area and area strain are

$$A=A_p\left(1+\sum _k\frac{{\tilde{h}}_k^2}{2}{k}^2{A}_p^{-2}\right)+\mathcal{O}\left[{\tilde{h}}_k^4\right],$$ (24)

$${\alpha }_A=\sum _k\frac{{\tilde{h}}_k^2}{2}{k}^2{A}_p^{-2}+\mathcal{O}\left[{\tilde{h}}_k^4\right].$$ (25)

An elastic model of the area deformation energy E_A goes as

$$E_A=\frac{K_A}{2}{A}_0{\alpha }_A^2.$$ (26)

In the linearized Monge gauge, the strain α_A is proportional to ${\tilde{h}}_k^2$. Thus, the compression energy proportional to K_A is fourth order in ${\tilde{h}}_k$ and, being small compared to the curvature energy (proportional to ${\tilde{h}}_k^2$), is typically neglected, including here. With bare tension σ, this component of the energy is α_Aσ. For an incompressible surface fluctuating on a variable projected area, the projected area and its strain can be computed by solving Eq. (24) for A_p,

$$A_p=A\left(1-\sum _k\frac{{\tilde{h}}_k^2}{2}{k}^2{A}^{-2}\right)+\mathcal{O}\left[{\tilde{h}}_k^4\right],$$ (27)

$${\alpha }_{A,p}=-\sum _k\frac{{\tilde{h}}_k^2}{2}{k}^2{A}^{-2}+\mathcal{O}\left[{\tilde{h}}_k^4\right],$$ (28)

where, to second order in ${\tilde{h}}_k$, the strains are equal in magnitude but opposite in sign. In this case, the energy is modulated by the external frame tension, $-\frac{\mathrm{d}F}{\mathrm{d}{A}_p}$. A consequence is that a shift in tension due to a mechanical imposition or due to a relaxation in the constituent number of lipids in the bilayer cannot be distinguished by second order fluctuations of ${\tilde{h}}_k$. However, either the in-surface or on-surface methodology will be more appropriate for one of these mechanisms depending on whether an increase in the area also increases the number of binding sites. The in-surface method has been designed with the case of area elasticity in mind. For example, the lipid flow we envision arises from relaxation of the spatially inhomogeneous tension implied by the Monge gauge.

2. Emergent external tension of the on-surface model

Negative tension enhances membrane fluctuations due to the increase in the area with ${\tilde{h}}_{\mathit{k}}$. As shown here, the coupling of particle position to the metric factor of the membrane is equivalent to an applied negative tension.

The internal tension is defined by

$$\sigma =\frac{\partial F}{\partial A}$$ (29)

and

$$F=-k_{\mathrm{B}}T\log Z,$$ (30)

where A is the area of the membrane. In the absence of a particle-dependent energy, each particle contributes a factor A to the partition function,

$$Z\propto A^N$$ (31)

and

$$F=F_0-N{k}_{\mathrm{B}}T\log \frac{A}{A_{\mathrm{p}}},$$ (32)

where F₀ is the component of the free energy not coupling particles to the area and A_p is the projected (frame) area used here as an arbitrary reference area used to define the standard state. The particle contribution to the tension is $-k_{\mathrm{B}}T\frac{N}{A}=-k_{\mathrm{B}}T\rho$.

In the in-surface model, on the other hand, the density is independent of the current shape of the membrane, and the presence of the particles does not change the surface tension of the membrane.

3. Altered projected particle density

The projected density distribution of the particles shows where the particles are expected to be on the membrane, given its current fluctuation. For the LB + FSBD and on-surface models, the probability in Cartesian coordinates is proportional to the metric factor, meaning that the particles are more likely to be where the metric is higher. This can be seen from the definition of the metric factor in Fig. 2: the area on the membrane is larger, leading to a higher projected density distribution. In our previous work,^8,27 where the particles are assumed to diffuse in the plane and not along the surface, the distribution is the normal Boltzmann distribution because the metric factor is not considered.

For the in-surface mechanism, the metric factor (and thus density) has spatially invariant value $g_{\mathrm{i}\mathrm{n}}\left(\left\{{\tilde{h}}_{\mathit{k}}\right\}\right)$ in terms of the internal coordinate system {x, y}. However, as established in VA, the metric for converting between the {x, y} system and the real space {r_x, r_y} system establishes the same expected particle distributions.

C. Dynamics of the on-surface model

In this section, the on-surface model for coupled equations of motion for particles diffusing on a fluctuating curved surface is derived. To determine how to propagate the Fourier amplitudes of the membrane and the particle positions in time, we begin with the Fokker-Planck equation associated with this system,^10,28,35,36

$$\frac{\partial \rho \left(\mathit{X},t\right)}{\partial t}=-{\nabla }_{\mathit{X}}\left[-\mathrm{\Gamma }\left({\nabla }_{\mathit{X}}{E}_{\mathrm{eff}}\right)\rho \left(\mathit{X},t\right)-k_{\mathrm{B}}T\mathrm{\Gamma }{\nabla }_{\mathit{X}}\rho \left(\mathit{X},t\right)\right],$$ (33)

where $\mathit{X}=\left[\left\{{\tilde{h}}_{\mathit{k}}\right\},\left\{{\mathit{r}}_i\right\}\right]$ (the set of all the degrees of freedom for the system) and $\mathrm{\Gamma }=\left[{\tilde{\mathrm{\Lambda }}}_{\mathit{k}},\mu {\left(g^{-1}\right)}_{\alpha \beta }\right]$ (the mobility factors for the membrane and particles, respectively). The mobility factor for the membrane, ${\tilde{\mathrm{\Lambda }}}_{\mathit{k}}=1/4\eta k$, is the Fourier representation of the Oseen tensor.³⁷ The mobility factor for the particles contains the metric tensor to accurately account for the membrane shape in the diffusion constant, which is related to the mobility through Einstein’s relation, D₀ = k_BTμ. The metric tensor will appear as an extra drift term of the Fokker-Planck equation for both dynamic variables. To ensure proper convergence, the flux

$$\mathit{J}=-\mathrm{\Gamma }\left({\nabla }_{\mathit{X}}{E}_{\mathrm{eff}}\right)\rho -k_{\mathrm{B}}T\mathrm{\Gamma }{\nabla }_{\mathit{X}}\rho$$ (34)

must be zero in thermal equilibrium,²⁸ where the probability distribution is

$$\rho \left(\left\{{\tilde{h}}_{\mathit{k}}\right\},\left\{{\mathit{r}}_i\right\}\right)=\frac{1}{Z}{\mathrm{e}}^{-E_{\mathrm{e}\mathrm{f}\mathrm{f}}/k_{\mathrm{B}}T}.$$ (35)

This can be confirmed by substituting Eq. (35) into Eq. (34), which yields zero flux.

The equations of motion for both ${\tilde{h}}_{\mathit{k}}$ and r_i can be determined using Eq. (33). The corresponding equations of motion are³⁵

$$\frac{\partial {\tilde{h}}_{\mathit{k}}}{\partial t}=-{\tilde{\mathrm{\Lambda }}}_{\mathit{k}}\left[{\left\{\frac{\delta E_{\mathrm{eff}}}{\delta h\left(\mathit{r}\right)}\right\}}_{\mathit{k}}\right]+{\tilde{\xi }}_{\mathit{k}}\left(t\right),$$ (36)

$$\frac{\mathrm{d}{r}_{i,\alpha }}{\mathrm{d}t}=-\mu {\left(g^{-1}\right)}_{\alpha \beta }{\partial }_{i,\beta }{E}_{\mathrm{eff}}+\sqrt{2D_0}{\left(g^{-1/2}\right)}_{\alpha \beta }{\mathit{\zeta }}_{i,\beta }\left(t\right),$$ (37)

for the membrane Fourier amplitudes and particle positions, respectively. Here, we have used the Einstein summation convention in which repeated indices are summed. The random noise term for the membrane is defined by⁸

$$\begin{array}{cc}\hfill ⟨\mathrm{R}\mathrm{e}\left({\tilde{\xi }}_{\mathit{k}}\right)⟩& =⟨\mathrm{I}\mathrm{m}\left({\tilde{\xi }}_{\mathit{k}}\right)⟩=0,\hfill \\ \hfill ⟨\mathrm{R}\mathrm{e}\left({\tilde{\xi }}_{\mathit{k}}\left(t\right)\right)\mathrm{R}\mathrm{e}\left({\tilde{\xi }}_{{\mathit{k}}^{\prime }}\left(t^{\prime }\right)\right)⟩& ={\tilde{\mathrm{\Lambda }}}_{\mathit{k}}{k}_{\mathrm{B}}T{L}^2\delta \left(t-t^{\prime }\right)\left({\delta }_{\mathit{k},{\mathit{k}}^{\prime }}+{\delta }_{-\mathit{k},{\mathit{k}}^{\prime }}\right),\hfill \\ \hfill ⟨\mathrm{I}\mathrm{m}\left({\tilde{\xi }}_{\mathit{k}}\left(t\right)\right)\mathrm{I}\mathrm{m}\left({\tilde{\xi }}_{{\mathit{k}}^{\prime }}\left(t^{\prime }\right)\right)⟩& ={\tilde{\mathrm{\Lambda }}}_{\mathit{k}}{k}_{\mathrm{B}}T{L}^2\delta \left(t-t^{\prime }\right)\left({\delta }_{\mathit{k},{\mathit{k}}^{\prime }}-{\delta }_{-\mathit{k},{\mathit{k}}^{\prime }}\right),\hfill \\ \hfill ⟨\mathrm{R}\mathrm{e}\left({\tilde{\xi }}_{\mathit{k}}\left(t\right)\right)\mathrm{I}\mathrm{m}\left({\tilde{\xi }}_{{\mathit{k}}^{\prime }}\left(t^{\prime }\right)\right)⟩& =0,\hfill \end{array}$$ (38)

and the random noise term for the particles is given by

$$⟨{\mathit{\zeta }}_{i,\alpha }\left(t\right){\mathit{\zeta }}_{j,\beta }\left(t^{\prime }\right)⟩={\delta }_{ij}{\delta }_{\alpha \beta }\delta \left(t-t^{\prime }\right).$$ (39)

Taking the quadratic approximation in Eqs. (36) and (37) leads to the following equations of motion:

$$\frac{\partial {\tilde{h}}_{\mathit{k}}}{\partial t}=-{\tilde{\mathrm{\Lambda }}}_{\mathit{k}}\left[\left(\sigma k^2+\kappa k^4\right){\tilde{h}}_{\mathit{k}}+k_{\mathrm{B}}T\sum _i\nabla h\left({\mathit{r}}_i\right)ı\mathit{k}{\mathrm{e}}^{-ı\mathit{k}\cdot {\mathit{r}}_i}\right]+{\tilde{\xi }}_{\mathit{k}}\left(t\right),$$ (40)

$$\begin{array}{l}\hfill \frac{\mathrm{d}{r}_{i,\alpha }}{\mathrm{d}t}=-D_0\left[{\partial }_{\alpha }{\partial }_{\gamma }h\left({\mathit{r}}_{\mathbf{i}}\right)\right]{\partial }_{\gamma }h\left({\mathit{r}}_i\right)+\sqrt{2D_0}\left(1-\frac{1}{2}{\partial }_{\alpha }h\left(\mathit{r}\right){\partial }_{\beta }h\left(\mathit{r}\right)\right){\mathit{\zeta }}_{i,\beta }\left(t\right).\\ \hfill \end{array}$$ (41)

The second equation (the equation of motion for the particles) is similar to Eq. (B8) in Appendix B; the difference lies with the quadratic approximation. The main difference between the method derived here and the LB + FSBD method described in Appendix B is the equation of motion for the membrane. In this method, there is an added drift term that does not appear in the LB + FSBD method; this drift ensures the proper probability distribution.

1. Altered diffusion

The projected diffusion constant of the particles is altered in different ways depending on the chosen method. In Refs. ⁹ and ¹⁰, the authors show that the projected diffusion is smaller than the bare diffusion in the membrane using a preaveraging approximation and numerical simulations using the LB + FSBD method. They find

$$\frac{D}{D_0}=\frac{1}{2}\left(1+⟨\frac{1}{g\left(\mathit{r}\right)}⟩\right)$$ (42)

for the effective diffusion using the preaveraging approximation. However, in Ref. 36, it was shown that using a simple area scaling analytical description is an accurate approximation as well. There, the expression for the effective diffusion is

$$\frac{D}{D_0}=⟨\frac{1}{\sqrt{g\left(\mathit{r}\right)}}⟩=⟨\frac{L_x{L}_y}{A_0}⟩,$$ (43)

where L_xL_y is the projected membrane area and A₀ is the actual membrane area. Both Eqs. (42) and (43) show that the projected diffusion is smaller than the bare diffusion in the membrane.

These expressions can be used to approximate the projected diffusion with the on-surface method as well. However, given that the membrane is affected by the presence of the particles in the on-surface mechanism, the values for the projected diffusion constant will not be the same as seen in Refs. ^{9, 10}, and ³⁶, where the LB + FSBD method is used.

Whereas with the on-surface model, diffusion is reduced according to the gradient of the metric, with the in-surface model, the metric is invariant up to second order in $\left\{{\tilde{h}}_k\right\}$. However, embedded particles will move with the flow of lipids. With the in-surface method, membrane fluctuations change the amount of area available to the system according to $g_{\mathrm{i}\mathrm{n}}\left(\left\{{\tilde{h}}_k\right\}\right)$. However, given that this function is spatially invariant up to second order in $\left\{{\tilde{h}}_k\right\}$, the global diffusion constant would be scaled depending on $\left\{{\tilde{h}}_k\right\}$. Choosing to modify diffusion thus does not affect the distribution of particles as it does with the on-surface method. In fact, it may be inconsistent with the diffusive mechanism to rescale diffusion for a deformed surface. For example, in an ensemble with fluctuating elastic area, the diffusion constant may increase for a stretched membrane if diffusion is based on a “hop” mechanism. That the diffusion of the in-surface model can be left unmodified is suggested by applying the preaveraging approximation, in which quantities ∂_αh and ∂_αu_β in Eqs. (8)–(12) are averaged to zero. However, given the assumptions of the preaveraging approximation, it is possible that violation of the separation of time scales may alter diffusion.

III. CONCLUSION

While both the on-surface and the in-surface models accurately couple the dynamics, given the desired distributions, they model different physical mechanisms leading to different consequences for each system. These ramifications are summarized in Table I for both in- and on-surface models, as well as for the LB + FSBD model (used in previous works^9–11 or see Appendix B) and a planar diffusion method. Three different aspects of the models are presented: effective surface tension (Sec. II B 2), projected density distribution (Sec. II B 3), and particle diffusion (Sec. II C 1). The last row in the table is some notes concerning other implications of the different models.

This article has focused on modeling coupled membrane-particle dynamics by choosing the dynamics to accurately produce the proper distributions for the total system. We have proposed a model that explicitly couples the dynamics of a membrane, described as a continuum, with pointwise particles that diffuse along the membrane surface. We call this diffusion “on” the surface, and as a result, the lateral entropy increases with surface area. In contrast to previous models,^9–11 the model presented here is consistent in its treatment of the membrane and particle dynamics.

Similarly, we comment on a consistent way to model diffusion “in” the surface, where lateral entropy should remain unchanged. For the in-surface method, the surface is parameterized such that the metric factor is spatially invariant at second order. When the ensemble is cast in terms of the finite binding sites on the surface, the particles’ contributions to the partition function [Eq. (17)] are independent of the membrane. This method does not result in a new tension term, consistent with modeling lipids whose entropy is already part of the underlying continuum model. However, evaluation of the position of a bound particle requires summing over all pairs of modes, which may significantly increase computational effort.

The identification of the proper change in tension with membrane binding makes a critical difference for membrane reshaping. For example, in Refs. ³⁸, small positive tensions dramatically increase the protein coverage necessary for tubulation of a membrane by green fluorescent protein (GFP). The current crowding mechanism uses the on-surface model, in which binding by GFP to specific lipids reduces the tension. However, the in-surface model is likely the appropriate choice.

We demonstrated how the diffusion on the surface model described in Sec. II C alters equilibrium and dynamic properties of the system. In Sec. II C 1, we reviewed two ways to approximate the projected diffusion constant for the particles compared to the bare diffusion in the membrane. The two approximations are a preaveraging of the metric^9,10,36 and an area scaling of the projected vs actual membrane area.³⁶ Both approximations to the predicted diffusion constant show that the projected diffusion constant is smaller than the bare diffusion constant in the membrane. The same approximations apply to the on-surface method; however, the absolute value may differ.

We have used a general particle model, which can easily be expanded upon by using different membrane-particle coupled energy terms to investigate more complex systems, bearing in mind how they are attached in order to accurately represent the dynamics of the system. For instance, one could look at peripheral or embedded membrane proteins which induce membrane curvature or systems of varying lipid compositions. These types of systems are of great interest in terms of membrane deformations and redistribution on the membrane.

APPENDIX A: THE “IN-SURFACE” LIPID FLOW FIELD

The Monge gauge parameterizes a surface weakly fluctuating in z with its height above the xy plane as h(x, y). The Fourier transformation of the metric for converting between the flat surface area, dxdy, and the fluctuating surface area of this element is

$$\begin{array}{ccc}\hfill \sqrt{g_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(x,y\right)}& =\hfill & 1+\frac{1}{2}\underset{\left\{x,y\right\}-\mathrm{i}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}}{\underbrace{\sum _{\mathit{k}}|\mathit{k}{|}^2{L}^{-4}{\tilde{h}}_{\mathit{k}}{\tilde{h}}_{-\mathit{k}}}}\hfill \\ \hfill & \hfill & +\underset{\left\{x,y\right\}-\mathrm{d}\mathrm{e}\mathrm{p}\mathrm{e}\mathrm{n}\mathrm{d}\mathrm{e}\mathrm{n}\mathrm{t}}{\underbrace{\frac{1}{2}\sum _{\mathit{k},{\mathit{k}}^{\prime }\ne -\mathit{k}}\mathit{k}\cdot {\mathit{k}}^{\prime }{L}^{-4}{\tilde{h}}_{\mathit{k}}{\tilde{h}}_{{\mathit{k}}^{\prime }}{\mathrm{e}}^{ı\left(\mathit{k}+{\mathit{k}}^{\prime }\right)\cdot \left\{x,y\right\}}}},\hfill \end{array}$$ (A1)

but here the spatially invariant and variant components of the metric are separated. Varying components are complicating factors for the detailed balance because they couple the metric to the particle’s instantaneous position. Instead of transforming the planar point from {x′, y′, 0} to {x′, y′, h(x′, y′)}, we insert an xy coordinate change,

$$\begin{array}{ll}\hfill \left\{x^{\prime },y^{\prime },0\right\}\to & \left\{\right.x^{\prime }+u_x\left(x^{\prime },y^{\prime }\right),y^{\prime }+u_y\left(x^{\prime },y^{\prime }\right),\hfill \\ \hfill & h\left(\right.x^{\prime }+u_x\left(x^{\prime },y^{\prime }\right),y^{\prime }+u_y\left(x^{\prime },y^{\prime }\right)\left.\right\}.\hfill \end{array}$$ (A2)

The functions u_x(x′, y′) and u_y(x′, y′) will depend on $\left\{{\tilde{h}}_{\mathit{k}}\right\}$, and so a particle’s position will “flow” as the membrane fluctuates independently of the dynamics propagating its reference coordinates {x′, y′}. To first order in the magnitude of u, the metric factor for converting between these two surface representations is

$$\sqrt{g_{\mathrm{o}\mathrm{n}\mathrm{t}\mathrm{o}\mathrm{i}\mathrm{n}}\left(x^{\prime },y^{\prime }\right)}=1+\frac{1}{2}{\partial }_{x^{\prime }}{u}_x\left(x^{\prime },y^{\prime }\right)+\frac{1}{2}{\partial }_{y^{\prime }}{u}_y\left(x^{\prime },y^{\prime }\right)+\mathcal{O}\left[u^2\right],$$ (A3)

computed from the square root of the metric determinant of the coordinate transformation. We now choose functions u_x(x′, y′) and u_y(x′, y′) that cancel the varying component of $\sqrt{g_{\mathrm{o}\mathrm{n}\mathrm{s}\mathrm{u}\mathrm{r}\mathrm{f}\mathrm{a}\mathrm{c}\mathrm{e}}\left(x,y\right)}$ at second order in $\left\{{\tilde{h}}_{\mathit{k}}\right\}$,

$$\begin{array}{ccc}\hfill \mathbf{u}\left(x^{\prime },y^{\prime }\right)& =\hfill & \frac{ı}{2}\sum _{\mathit{k},{\mathit{k}}^{\prime }\ne -\mathit{k}}\exp \left[ı\left(\mathit{k}+{\mathit{k}}^{\prime }\right)\cdot \left\{x^{\prime },y^{\prime }\right\}\right]\hfill \\ \hfill & \hfill & \times \frac{\mathit{k}\cdot {\mathit{k}}^{\prime }}{\left(\mathit{k}+{\mathit{k}}^{\prime }\right)\cdot \left(\mathit{k}+{\mathit{k}}^{\prime }\right)}\left(\mathit{k}+{\mathit{k}}^{\prime }\right)L^{-4}{\tilde{h}}_k{\tilde{h}}_{k^{\prime }};\hfill \end{array}$$ (A4)

here the x and y components of u(x′, y′) are determined by the vector components of k and k′. The field u is second order in $\left\{{\tilde{h}}_k\right\}$, so by terminating the expansion at first order in u in Eq. (A3), we have kept terms up to second order in $\left\{{\tilde{h}}_k\right\}$. The total square root of the metric g_in for evaluation is now

$$\sqrt{g_{\mathrm{i}\mathrm{n}}\left(x^{\prime },y^{\prime }\right)}=1+\frac{1}{2}\underset{\mathrm{i}\mathrm{n}\mathrm{v}\mathrm{a}\mathrm{r}\mathrm{i}\mathrm{a}\mathrm{n}\mathrm{t}}{\underbrace{\sum _{\mathit{k}}|\mathit{k}{|}^2{L}^{-4}{\tilde{h}}_{\mathit{k}}{\tilde{h}}_{-\mathit{k}}}}+\mathcal{O}\left[{\left\{{\tilde{h}}_{\mathit{k}}\right\}}^3\right],$$ (A5)

that is, only the invariant portion is retained to second order in $\left\{{\tilde{h}}_{\mathit{k}}\right\}$.

APPENDIX B: LAPLACE-BELTRAMI WITH FOURIER SPACE BROWNIAN DYNAMICS FORMALISM

In Table I, the models derived in this paper are compared to the “Laplace-Beltrami with Fourier Space Brownian Dynamics” method. The equation of motion for particles diffusing on a curved surface is derived using this method here; in comparison with the on-surface method defined in Sec. II C, this method only applies to the particles. In the LB + FSBD method,^9–11 the Laplace operator in the Fokker-Planck equation is replaced by the Laplace-Beltrami operator, which acts on a scalar function f as

$${\nabla }_{LB}^{2}f=\frac{1}{\sqrt{g\left(\mathit{r}\right)}}{\partial }_{\alpha }\left(\sqrt{g\left(\mathit{r}\right)}{\left(g^{-1}\right)}_{\alpha \beta }{\partial }_{\beta }f\right).$$ (B1)

The Fokker-Planck equation that describes particle diffusion on a curved surface is

$$\frac{\partial \rho \left(\mathit{r},t\right)}{\partial t}=D_0\frac{1}{\sqrt{g\left(\mathit{r}\right)}}{\partial }_{\alpha }\left(\sqrt{g\left(\mathit{r}\right)}{\left(g^{-1}\right)}_{\alpha \beta }{\partial }_{\beta }\rho \left(\mathit{r},t\right)\right),$$ (B2)

using Eq. (B1). This assumes that the probability density is normalized as $\int \mathrm{d}\mathit{r}\sqrt{g\left(\mathit{r}\right)}\rho \left(\mathit{r},t\right)=1$. However, it can be normalized such that ∫drρ(r, t) = 1, which defines ρ as the projected density conveniently. This leads to a slight change in how the Laplace-Beltrami operator is expressed within the Fokker-Planck equation,

$$\frac{\partial \rho \left(\mathit{r},t\right)}{\partial t}=D_0{\partial }_{\alpha }\left(\sqrt{g\left(\mathit{r}\right)}{\left(g^{-1}\right)}_{\alpha \beta }{\partial }_{\beta }\frac{1}{\sqrt{g\left(\mathit{r}\right)}}\rho \left(\mathit{r},t\right)\right).$$ (B3)

The general form of the Fokker-Planck equation is

$$\frac{\partial \rho \left(\mathit{r},t\right)}{\partial t}={\partial }_{\alpha }\left[{\nu }_{\alpha }\rho \left(\mathit{r},t\right)+{\mathrm{\Gamma }}_{\alpha \beta }{\partial }_{\beta }\rho \left(\mathit{r},t\right)\right],$$ (B4)

which has a corresponding Langevin equation of motion of the form

$$\frac{\mathrm{d}{r}_{\alpha }}{\mathrm{d}t}={\nu }_{\alpha }+\sqrt{2}{\mathrm{\Gamma }}_{\alpha \beta }^{1/2}{\zeta }_{\beta }.$$ (B5)

The drift, ν, and the mobility, Γ_αβ, can be determined from Eqs. (B3) and (B4) by identifying the factors proportional to the density (the drift) and its gradient (the mobility),

$${\nu }_{\alpha }=-D_0{\left(g^{-1}\right)}_{\alpha \beta }\left[{\partial }_{\beta }{\partial }_{\gamma }h\left(\mathit{r}\right)\right]\frac{{\partial }_{\gamma }h\left(\mathit{r}\right)}{g\left(\mathit{r}\right)},$$ (B6)

$${\mathrm{\Gamma }}_{\alpha \beta }=D_0{\left(g^{-1}\right)}_{\alpha \beta }.$$ (B7)

Therefore, the equation of motion for a particle diffusing on a curved surface is

$$\frac{\mathrm{d}{r}_{\alpha }}{\mathrm{d}t}=-D_0{\left(g^{-1}\right)}_{\alpha \beta }\left[{\partial }_{\beta }{\partial }_{\gamma }h\left(\mathit{r}\right)\right]\frac{{\partial }_{\gamma }h\left(\mathit{r}\right)}{g\left(\mathit{r}\right)}+\sqrt{2D_0}{\left(g^{-1/2}\right)}_{\alpha \beta }{\zeta }_{\beta }.$$ (B8)

As noted by Sigurdsson et al.,²⁸ using this equation of motion in conjunction with a membrane equation of motion [Eq. (36)] that does not take into account E_eff, but instead uses E_m, leads to an inconsistent probability distribution. The methods described in the main text are our solutions to this.