Quantification of Curvature Sensing Behavior of Curvature-Inducing Proteins on Model Wavy Substrates

Abstract

Curvature-inducing proteins are involved in a variety of membrane remodeling processes in the cell. Several in vitro experiments have quantified the curvature sensing behavior of these proteins in model lipid systems. One such system consists of a membrane bilayer laid atop a wavy substrate (Hsieh in Langmuir 28:12838–12843, 2012). In these experiments, the bilayer conforms to the wavy substrate, and curvature-inducing proteins show preferential segregation on the wavy membrane. Using a mesoscale computational membrane model based on the Helfrich Hamiltonian, here we present a study which analyzes the curvature sensing characteristics of this membrane-protein system, and elucidates key physical principles governing protein segregation on the wavy substrate and other in vitro systems. In this article we compute the local protein densities from the free energy landscape associated with membrane remodeling by curvature-inducing proteins. In specific, we use the Widom insertion technique to compute the free energy landscape for an inhomogeneous system with spatially varying density and the results obtained with this minimal model show excellent agreement with experimental studies that demonstrate the association between membrane curvature and local protein density. The free energy-based framework employed in this study can be used for different membrane morphologies and varied protein characteristics to gain mechanistic insights into protein sorting on membranes.

Graphical Abstract

Introduction

Remodelling of membrane curvature is an essential part of several cellular processes ranging from cell division and motility to cargo trafficking and growth (McMahon and Boucrot 2015; Jarsch et al. 2016). Among several mechanisms postulated for membrane remodelling, a class of peripheral membrane proteins capable of generating membrane curvature play a central role (Zimmerberg and Kozlov 2006; Shibata et al. 2009; Kozlov et al. 2014). Curvature-inducing proteins are also capable of sensing membrane curvature (Peter et al. 2004; Bhatia et al. 2009; Baumgart et al. 2011; Martyna et al. 2016; Tsai et al. 2021). This duality is thought to enable cells in sorting these proteins to curved regions using their curvature sensing behavior, and further augmenting the change in membrane curvature. For instance, dynamin GTPases, key drivers of membrane scission (Ferguson and De Camilli 2012), have been shown in vitro to favor binding on higher curvature membrane tubes as compared to liposomes (Roux et al. 2010). This property of dynamin as a curvature sensor provides a possible mechanism by which cells efficiently recruit dynamin proteins to the constricted neck region of mature clathrin-bound buds to induce membrane fission for detaching the vesicle.

Several in vitro membrane systems have been utilized to delineate the complex process of curvature sensing employed by proteins (Has and Das 2021). One such model involves pulling a membrane tether from a Giant Unilammelar Vesicle (GUV) (Tian and Baumgart 2009; Heinrich et al. 2010; Capraro et al. 2010) where the positive curvature-inducing proteins were found to preferentially translocate to highly curved tether region of the membrane. However, a common shortcoming of GUV-based and other experimental techniques such as single liposome curvature (SLiC) assay (Bhatia et al. 2009) is that sensitivity to only positive curved surfaces can be explored. Methodology proposed by Hsieh et al. (2012) overcomes this limitation by using a lipid bilayer atop a predefined wavy substrate surface. In this wavy substrate model, the lipid bilayer conforms to the underlying wavy glass substrate and thereby, creates a nonuniform curvature profile. They showed that positive curvature-inducing proteins (Epsin N-terminal homology (ENTH) and N-terminal Bin-Amphiphysin-Rvs (N-BAR) domains) localized to the peaks of the wave, while negative curvature-inducing proteins (the cholera toxin subunit B (CTB)) preferred the valleys. Furthermore, experimental method used in Hsieh et al. (2012) was able to quantify curvature sensing behavior over a mean curvature range an order of magnitude smaller than previous experiments measuring protein segregation on liposome tethers.

In this article, we seek to characterize the curvature sensitive segregation of proteins within the framework of a mesoscale computational membrane model to elucidate, through free energy considerations, the factors driving curvature sensing by peripheral membrane proteins. We performed Monte Carlo based simulations for a sinusoidal membrane geometry to quantify curvature sorting for both positive and negative curvature-inducing proteins. Our results delineate the spatially varying density using a free energy landscape based perspective for the curvature sensing behavior of proteins.

Methods

Continuum Mesoscale Membrane Model

Similar to the mesoscale membrane model in our previous works (Ramakrishnan et al. 2014; Tourdot et al. 2014a, b), the membrane is modeled as a thin elastic sheet consisting of a triangle mesh with $T$ triangles, constructed with $N$ vertices and $L$ links that connect the vertices (Ramakrishnan et al. 2010). Each vertex is modeled as a hardsphere of radius $a_0$ and self-avoidance is imposed by restricting the link length $l$ to be in the range $a_0\le l\le \sqrt{3}{a}_0$. We note that the length scale in the model is set by the value of $a_0$. For biological membranes whose thickness is negligible when compared to its lateral dimensions, the thermodynamic behavior of the membrane is well captured by the elastic energy functional given by

$$\mathscr{H}=\sum _{v=1}^N\left\{\frac{\kappa }{2}{\left(C_{1,v}+C_{2,v}-H_{0,v}\right)}^2+{\sigma }_{\text{bare}}\right\}{A}_v,$$ (1)

which is a discretized form of the Canham-Helfrich Hamiltonian (Helfrich 1973); where the material properties $\kappa$ and ${\sigma }_{\text{bare}}$ represent the bending rigidity and the bare surface tension respectively. $C_{1,v}$ and $C_{2,v}$ are the curvatures at vertex $v$ along the two principle axis, computed as in (Ramakrishnan et al. 2010). $H_{0,v}$ is a spontaneous curvature field at vertex $v$ which captures the curvature inducing interactions between the protein and membrane (see section 2.2 for details). This approach of treating the effect of the curvature-inducing protein as a curvature field in the continuum field formulation has been utilized in prior studies (Weinstein and Radhakrishnan 2006; Agrawal et al. 2010; Liu et al. 2012; Zhao et al. 2013; Ramakrishnan et al. 2018). The core methodology of dynamically triangulated Monte Carlo (DTMC) framework used for evolving the membrane is same as that used in our previous studies (Tourdot et al. 2014a, b, 2015).

Membrane-Protein Model

In this study, the curvature induced in the membrane at ${\overrightarrow{r}}_m$ due to a protein field at ${\overrightarrow{r}}_p$ is represented as,

$$H_0({\overrightarrow{r}}_m,{\overrightarrow{r}}_p)=C_0ℱ({\overrightarrow{r}}_m,{\overrightarrow{r}}_p),$$ (2)

where $C_0$ is the induced membrane curvature at ${\overrightarrow{r}}_m={\overrightarrow{r}}_p$. We choose this deformation profile $ℱ({\overrightarrow{r}}_m,{\overrightarrow{r}}_p)$ to be a Gaussian function. Thus, a radially symmetric curvature profile has the form

$${ℱ}_{\text{iso}}(r)=\text{exp}\left(-\frac{r^2}{{ϵ}^2}\right).$$ (3)

where $r=\mid {\overrightarrow{r}}_m-{\overrightarrow{r}}_p\mid$ and the ${ϵ}^2∕2$ is the variance. The form of the curvature field and the numerical value of the field-parameters has been justified by performing molecular dynamics simulations at the atomic or near-atomic (coarsegrained) scales; see Tourdot et al. (2014a) where the form in Eq. (3) is also justified.

Each protein field in our model is associated with a vertex and each vertex can have at most one such field. To account for enhancement of membrane spontaneous curvature when multiple protein fields are in vicinity of each other, we use a simple additive rule involving linear addition of each field’s curvature contributions at a given membrane location with a truncation at an upper threshold value, defined as

$$H_0({\overrightarrow{r}}_m)=\text{min}\left(2C_0,\sum _{p=1}^{n_p}{H}_0({\overrightarrow{r}}_m,{\overrightarrow{r}}_p)\right),$$ (4)

A Snapshot of the simulated membrane-protein system in Fig. 1a shows the spontaneous curvature induced by protein fields during the simulation.

Fig. 1.

Wavy membrane model with curvature-inducing protein fields. a Snapshot during the simulation of membrane having $n_p=10$ and initialized with $A=2a_0$ and $\lambda =40a_0$ showing the spontaneous curvature induced by protein fields. b Two model wavy substrate systems which are modeled with a periodic sine function with an amplitude $A=2a_0$. Systems have a wavelength of $20a_0$ and $40a_0$ respectively. The snapshots shown are colored by their mean curvature $H$. Note that $a_0$ is the characteristic length scale of the model. c Mean curvature $H$ along the $x$-dimension for initial wavy membrane having with $A=2a_0$ and $\lambda =40a_0$

Widom Field (or test particle) Insertion Method

The change in the free energy of the membrane due to protein binding can be determined using Widom field insertion method (Widom 1963) in terms of the excess chemical potential (${\mu }_{\text{ex}}$) for $n_{{}_P}$ protein fields. Widom insertion samples the excess chemical potential by randomly inserting a protein field, and determining the energy difference due to the extra field. In this article, we follow the inhomogeneous Widom insertion approach from our prior studies (Tourdot et al. 2014b, 2015) to compute the spatially dependent excess chemical potentials .

If $\mathbf{r}$ denotes a state point in the configurational phase space, then ${\mu }_{\text{ex}}$ for spatially inhomogeneous membrane having $n_{{}_P}$ protein fields is given by (Frenkel and Smit 2002)

$${\mu }_{\text{ex}}(\mathbf{r})=-k_{B}T\text{ln}\langle \text{exp}(-\beta \Delta \mathscr{H}(\mathbf{r})){\rangle }_{n_{{}_P}},$$ (5)

where $\beta =(k_{B}T{)}^{-1}$ and the ensemble average $\langle \cdot \rangle$ is taken over the phase space defined by the membrane and the $n_{{}_P}$ protein fields. Here, $\Delta \mathscr{H}(\mathbf{r})$ denotes change in membrane energy due to insertion of $(n_{{}_P}+1{)}^{th}$ protein field at $\mathbf{r}$.

The total chemical potential ($\mu$) of the membrane can be decomposed into contributions from ideal and excess parts, written as

$$\mu ={\mu }_{\text{id}}(\rho )+{\mu }_{\text{ex}}(\mathbf{r}),$$ (6)

where the ideal part can be calculated from the protein density $\rho$ as ${\mu }_{\text{id}}=k_{B}T$ In $\rho$. Note that here we have only considered the configurational component of the ideal part and not included contributions from the internal degrees of freedom. At equilibrium, the chemical potential $\mu$ is constant throughout the system. Hence, the density has to be spatially inhomogeneous and it can be written as

$$\rho (\mathbf{r})∕{\rho }_0=\langle \text{exp}(-\beta \Delta \mathscr{H}(\mathbf{r})){\rangle }_{n_p},$$ (7)

where ${\rho }_0=\text{exp}(\beta \mu )$. Note that Widom insertion technique is more suitable to probe chemical potentials in dilute systems while its applicability to systems with large protein concentration is limited (Tourdot et al. 2014b).

Wavy Substrate System

A wavy substrate supported bilayer is simulated by initializing a membrane patch (Sect. 2.1) as a sinusoid along one dimension with a prescribed amplitude ($A$) and wavelength ($\lambda$). The membrane has to be randomly pinned to this sine wave to ensure that the wavy geometry is maintained as the membrane undergoes fluctuations during simulation. Snapshots of the model wavy substrate initial conditions are shown in Fig. 1b. For $A=0.2a_0$ and $\lambda =40a_0^{-1}$ used to initialize the wavy geometry of the membrane, Fig. 1c shows how the mean curvature ($H$) varies along the x-dimension. In case of a 1-dimensional sinusoidal surface $\left(f(x)=A\sin \left(\frac{2\pi x}{\lambda }\right)\right)$, the mean curvature can be derived as a function of $x$ (see supplementary information, Sect. S1). The equation for $H(x)$ is given by

$$H(x)=\frac{A{\left(\frac{2\pi }{\lambda }\right)}^2\sin \left(\frac{2\pi x}{\lambda }\right)}{2{\left(1+{\left[A\left(\frac{2\pi }{\lambda }\right)\cos \left(\frac{2\pi x}{\lambda }\right)\right]}^2\right)}^{3∕2}}.$$ (8)

Simulation Parameters

The results presented here are for a membrane patch with $N=1600$ vertices, ${\sigma }_{\text{bare}}=0$ and a vertex hard sphere radius of $a_0=10\text{nm}$. The initial link length $l$ is taken as $1.3a_0$. The amplitude and wavelength for the sine wave used to initialize the membrane are taken as $A=2a_0$ and $\lambda =40a_0$, and ~ 6% of the vertices are pinned to ensure that the wavy geometry is maintained for the membrane. The mean curvature range analyzed by the simulation is −0.0025 nm⁻¹ to 0.0025 nm⁻¹ which is similar to what was used in the wavy substrate experiments (Hsieh et al. 2012).

In this study, to obtain free energy landscape along $x$-dimension, $\mathbf{r}$ in Eq. 5 is binned (histogrammed) based on the value of coordinate $x$ of the vertex where the test protein field is inserted. Each simulation is run for 5 million Monte Carlo (MC) steps to ensure sufficient sampling for Widom insertion calculations of ${\mu }_{\text{ex}}$. During the first 0.5 million MC steps, the membrane is allowed to equilibirate. Thereafter, Widom test protein field is inserted every 100 MC steps at randomly chosen spatial location (vertex) and the value for $exp(-\beta \Delta \mathscr{H})$ is recorded for every insertion move. Note that the ${\mu }_{\text{ex}}$ values reported here are averaged over four replicate ensembles.

Results

Excess Chemical Potential along the Wavy Membrane

We employ inhomogeneous Widom insertion method to quantify the equilibrium chemical potential along the wave, by sampling ${\mu }_{\text{ex}}$ in several windows along the $x$-axis. The spatially dependent excess chemical potential, ${\mu }_{\text{ex}}(x)$, is shown in Fig. 2a for a range of $C_0$ values with a fixed ${ϵ}^2$ of 2.3 $a_0^2$. It can be seen that for proteins with positive spontaneous curvature field (i.e. $C_0>0$) ${\mu }_{\text{ex}}$ is negative in the region around $x=10a_0$ which corresponds to the peak area of the sinusoidal membrane. In contrast, negative curvature-inducing proteins ($C_0<0$) have negative ${\mu }_{\text{ex}}$ at $x\approx 30$ where the valley of the wavy membrane exists. The locations of minima in the ${\mu }_{\text{ex}}$ landscape corresponds to thermodynamically favorable position for proteins. Therefore, our model predicts that positive curvature-inducing proteins such as ENTH and BAR domain proteins prefer the positively curved regions of the wavy substrate while the negative curvature-inducing proteins such as IBAR, Exo70, and CTB favor the negatively curved regions of the membrane. These observations are in accordance with the experiments by Hseih and coworkers where they demonstrated on a solid-supported wavy membrane ENTH and N-BAR domains preferentially partition into positively curved regions whereas CTB partitions into negatively curved regions (Hsieh et al. 2012). In effect, the excess chemical potential difference along the wave can be considered as defining a driving force for curvature sensing by proteins.

Fig. 2.

Excess chemical potential obtained with inhomogenenous Widom insertion method for different values of a $C_0$, b ${ϵ}^2$, c $\kappa$, and d $n_{{}_P}$. All panels have parameter values as $C_0=0.2a_0^{-1}$, ${ϵ}^2=2.3a_0^2$, $\kappa =10k_{B}T$, and $n_{{}_P}=2$ unless stated otherwise

Another trend visible from our simulations is the increased difference between ${\mu }_{\text{ex}}$ of valley and peak (i.e. more driving force for migration to preferred curvature region) for higher values of $C_0$ (Fig. 2a), ${ϵ}^2$ (Fig. 2b) and $\kappa$ (Fig. 2c). This points to a stronger curvature sensing behavior for larger values of protein field parameters ($C_0$, ${ϵ}^2$) and bending stiffness. The Widom insertion calculations discussed so far are done near infinite dilution ($n_{{}_P}=2$). We also performed simulations for some higher values of $n_{{}_P}$. It can be seen in Fig. 2d that for concentrations ranging from $n_{{}_P}=2$ to $n_{{}_P}=14$ the ${\mu }_{\text{ex}}$ landscape for curvature sensing is not altered significantly. We note that only moderate number of protein fields are considered to ensure that free energy landscape calculations for curvature sensing are not effected by significant changes in membrane morphology (like tubulation) at higher curvature inducing protein concentrations. This extensive membrane curvature change after a critical value of $n_{{}_P}$ has been described in our previous studies (Tourdot et al. 2015; Ramakrishnan et al. 2018).

Comparison of Normalized Protein Density with Experiments

Due to the spatial dependence in excess chemical potential, the protein density has to be inhomogeneous along the wavy membrane to ensure that the total chemical potential is constant throughout the system at equilibrium (see Sect. 2.3). We make use of the Widom insertion method to determine the protein density on the membrane using Eq. 7. To quantify the density with respect to membrane curvature, we use the direct mapping between mean curvature $H$ and $x$ coordinate (Fig. 1c). We note that the pinning of the membrane in our system (described in Sect 2.4) keeps the wavy geometry of the membrane along $x$ approximately similar to the initialized sinusoidal geometry throughout the simulation. Membrane pinning can be seen as a biophysical abstraction of membrane adhesion to underlying cortex (through cytoskeleton-membrane linker proteins) or to surrounding ECM. While in this study we have used a small amount of pinning to maintain wavy nature of the membrane, our previous studies have delineated how membrane anchorage can affect the emergent membrane morphology (Kandy and Radhakrishnan 2019), and the impact of pinning on nanoparticle binding to membrane (Farokhirad et al. 2021).

We test our model predictions of normalized (with respect to the average density of the system) density against the normalized fluorescence intensity of peripheral membrane proteins on solid-supported wavy membranes reported by Hsieh et al. (2012). In the experiments by Hseih and coworkers, ENTH, BAR and CTB were fluorescent tagged and their respective distributions with respect to the membrane curvature on a wavy substrate was quantified in terms of normalized fluorescence intensity. Therefore, normalized intensity can be used a measure of protein density along the wavy membrane, and provides a direct comparison between our computational results and the experimental data.

Figure 3 depicts the spatially inhomogeneous normalized density of proteins with respect to the mean curvature for different values of $C_0$, ${ϵ}^2$ and $\kappa$. It can be seen that the protein density varies monotonically with $H$. The positive curvature-inducing proteins show an increase in normalized density with increasing membrane curvature whereas for negative curvature-inducing proteins, the normalized density decreases with increasing $H$. In addition to showing analogous monotonic dependence on membrane curvature, the normalized density values are also found to be in good quantitative agreement with the experimental measures of normalized fluorescence intensity (Fig. 3a). The degree of density scaling with $H$ increases with higher values of $C_0$ (Fig. 3a), ${ϵ}^2$ (Fig. 3b) and $\kappa$ (Fig. 3c). This can be explained by the stronger curvature sensing behavior for larger Gaussian field parameters and bending stiffness due to greater variation in excess chemical potential across $x$ (as discussed in Sect. 3.1; Fig. 2). Therefore, our results indicate that stronger curvature-inducing proteins (i.e. higher $C_0$ and ${ϵ}^2$) have better curvature sensing capabilities, and also that the extent of curvature-based sorting of proteins is greater on a more rigid membrane (higher $\kappa$).

Fig. 3.

Normalized protein density obtained with inhomogenenous Widom insertion method for different values of a $C_0$, b ${ϵ}^2$, and c $\kappa$. The filled symbols with solid lines correspond to experimental values (expt) of normalized fluorescence intensity for ENTH, N-BAR and CTB reported in Hsieh et al. (2012). Simulations data (sim) in all panels is depicted for $C_0=0.1a_0^{-1}$, ${ϵ}^2=2.3a_0^2$, $\kappa =5k_{B}T$, and $n_{{}_P}=2$ unless otherwise stated

Analytic Approximation for Excess Chemical Potential

The excess chemical potential can be analytically solved for the wavy membrane system in the zero temperature limit (i.e. no undulations in the membrane). At infinite dilution and no fluctuations limit, ${\mu }_{\text{ex}}(x)$ can be simplified to

$${\mu }_{\text{ex}}(x)=\frac{\kappa \pi {ϵ}^2{C}_0^2}{4}-2\kappa {\int }_{S}H(x^{\prime })H_0(x^{\prime },y^{\prime },x,y)d{x}^{\prime }d{y}^{\prime },$$ (9)

where $H_0\left(x^{\prime },y^{\prime },x,y\right)$ is the curvature induced at $(x,y)$ by a protein field at ($x^{\prime }$, $y^{\prime }$) and $H(x^{\prime })$ is the mean curvature of the sinusoidal wave defined in Eq. (8). Detailed derivation of the Eq. 9 is presented in the supplementary information (section S2).

However, due to a highly non-trivial integrand, Eq. 9 cannot be solved to obtain a closed form solution. But, if the protein curvature field is approximated as a point source of curvature-field or a Dirac delta function ($H_0=C_0\delta (r)$), then the integral can be solved and it is possible to obtain a closed form analytical expression for ${\mu }_{\text{ex}}(x)$. Thus, as a discrete approximation of the Gaussian field (Eq. 2), we can take the curvature field for protein as point source of spontaneous curvature given by

$$H_0({\overrightarrow{r}}_m,{\overrightarrow{r}}_p)=C_0\delta (r),$$ (10)

where $r=\mid {\overrightarrow{r}}_m-{\overrightarrow{r}}_p\mid$. Using Eq. 10, we can simplify Eq. 9 as

$${\mu }_{\text{ex}}(x)=\frac{\kappa \pi C_0^2}{2A_{vertex}}-2\kappa C_{0}H(x),$$ (11)

$$\begin{gathered} {\mu }_{\text{ex}}(x)=\frac{\kappa \pi C_0^2}{2A_{vertex}} \\ -\kappa C_0\frac{A{\left(\frac{2\pi }{\lambda }\right)}^2\sin \left(\frac{2\pi x}{\lambda }\right)}{{\left(1+{\left[A\left(\frac{2\pi }{\lambda }\right)\cos \left(\frac{2\pi x}{\lambda }\right)\right]}^2\right)}^{3∕2}}. \end{gathered}$$ (12)

Here, $A_{vertex}=\sqrt{3}(1.3a_0{)}^2∕2$ is the area per vertex in our discrete triangular mesh (described in Sect. 2.1) and $1.3a_0$ is the link length value used in the simulations. The factor ($A_{vertex}$) arises due to the discrete approximation to Dirac delta function. Since in the zero temperature limit ${\mu }_{\text{ex}}$ is equivalent to $\Delta \mathscr{H}$ (described in Sect. S2), the first term in Eq. 12 can be interpreted as representative of energy required to insert a protein on a flat membrane. The second term would then correspond to a decrease in the energy required for adding a protein on the wavy membrane based on how well the curvature profile ($C_0$) of the protein matches the background curvature of the sinusoidal wave. Figure 4 shows the comparison between normalized density values evaluated using zero temperature limit expression for ${\mu }_{\text{ex}}$ in Eq. 12 and the density values obtained from our simulations. The monotonic trend for analytically determined normalized density with respect to membrane curvature is similar to the curvature dependence of normalized density obtained from simulations. However, the extent of curvature sorting based on analytical values is much lower compared to simulation based prediction. This difference indicates that membrane undulations at finite temperature result in more preferential curvatures (i.e. better matching with the curvature profiles of protein fields) in the hill (and valley) region as compared to the background curvature of (sinusoidal) membrane which, though having favorable curvature regions (hill and valley), is almost an order of magnitude lower than the curvature associated with protein fields. Therefore, even though the undulations that can disturb the wavy membrane geometry itself are avoided by small amount of membrane pinning in simulation, the correction term in excess chemical potential due to fluctuations in membrane at finite temperature can lead to higher curvature sorting observed in simulations as compared to zero temperature limit.

Fig. 4.

Comparison of normalized protein density evaluated analytically in the zero temperature limit and infinite dilution (Eq. 12) with the values obtained through simulation for different values of $C_0$. The parameters for the simulations data (sim) depicted by open symbols are ${ϵ}^2=2.3a_0^2$, $\kappa =5k_{B}T$, and $n_{{}_P}=2$. Eq. 12 (eqn) represented by different line styles is solved for $\kappa =5k_{B}T$. Note that the units of $C_0$ are $a_0^{-1}$ for sim and $a_0$ for eqn due to the discrete approximation of curvature field to Dirac delta function

Conclusion

In this work, we utilize a mesoscale computational model for wavy membrane incorporated with curvature-inducing proteins as spontaneous curvature fields to yield a curvature sensing free energy landscape for the proteins. Consistent with (Hsieh et al. 2012), our model shows proteins favoring certain regions of the membrane (positively or negatively curved) depending on their curvature inducing profile. We posit that this curvature sensing behavior can be explained from a thermodynamic perspective using free energy landscape. Our model predicts key factors, including the strength and extent of the spontaneous curvature field, and the bending stiffness of the membrane, which define the curvature sensing behavior of proteins. This study outlines a free energy-based framework to gain mechanistic understanding into protein sorting on membranes, and can be implemented for different membrane morphologies. The thermodynamic based description of curvature sensing can be also be extended to study the effect of protein specific characteristics (such as anisotropic curvature deformation (Lai et al. 2012; Simunovic et al. 2013)) on their curvature sensing behavior through changes in free energy landscape.