Binding of anisotropic curvature-inducing proteins onto membrane tubes

Protein binding on a cylindrical membrane tube is considered as depicted in Fig. 1(a). The membrane is in a fluid phase and the surface area $A$ is fixed. The radius and length of the tube are $R_{\rm cy}$ and $L_{\rm cy}$: $A=2\pi R_{\rm cy}L_{\rm cy}$. The tube volume can be freely changed. This corresponds to the tubular region of a tethered vesicle in the limit condition, in which the tube volume is negligibly small ($\pi R_{\rm cy}^{2}L_{\rm cy}\ll V$, where $V$ is the vesicle volume) Smith et al. (2004); Noguchi (2021b). Proteins can align in the membrane surface. To quantify it, the degree of the orientational order is calculated as

where $\langle...\rangle$ is the ensemble average, and $\theta_{\rm ps}$ is the angle between the major protein axis and the ordered direction. The angles between the ordered direction and the azimuthal direction of the cylinder and between the major protein axis and azimuthal direction are $\theta_{\rm sc}$ and $\theta_{\rm pc}$, respectively, with $\theta_{\rm ps}=\theta_{\rm pc}+\theta_{\rm sc}$ (see Fig. 1(a)). Experimentally, the oritational order $S_{z}$ along the tube ($z$) axis is more easily measured: for $\theta_{\rm sc}=0$ and $\pi/2$, $S_{z}=-S$ and $S_{z}=S$, respectively.

The bound protein is approximated to have laterally an elliptic shape with an aspect ratio of $d_{\rm el}=\ell_{1}/\ell_{2}$ and area $a_{\rm p}=\pi\ell_{1}\ell_{2}/4$, where $\ell_{1}$ are $\ell_{2}$ are the lengths in the major and minor axes, respectively. An orientation-dependent excluded-volume interaction is considered between proteins. When two proteins are perpendicularly oriented, the excluded area $A_{\rm exc}$ between them is larger than the parallel pairs, as shown in Fig. 1(b). This area $A_{\rm exc}$ is expressed as $A_{\rm exc}=B_{0}+B_{2}(\cos^{2}(\theta_{\rm pp})-1/2)+O(\cos^{4}(\theta_{\rm pp}))$ by a Taylor expansion, where $\theta_{\rm pp}$ is the angle between the major axes of two ellipses. In our previous study Tozzi et al. (2021), the values of $B_{0}$ and $B_{2}$ are calculated by a two-parameter fit. In this study, the one-parameter fit of $A_{\rm exc}=[4-b_{\rm exc}(\cos^{2}(\theta_{\rm pp})-1)]a_{\rm p}$ is used, since the minimum value $A_{\rm exc}^{\rm min}=4a_{\rm p}$ is obtained at the parallel pairs ($\theta_{\rm pp}=0$) for any ratio of $d_{\rm el}$: $b_{\rm exc}=0.840$, $1.98$, $3.44$, and $7.61$ at $d_{\rm el}=2$, $3$, $4$, and $7$, respectively. The effective excluded area is represented by $A_{\rm eff}=\lambda A_{\rm exc}$. The parameter $\lambda$ is a function of the protein density and $\lambda=1/2$ at a low-density limit Nascimento et al. (2017). At the close-packed condition, the area fraction $\phi$ of the bound protein has the maximum: $\phi_{\rm max}=a_{\rm p}/\lambda A_{\rm exc}^{\rm min}=1/4\lambda=\pi/2\sqrt{3}\approx 0.907$ in two-dimensional space Tanemura and Matsumoto (1997). For simplicity, we use a constant value, $\lambda=1/3$, as in our previous study Tozzi et al. (2021); Roux et al. (2021), i.e., $\phi_{\rm max}=0.75$. In this study, we consider no attractive interactions between the proteins and focus on isotropic and nematic phases, such that smectic and crystal phases are not in the scope.

The bending energy of the bare (unbound) membrane is given by

where $C_{1}$ and $C_{2}$ are the principal curvatures ($C_{1}=1/R_{\rm cy}$ and $C_{2}=0$ for the cylinder). The unbound membrane has a bending rigidity $\kappa_{\rm d}$ and zero spontaneous curvature. The tubular membrane is connected to a large lipid reservoir, and the area difference elasticity Seifert (1997); Svetina (2009) is negligible. The bound protein gives an additional bending energy as $\langle U\rangle=U_{\rm mb}+N_{\rm p}\langle U_{\rm p}\rangle$, where $N_{\rm p}=\phi A/a_{\rm p}$ is the number of the bound protein and $U_{\rm p}$ is the bending energy of one protein. The protein has an anisotropic bending energy:

where $C_{\ell 1}$ and $C_{\ell 2}$ are curvatures along the major and minor axes of the proteins, respectively. $\kappa_{\rm p}$ and $C_{\rm p}$ are the bending rigidity and spontaneous curvature along the major protein axis, respectively, and $\kappa_{\rm side}$ and $C_{\rm side}$ are those along the minor axis (side direction). Here, $\kappa_{\rm side}=0$ is used unless otherwise specified.

The free energy $F_{\rm p}$ of the bound proteins is expressed as

where $\Theta(x)$ denotes the unit step function, $k_{\rm B}T$ is the thermal energy, and $\beta=1/k_{\rm B}T$. The factor $g$ expresses the effect of the orientation-dependent excluded volume, where $b_{0}=B_{0}\lambda/a_{\rm p}=(4+b_{\rm exc}/2)\lambda$ and $b_{2}=-B_{2}\lambda/a_{\rm p}=b_{\rm exc}\lambda$. Unoverlapped states exist at $g>0$. Since $w(\theta_{\rm ps})$ is the weight of each protein orientation, the ensemble average of a quantity $\chi$ is given by

The quantities $\Psi$ and $\bar{\Psi}$ are the symmetric and asymmetric components of the nematic tensor, respectively, and are determined by Eq. (1) and $\langle\sin(\theta_{\rm ps})\cos(\theta_{\rm ps})\rangle=0$ using Eq. (11). When the nematic order is parallel to one of the directions of the membrane principal curvatures ($\theta_{\rm sc}=0$ or $\pi/2$), $\bar{\Psi}=0$. The free energy minimum is calculated from $\partial f_{\rm p}/\partial S=\partial f_{\rm p}/\partial\theta_{\rm sc}=0$. More detail is described in Ref. 42.

In this study, we examine the axial force $f_{\rm ex}$ and the equilibrium of the protein binding and unbinding. In experiments, an external force $f_{\rm ex}$ is imposed by optical tweezers and micropipette in order to extend a membrane tube from a liposome. The free energy is give by $F=F_{\rm p}+U_{\rm mb}-f_{\rm ex}L_{\rm cy}$. This force $f_{\rm ex}$ is balanced with the membrane axial force and is obtained by $\partial F/\partial L_{\rm cy}=0$ as

The last term $f_{\rm mb}$ represents the force of the bare membrane tube ($\phi=0$),

where $f_{0}=2\pi\kappa_{\rm d}C_{\rm p}$ is the force at $R_{\rm cy}C_{\rm p}=1$ and is used as the unit hereafter.

The proteins bind and unbind the membrane with the binding chemical potential $\mu$. The equilibrium of the binding and unbinding is obtained by minimizing $F-\mu N_{\rm p}$. Hence, the equilibrium protein density is calculated from $\mu=a_{\rm p}\partial f_{\rm p}/\partial\phi$. Here, the number $N_{\rm lip}$ of the lipids and the area $A$ remain constant, so that the ensemble is changed from the $N_{\rm p}N_{\rm lip}AT$ ensemble to $\mu N_{\rm lip}AT$ ensemble.

Unless otherwise specified, we use $d_{\rm el}=3$ and $a_{\rm p}C_{\rm p}^{2}=0.26$, which correspond to the N-BAR domain ($\ell_{1}=15$ nm $\ell_{2}=5$ nm, and $1/C_{\rm p}=15$ nm) Roux et al. (2021). Another area ratio $a_{\rm p}C_{\rm cy}^{2}=0.26$ is used to examine $C_{\rm p}$ dependence. The detail of the numerical methods is described in Appendix A.

For flat membranes, the proteins exhibit a first-order transition from a randomly oriented state ($S=0$) to an ordered state ($S>0$) with increasing protein density $\phi$, as shown in Fig. 2(a). This transition density decreases as the elliptic ratio $d_{\rm el}$ increases, and the same behavior is obtained for spherical membranes Tozzi et al. (2021). For cylindrical membranes, the proteins are oriented on average even at $\phi\to 0$ (see Figs. 2(b)–(e)). The maximum density $\phi_{\rm lim}(S)$ is given by

that is independent of the membrane curvature ($\phi_{\rm max}=\phi_{\rm lim}(1)$). The straight line ($S=0$) for $0<\phi<\phi_{\rm lim}(0)$ in Fig. 2(a) is divided into two, and the right branch remains even at large tube curvatures, $1/R_{\rm cy}$, although it has a high energy with a narrow width of $\phi$ (see dashed lines at $\phi\simeq 0.6$ in Figs. 2(b)–(e)). Meanwhile, the left branch connects to the upper branch at $1/R_{\rm cy}C_{\rm p}\gtrsim 0.1$ (see Fig. 2(b)). Note that it is separated at $1/R_{\rm cy}C_{\rm p}=0.01$ (data not shown). At $1/R_{\rm cy}C_{\rm p}\leq 1$, the proteins prefer to align to the azimuthal direction ($\theta_{\rm pc}=0$), while the thermal fluctuations disturb it. Hence, the proteins are more ordered (higher $S$) at higher $\kappa_{\rm p}$ (see Fig. 3(a)). At high density $\phi$ (close to $\phi_{\rm max}$), $S$ is dominantly determined by the orientation-dependent excluded volume and the effects increase with increasing $d_{\rm el}$ (see Fig. 3).

At $1/R_{\rm cy}C_{\rm p}>1$, the protein preferred direction is tilted either to a positive or negative angle of $\theta_{\rm pc}=\pm\arccos(\sqrt{R_{\rm cy}C_{\rm p}})$. At low $\phi$, the positive and negative angles simultaneously exist, so that the proteins exhibit a symmetric distribution with $\theta_{\rm sc}=0$ (see Fig. 4). In contrast, at high $\phi$, these two angles cannot coexist at the same time owing to the large excluded-volume interactions between them (see the right distributions in Figs. 4(c) and (d)). The transitions between these two states are the second order and first order for $1/R_{\rm cy}C_{\rm p}\leq 1.3$ and $1/R_{\rm cy}C_{\rm p}\geq 1.35$ (see Figs. 4(a) and (b)), respectively. The two states coexist at $\phi\simeq 0.55$ and $1/R_{\rm cy}C_{\rm p}=1.5$ as shown in Fig. 4(d). Correspondingly, the $S$–$\phi$ curves exhibit discrete changes of the slope and position (see Figs. 2(d) and (e)), respectively. In the case of the second-order transition, the excluded-volume interactions push the protein into the azimuthal direction leading to the angular distribution of a single peak near the transition point (see the data at $\phi=0.6$ in Fig. 4(c)), so that the symmetric peak continuously changes to an asymmetric peak at the transition.

Figure 5 shows the tube curvature $1/R_{\rm cy}$ dependence of the orientational degree $S_{z}$ along the tube axis, protein free-energy, and axial force $f_{\rm ex}$. The preferred orientation changes from $\theta_{\rm sc}=0$ to $\pi/2$ at $1/R_{\rm cy}C_{\rm p}=2$, so that $S_{z}$ changes from negative values ($S_{z}=-S$) to positive values ($S_{z}=S$). Interestingly, the orientational order ($S=|S_{z}|$) has a maximum at $1/2<1/R_{\rm cy}C_{\rm p}<1$, i.e., less than the matching curvature $1/R_{\rm cy}C_{\rm p}=1$ (see Figs. 5(a) and 2). Oppositely, the curvature $C_{\rm e}$ of the free-energy minimum is higher than the matching curvature (see Fig. 5(b)). These are determined by the competition between the orientational entropy and bending energy. The curvature ($C_{\rm order}$) of the maximum order and $C_{\rm e}$ decrease with increasing $\phi$ and $\kappa_{\rm p}$ (see Fig. 6); at $\kappa_{\rm p}\to\infty$, $C_{\rm order}\to 1/2R_{\rm cy}$, in which the strength of the approximately harmonic potential of $U_{\rm p}$ for $\theta_{\rm pc}\ll 1$ (i.e., $\partial^{2}U_{\rm p}/\partial\theta_{\rm pc}^{2}|_{\theta_{\rm pc}=0}$) is maximum. The amplitudes of the order and the depth of free-energy minimum increase with increasing $\kappa_{\rm p}$ (see Figs. 5(a) and (b), respectively). Corresponding to the larger energy change in $f_{\rm p}$, the axial force $f_{\rm ex}$ deviates more from the value of the bare membrane ($f_{\rm mb}$ given by Eq. (13)) at higher $\kappa_{\rm p}$ (see Fig. 5(c)). Spontaneously formed membrane tubes require no axial force (i.e., $f_{\rm ex}=0$). The generation curvature $C_{\rm g}$ of this spontaneous tube increases with increasing $\kappa_{\rm p}$ and $\phi$ (see the inset of Fig. 5(c)), i.e., a narrower tube is generated. Note that $C_{\rm g}$ also depends on the bending rigidity $\kappa_{\rm d}$ of the bare membrane in contrast to $C_{\rm order}$ and $C_{\rm e}$.

Next, we examine the effects of the protein bending energy in the side direction (see Fig. 7). At zero side spontaneous curvature ($C_{\rm side}=0$), the proteins more align in the azimuthal direction with increasing $\kappa_{\rm side}$, since the curvature $C_{\ell 2}$ in the side direction becomes closer to $C_{\rm side}$. This effect is pronounced at narrow tubes, whereas it is negligible for $1/R_{\rm cy}C_{\rm p}<1$. For a negative value of $C_{\rm side}$, $C_{\rm order}$ and $C_{\rm e}$ become close to $C_{\rm p}$, and the proteins more align in a wider range of $1/R_{\rm cy}$. For a positive value of $C_{\rm side}$, $C_{\rm order}$ and $C_{\rm e}$ become deviated from $C_{\rm p}$, and the proteins less align. The generation curvature slightly increases with increasing $C_{\rm side}$: $C_{\rm g}=0.412$, $0.421$, and $0.448$ at $C_{\rm side}=-1$, $0$, and $1$, respectively, for the condition used in Fig. 7.

We have fixed the protein curvature $C_{\rm p}$ until here. Figure 8 shows the effects of $C_{\rm p}$ variation with maintaining the other parameters. As $C_{\rm p}$ changes from null to $1/R_{\rm cy}$, the nematic orientation changes from the axial direction ($\theta_{\rm sc}=\pi/2$) to the azimuthal direction ($\theta_{\rm sc}=0$) (see the left half of Fig. 8(c)). This change becomes steeper at higher $\kappa_{\rm p}$. During this change, the axial force $f_{\rm ex}$ is almost constant, although a small peak appears for high $\kappa_{\rm p}$ and/or high $\phi$ (see the left regions of Figs. 8(a) and (b)). This is due to little change in the bending energy, because the proteins can find their preferred curvature by adjusting their orientation. For $C_{\rm p}\gtrsim 1/R_{\rm cy}$, $f_{\rm ex}$ almost linearly decreases, and the slope increases with increasing $\kappa_{\rm p}$ and $\phi$ (see the right region of Figs. 8(a) and (b)). These dependencies qualitatively agree with the results of our previous meshless membrane simulations Noguchi (2014, 2015, 2016b). A quantitative comparison is described in Sec. III.2.

Finally, we examine the equilibrium of the protein binding and unbinding. As the binding chemical potential $\mu$ increases, more proteins bind onto the membrane. The protein binding exhibits a first-order transition from a wide tube with low $\phi$ to a narrow tube with high $\phi$ at small force, $f_{\rm ex}<f_{0}$ (see Fig. 9). This transition agrees with the observation of the coexistence of two tubes with different $R_{\rm cy}$ and $\phi$ in the experiments of an I-BAR protein Prévost et al. (2015). The force-dependence curves shown in Fig. 9 are asymmetric and exhibit weak dependence at $f_{\rm ex}>f_{0}$, owing to the adjustment of the protein orientation that reduces a change in the protein bending energy (see Fig. 9(c)). These behaviors are different from the binding of proteins with an isotropic spontaneous curvature Noguchi (2021b), where the $f_{\rm ex}$–$1/R_{\rm cy}$ and$f_{\rm ex}$–$\phi$ curves are point symmetric and reflection symmetric to $f_{\rm ex}=f_{0}$, respectively.

The protein binding has a maximum in the variation of the tube curvature (compare Figs. 9(a) and (b)). This curvature is called sensing curvature (denoted $C_{\rm s}$) and can be calculated from $\partial\phi/\partial(1/R_{\rm cy})=0$. Interestingly, $C_{\rm s}$ is varied by $\mu$ and $\kappa_{\rm p}$ (see Fig. 6). For low $\phi$ at low $\mu$, $C_{\rm s}$ approach $C_{\rm e}$, since the excluded volume gives negligible effects. For high $\mu$ ($\phi\gtrsim 0.5$ in Fig. 6(a)), $C_{\rm s}$ becomes lower than $C_{\rm p}$, and $\phi$ has a broad peak. A similar $C_{\rm s}$ dependence on the tube curvature has been reported in the experiments of the BAR proteins Prévost et al. (2015); Tsai et al. (2021). It indicates the anisotropic interaction of the BAR proteins.

The asymmetry of the force-dependence curves is caused not by the orientation-dependent excluded volume but by the anisotropy of the protein bending energy. To clearly show it, the force-dependence curves for the elliptic proteins with an isotropic spontaneous curvature $C_{\rm iso}$ are plotted in Fig. 10. The proteins have a bending energy

instead of $U_{\rm p}$. Note that the anisotropic bending energy $U_{\rm p}$ with $\kappa_{\rm p}=\kappa_{\rm side}$ and $C_{\rm p}=C_{\rm side}$ does not coincide to $U_{\rm iso}$ except for the case of $\theta_{\rm pc}=0$ or $\pi/2$. The $f_{\rm ex}$–$1/R_{\rm cy}$ and$f_{\rm ex}$–$\phi$ curves become point symmetric and reflection symmetric to $f_{\rm ex}=f_{0}$, respectively, and the first-order transitions occur both at small and large forces symmetrically. The transition points are almost constant for a variation in $\mu$. This is due to the excluded-volume dependence on the protein orientation, since the transition points move outwards in the case of orientation-independent excluded volume Noguchi (2021b).

A fluid membrane is represented by a self-assembled single-layer sheet of $N$ particles. The position and orientational vectors of the $i$-th particle are ${\bm{r}}_{i}$ and ${\bm{u}}_{i}$, respectively. The membrane particles interact with each other via a potential $U=U_{\rm{rep}}+U_{\rm{att}}+U_{\rm{bend}}+U_{\rm{tilt}}$. The potential $U_{\rm{rep}}$ is an excluded volume interaction with diameter $\sigma$ for all pairs of particles. The solvent is implicitly accounted for by an effective attractive potential $U_{\rm{att}}$. The details of the meshless membrane model and protein rods are described in Ref. 66 and Refs. 57; 59, respectively. We employ the parameter sets used in Ref. 59.

The bending and tilt potentials are given by $U_{\rm{bend}}/k_{\rm B}T=(k_{\rm{bend}}/2)\sum_{i<j}({\bm{u}}_{i}-{\bm{u}}_{j}-C_{\rm{bd}}\hat{\bm{r}}_{i,j})^{2}w_{\rm{cv}}(r_{i,j})$ and $U_{\rm{tilt}}/k_{\rm B}T=(k_{\rm{tilt}}/2)\sum_{i<j}[({\bm{u}}_{i}\cdot\hat{\bm{r}}_{i,j})^{2}+({\bm{u}}_{j}\cdot\hat{\bm{r}}_{i,j})^{2}]w_{\rm{cv}}(r_{i,j})$, respectively, where ${\bm{r}}_{i,j}={\bm{r}}_{i}-{\bm{r}}_{j}$, $r_{i,j}=|{\bm{r}}_{i,j}|$, $\hat{\bm{r}}_{i,j}={\bm{r}}_{i,j}/r_{i,j}$, $w_{\rm{cv}}(r_{i,j})$ is a weight function. The spontaneous curvature $C_{0}$ of the membrane is given by $C_{0}\sigma=C_{\rm{bd}}/2$. Shiba and Noguchi (2011) In this study, $C_{0}=0$ and $k_{\rm{bend}}=k_{\rm{tilt}}=10$ are used except for the membrane particles belonging to the protein rods.

An anisotropic protein and membrane underneath it are together modeled as a rod that is a linear chain of $N_{\rm{sg}}$ membrane particles Noguchi (2014). We use $N_{\rm{sg}}=5$ and $10$ with the density $\phi=N_{\rm{sg}}N_{\rm{rod}}/N=0.167$. The protein rods have spontaneous curvatures $C_{\rm{rod}}$ along the rod axis and have no spontaneous (side) curvatures perpendicular to the rod axis. The protein-bound membrane are more rigid than the bare membrane: the values of $k_{\rm{bend}}$ and $k_{\rm{tilt}}$ are $k_{\rm r}$ times higher than those of the bare membrane.

The membrane has mechanical properties that are typical of lipid membranes: the bare membrane has a bending rigidity $\kappa/k_{\rm B}T=16.1\pm 0.02$, area of the tensionless membrane per particle $a_{0}/\sigma^{2}=1.2778\pm 0.0002$, area compression modulus $K_{A}\sigma^{2}/k_{\rm B}T=83.1\pm 0.4$, edge line tension $\Gamma\sigma/k_{\rm B}T=5.73\pm 0.04$ Noguchi (2014), and the Gaussian modulus $\bar{\kappa}/\kappa=-0.9\pm 0.1$ Noguchi (2019b). The bending rigidity is calculated by Eq. (13), which is slightly greater than the value ($15\pm 1$) estimated by thermal undulation Shiba and Noguchi (2011). The membrane tube with a length of $L_{\rm cy}$ is connected by the periodic boundary, and the tube volume can be freely varied. Molecular dynamics with a Langevin thermostat is employed Shiba and Noguchi (2011); Noguchi (2011). The dependence on the rod curvature $C_{\rm rod}$ was calculated at $L_{\rm cy}=48\sigma$ and $N=2400$ in Ref. 59 using the replica-exchange method Hukushima and Nemoto (1996); Okamoto (2004). The dependence on the tube radius was calculated at $k_{\rm r}=4$ and $N=4800$ in this study.

Figure 11 and Figs. 12,13 show the simulation and theoretical results for the short and long protein rods ($N_{\rm sg}=5$ and $10$), respectively. Since the simulated proteins do not have an elliptic shape and are flexible, the protein parameters are adjusted as follows. For the short rods, we used the orientational degree $S_{z}$ at $C_{\rm rod}\sigma=0.15$ (the second line from the top in Fig. 11(d)) for a fit and obtained $\kappa_{\rm p}=30k_{\rm B}T$ and $C_{\rm rod}/C_{\rm p}=2$ for $a_{\rm p}=N_{\rm sg}a_{0}$ and $d_{\rm el}=3$. This parameter set reproduces the simulation data of $S_{z}$ and $f_{\rm ex}$ at different values of $C_{\rm rod}$ very well. Thus, this theory can quantitatively describe the behavior of the short proteins.

However, less agreement is obtained for the long rods of $N_{\rm sg}=10$ (see Figs. 12 and 13). It is due to the protein assembly induced by the membrane-mediated attractive interactions between the proteins (see the snapshots in Figs. 12(a)–(c)). At a high rod curvature ($C_{\rm rod}\sigma=0.25$), the proteins assemble in the azimuthal direction, and the membrane deforms into an elliptic tube, as shown in Fig. 12(c). For longer (narrower) and shorter (wider) tubes, cylindrical and triangular shapes are formed (see Movie 1 in ESI). Thus, large negative values of $S_{z}$ (Fig. 12(e)) and nonmonotonic dependence of $f_{\rm ex}$ for $C_{\rm rod}R_{\rm cy}\gtrsim 2.5$ ($C_{\rm rod}\sigma\gtrsim 0.25$) at $k_{\rm r}=4$ (Fig. 13(a)) are obtained. In the elliptic and triangular membranes, the proteins align in the azimuthal direction, so that their stabilities can be analyzed by assuming a fixed protein orientation as reported in Ref. 41. More detail of this assembly is described in Refs. 57; 41; 59.

For a lower rod curvature ($C_{\rm rod}\sigma\leq 0.2$) of the long rods at $k_{\rm r}=4$, the azimuthal assembly does not occur, but clusters of a few proteins appear as shown in Figs. 12(a) and (b). We fitted the linear-decrease region of the force-dependence curve in Fig. 13(a) at $C_{\rm rod}R_{\rm cy}>1$ and obtained $\kappa_{\rm p}/k_{\rm B}T=60$, $90$, $120$, and $150$ with $C_{\rm rod}/C_{\rm p}=2.5$, $2.05$, $1.7$, and $1.5$ for $k_{\rm r}=2$, $4$, $8$, and $12$, respectively, at $a_{\rm p}=N_{\rm sg}a_{0}$ and $d_{\rm el}=7$. The orientational orders $S_{z}$ calculated by these parameter sets show quantitative deviation from the simulation data, although they capture qualitative behavior (see Figs. 12(e) and 13(b)). Moreover, the other regions of the force-dependence curves have quantitative differences: the heights of peaks at $C_{\rm rod}R_{\rm cy}<1$ in Fig. 13(a) deviate from the simulation values, and the slopes at $\sigma/R_{\rm cy}<0.1$ in Fig. 12(d) are different. Although the present theory assumes the uniform lateral distribution of the proteins, the protein clusters can bend the membrane more strongly as demonstrated by the formation of the elliptic tube. Therefore, we consider that the clusters effectively work as large or rigid proteins. The greater values of $\kappa_{\rm p}$ and $C_{\rm p}$ obtained by the fits support this mechanism. Thus, for a quantitative prediction of a long protein (i.e., a large elliptic ratio $d_{\rm el}$), it is significant to include the effects of the protein clusters.