# Estimation of anisotropic bending rigidities and spontaneous curvatures of
crescent curvature-inducing proteins from tethered-vesicle experimental data

Hiroshi Noguchi<EMAIL_ADDRESS>Institute for Solid State Physics,
University of Tokyo, Kashiwa, Chiba 277-8581, Japan  Nikhil Walani
Universitat Politèdcnica de Catalunya-BarcelonaTech, 08034 Barcelona, Spain
Present address: Department of Applied Mechanics, IIT Delhi, Hauz Khas, New
Delhi, 110016, India  Marino Arroyo  Universitat Politèdcnica de Catalunya-
BarcelonaTech, 08034 Barcelona, Spain  Institute for Bioengineering of
Catalonia (IBEC), The Barcelona Institute for Science and Technology (BIST),
08028 Barcelona, Spain  Centre Internacional de Mètodes Numèrics en Enginyeria
(CIMNE), 08034 Barcelona, Spain

###### Abstract

The Bin/amphiphysin/Rvs (BAR) superfamily proteins have a crescent binding
domain and bend biomembranes along the domain axis. However, their anisotropic
bending rigidities and spontaneous curvatures have not been experimentally
determined. Here, we estimated these values from the bound protein densities
on tethered vesicles using a mean-field theory of anisotropic bending energy
and orientation-dependent excluded volume. The dependence curves of the
protein density on the membrane curvature are fitted to the experimental data
for the I-BAR and N-BAR domains reported by C. Prévost et al. Nat. Commun. 6,
8529 (2015) and F.-C. Tsai et al. Soft Matter 17, 4254 (2021), respectively.
For the I-BAR domain, all three density curves of different chemical
potentials exhibit excellent fits with a single parameter set of anisotropic
bending energy. When the classical isotropic bending energy is used instead,
one of the curves can be fitted well, but the others exhibit large deviations.
In contrast, for the N-BAR domain, two curves are not well-fitted
simultaneously using the anisotropic model, although it is significantly
improved compared to the isotropic model. This deviation likely suggests a
cluster formation of the N-BAR domains.

##  I Introduction

In living cells, membrane morphology is regulated by the binding and unbinding
of curvature-inducing proteins McMahon and Gallop (2005); Suetsugu  _et al._
(2014); Johannes  _et al._ (2015); Brandizzi and Barlowe (2013); Hurley  _et
al._ (2010); McMahon and Boucrot (2011); Baumgart  _et al._ (2011); Has and
Das (2021). Some types of these proteins bend a membrane in a laterally
isotropic manner and generate spherical membrane buds Johannes  _et al._
(2015); Brandizzi and Barlowe (2013); Hurley  _et al._ (2010); McMahon and
Boucrot (2011). In contrast, the Bin/amphiphysin/Rvs (BAR) superfamily
proteins have a crescent binding domain (BAR domain) and bend membranes along
the BAR domain axis, generating cylindrical membrane tubes McMahon and Gallop
(2005); Suetsugu  _et al._ (2014); Johannes  _et al._ (2015); Itoh and De
Camilli (2006); Masuda and Mochizuki (2010); Mim and Unger (2012); Frost  _et
al._ (2008). Several types of BAR domains are known: N-BARs and F-BARs bend
membranes positively, but I-BARs bend them in the opposite direction.

These curvature-inducing proteins can sense membrane curvature; that is, their
binding onto membranes depends on the local membrane curvatures. Tethered
vesicles have been widely used to observe the curvature sensing experimentally
Baumgart  _et al._ (2011); Has and Das (2021); Sorre  _et al._ (2012); Prévost
_et al._ (2015); Tsai  _et al._ (2021); Rosholm  _et al._ (2017); Aimon  _et
al._ (2014); Yang  _et al._ (2022); Roux  _et al._ (2010); Moreno-Pescador
_et al._ (2019); Larsen  _et al._ (2020). A vesicle is pulled by optical
tweezers and a micropipette to form a narrow membrane tube (tether). The tube
radius can be controlled by adjusting the position of the optical tweezers.
The curvature sensing of BAR proteins Baumgart  _et al._ (2011); Has and Das
(2021); Sorre  _et al._ (2012); Prévost  _et al._ (2015); Tsai  _et al._
(2021), G-protein coupled receptors (GPCRs) Rosholm  _et al._ (2017), ion
channels Aimon  _et al._ (2014); Yang  _et al._ (2022), dynamin Roux  _et al._
(2010), annexins Moreno-Pescador  _et al._ (2019), and Ras proteins Larsen
_et al._ (2020) have been reported. Additionally, the curvature sensing has
been detected by the protein binding onto different sizes of spherical
vesicles Larsen  _et al._ (2020); Hatzakis  _et al._ (2009); Zeno  _et al._
(2019).

Evaluating the mechanical properties of these proteins is crucial for
quantitatively understanding their curvature generation and sensing. The aim
of this study is to determine the anisotropic bending rigidity and spontaneous
curvature of BAR proteins from experimental data of tethered vesicles. In
previous studies Sorre  _et al._ (2012); Prévost  _et al._ (2015); Tsai  _et
al._ (2021), the bending rigidity and spontaneous curvature of BAR proteins
have been estimated using the Canham–Helfrich theory Canham (1970); Helfrich
(1973). However, this theory is formulated for laterally isotropic fluid
membranes; thus, the anisotropy of the proteins is not considered. Recently,
we developed a mean-field model for anisotropic bending energy and entropic
interactions Tozzi  _et al._ (2021); Roux  _et al._ (2021); Noguchi  _et al._
(2022). Orientational fluctuations are included based on Nascimentos’ theory
for three-dimensional liquid crystals Nascimento  _et al._ (2017). The first-
and second-order transitions between isotropic and nematic phases are obtained
with increasing protein density in narrow membrane tubes Noguchi  _et al._
(2022). In the present study, we use this theoretical model to estimate the
anisotropic bending rigidity and spontaneous curvature. The experimental data
for the I-BAR domain of IRSp53 and the N-BAR domain of amphiphysin 1 reported
in Refs. 14 and 15, respectively, are used for the estimation. In theoretical
studies, a protein is often assumed to be a rigid body Dommersnes and Fournier
(1999, 2002); Schweitzer and Kozlov (2015); Kohyama (2019). The membrane-
mediated interaction of two rigid proteins qualitatively reproduces that of
two flexible proteins obtained by meshless membrane simulations, but the
amplitude is overestimated Noguchi and Fournier (2017). Hence, the estimation
of the bending rigidity is also important for evaluating the interaction
between proteins.

The mean-field theory and fitting method are described in Sec. II. Secs. III
and IV present and discuss the fitting results for the I-BAR and N-BAR
domains, respectively. Additionally, the results of the isotropic and
anisotropic protein models are compared. Sec. V concludes the paper.

Figure 1:  Schematic of a tethered vesicle and protein binding. (a)
Experimental setup of the tethered vesicle. The proteins are bound in the
tether and spherical vesicle regions with bound densities of
$\phi_{\mathrm{cy}}$ and $\phi_{\mathrm{v}}$, respectively. The angles between
the nematic direction S, the azimuthal direction, and/or the protein axis are
shown in the bottom panel. (b) Binding and unbinding of BAR domains. N-BAR and
I-BAR domains bind onto the outer and inner surfaces of the vesicle,
respectively. (c) Excluded-volume interactions between proteins. A
perpendicular protein pair has a larger excluded area on the membrane surface
(represented by thick dashed lines) than a parallel pair (compare the left and
right panels).

##  II Theory

A cylindrical membrane tube (tether) protrudes from the spherical vesicle, as
depicted in Fig. 1(a). The tether length $L_{\mathrm{cy}}$ and radius
$R_{\mathrm{cy}}$ are controlled by force $f_{\mathrm{ex}}$ generated by
optical tweezers and a micropipette. The membrane is in a fluid phase and is
homogeneous. The radius $R_{\mathrm{v}}$ of the spherical region is on a
$\mu$m scale; thus, the membrane can be approximated as flat. The subscripts v
and cy represent the quantities in the spherical and tether regions,
respectively. The total membrane area $A$ is fixed and the membrane area
inside of the micropipette is assumed to be constant. The tether area
$A_{\mathrm{cy}}=2\pi R_{\mathrm{cy}}L_{\mathrm{cy}}$ is approximated as a
constant, since the tube volume is negligibly small Smith  _et al._ (2004);
Noguchi (2021a). The protein density $\phi$ is the local area fraction covered
by the bound proteins ($\phi_{\mathrm{v}}$ and $\phi_{\mathrm{cy}}$ represent
the densities in the spherical and tether regions, respectively). N-BAR and
I-BAR proteins bind onto the outer and inner surfaces, respectively (see Fig.
1(b)). Here, the curvature direction is defined as outward following the
membrane curvature. Hence, the proteins binding to the inner surface have the
opposite sign of curvature from the protein viewpoint (see I-BAR in Fig.
1(b)).

###  II.1 Isotropic proteins

First, we describe the mean-field theory of proteins that bend membranes
isotropically (no preferred lateral direction). The bending energy is given as
follows: Noguchi (2021b, 2022a)

$\displaystyle F_{\mathrm{cv}}$ $\displaystyle=$ $\displaystyle
4\pi\bar{\kappa}_{\mathrm{d}}(1-g_{\mathrm{ves}})+\int{\mathrm{d}}A\
\Big{\\{}2\kappa_{\mathrm{d}}H^{2}(1-\phi)$ (1)
$\displaystyle+\frac{\kappa_{\mathrm{pi}}}{2}(2H-C_{0})^{2}\phi+(\bar{\kappa}_{\mathrm{pi}}-\bar{\kappa}_{\mathrm{d}})K\phi\Big{\\}},$

where $g_{\mathrm{ves}}$ represents the genus of the vesicle
($g_{\mathrm{ves}}=0$ for tethered vesicles). $H=(C_{1}+C_{2})/2$ and
$K=C_{1}C_{2}$ represent the mean and Gaussian curvatures of each position,
respectively, with $C_{1}$ and $C_{2}$ being the principal curvatures. The
bare (protein-unbound) membrane has a bending rigidity of
$\kappa_{\mathrm{d}}$, zero spontaneous curvature, and saddle-splay modulus of
$\bar{\kappa}_{\mathrm{d}}$ (also called the Gaussian modulus) in the
Canham–Helfrich theory Canham (1970); Helfrich (1973); Safran (1994). The
bound membrane has a bending rigidity of $\kappa_{\mathrm{pi}}$, finite
spontaneous curvature $C_{0}$, and saddle-splay modulus of
$\bar{\kappa}_{\mathrm{pi}}$. The first term of eqn (1) represents the
integral over the Gaussian curvature $K$. Note that the curvature mismatch
model Tsai  _et al._ (2021); Has and Das (2021); Prévost  _et al._ (2015);
Rosholm  _et al._ (2017) and spontaneous curvature model Tsai  _et al._
(2021); Has and Das (2021); Ramaswamy  _et al._ (2000); Tozzi  _et al._ (2019)
are subsets of the present model for
$\kappa_{\mathrm{pi}}>\kappa_{\mathrm{d}}$ and
$\kappa_{\mathrm{pi}}=\kappa_{\mathrm{d}}$, respectively Noguchi (2021b). For
$\kappa_{\mathrm{pi}}<\kappa_{\mathrm{d}}$, the proteins exhibit curvature
sensing but do not have a non-passive curvature-generation capability Noguchi
(2022a).

The membrane free energy $F$ consists of the binding energy and mixing entropy
in addition to the bending energy $F_{\rm cv}$,

$F=F_{\mathrm{cv}}+\int{\mathrm{d}}A\
\Big{\\{}-\frac{\mu}{a_{\mathrm{p}}}\phi+\frac{k_{\mathrm{B}}T}{a_{\mathrm{p}}}[\phi\ln(\phi)+(1-\phi)\ln(1-\phi)]\Big{\\}},$
(2)

where $a_{\mathrm{p}}$ represents the area covered by one protein and
$k_{\mathrm{B}}T$ represents the thermal energy. The maximum number of bound
proteins is $A/a_{\mathrm{p}}$. The first and second terms in the integral of
eqn (2) represent the protein-binding energy with the chemical potential $\mu$
and the mixing entropy of bound proteins, respectively. Here, we neglect the
inter-protein interaction energy ($\sim\phi^{2}$) Noguchi (2021b, a, 2022a),
since we consider low protein densities in this study.

In thermal equilibrium, the protein density $\phi$ is locally determined for
each membrane curvature: Noguchi (2021b, 2022a)

$\displaystyle\phi$ $\displaystyle=$
$\displaystyle\frac{1}{1+\exp(w_{\mathrm{b}})},$ (3) $\displaystyle
w_{\mathrm{b}}$ $\displaystyle=$ $\displaystyle-\frac{\mu}{k_{\mathrm{B}}T}$
$\displaystyle+\frac{a_{\mathrm{p}}}{k_{\mathrm{B}}T}\Big{(}2\kappa_{\mathrm{dif}}H^{2}+\bar{\kappa}_{\mathrm{dif}}K-2\kappa_{\mathrm{pi}}C_{0}H+\frac{\kappa_{\mathrm{pi}}C_{0}^{2}}{2}\Big{)},$

where $\kappa_{\mathrm{dif}}=\kappa_{\mathrm{pi}}-\kappa_{\mathrm{d}}$ and
$\bar{\kappa}_{\mathrm{dif}}=\bar{\kappa}_{\mathrm{pi}}-\bar{\kappa}_{\mathrm{d}}$.
Since the curvature of the spherical region of the tethered vesicles is
approximated as $H=K=0$, the protein density in the spherical region is given
as
$\phi_{\mathrm{v}}=1/\\{1+\exp[(-\mu+a_{\mathrm{p}}\kappa_{\mathrm{pi}}C_{0}^{2}/2)/k_{\mathrm{B}}T]\\}$.
Hence, the protein density $\phi_{\mathrm{cy}}$ in the tether regions is given
as

$\phi_{\mathrm{cy}}=\frac{1}{1+\frac{1-\phi_{\mathrm{v}}}{\phi_{\mathrm{v}}}\exp\big{[}\frac{a_{\mathrm{p}}}{k_{\mathrm{B}}T}\big{(}\frac{\kappa_{\mathrm{dif}}}{{2R_{\mathrm{cy}}}^{2}}-\frac{\kappa_{\mathrm{pi}}C_{0}}{R_{\mathrm{cy}}}\big{)}\big{]}}.$
(5)

When the membrane has the sensing curvature $C_{\mathrm{s}}$,
$\phi_{\mathrm{cy}}$ is maximized. The sensing curvature is obtained from
$\phi_{\mathrm{cy}}/(1/R_{\mathrm{cy}})=0$:

$C_{\mathrm{s}}=\frac{\kappa_{\mathrm{pi}}C_{0}}{\kappa_{\mathrm{dif}}}.$ (6)

For $\kappa_{\mathrm{dif}}\neq 0$, the numerator of the second term in
parentheses can be replaced with $\kappa_{\mathrm{dif}}C_{\mathrm{s}}$, so
that $\kappa_{\mathrm{dif}}$ and $C_{\mathrm{s}}$ can be used as fitting
parameters. Here, the protein densities of tethered vesicles are independent
of $\bar{\kappa}_{\mathrm{dif}}$, since $R_{\mathrm{v}}\gg R_{\mathrm{cy}}$.
Note that at $R_{\mathrm{v}}\sim R_{\mathrm{cy}}$, $\phi_{\mathrm{v}}$ is also
dependent on $\bar{\kappa}_{\mathrm{dif}}$ Noguchi (2022a). Thus, the density
difference between small vesicles and tethers of the same mean curvature
reported in Ref. 21 may be caused by this Gaussian curvature dependence. Other
isotropic proteins, such as GPCRs Rosholm  _et al._ (2017) and ion channels
Aimon  _et al._ (2014); Yang  _et al._ (2022), may also have similar
dependences.

For the low-density limit ($\phi_{\mathrm{cy}}\ll 1$), the density ratio is
expressed by the exponential function Prévost  _et al._ (2015); Noguchi
(2021a)

$\frac{\phi_{\mathrm{cy}}}{\phi_{\mathrm{v}}}=\exp\Big{[}-\frac{a_{\mathrm{p}}}{k_{\mathrm{B}}T}\Big{(}\frac{\kappa_{\mathrm{dif}}}{{2R_{\mathrm{cy}}}^{2}}-\frac{\kappa_{\mathrm{pi}}C_{0}}{R_{\mathrm{cy}}}\Big{)}\Big{]}.$
(7)

The ratio $\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$ is independent of
$\phi_{\mathrm{v}}$ in this limit. Details regarding isotropic-protein binding
on tethered vesicles are described in Ref. 36.

###  II.2 Anisotropic proteins

Anisotropies of the protein bending energy and excluded volume are considered.
The lateral shape of a bound protein is approximated as an ellipse with major
and minor axis lengths of $\ell_{1}$ and $\ell_{2}$, respectively. The aspect
ratio is $d_{\mathrm{el}}=\ell_{1}/\ell_{2}$, and the area is
$a_{\mathrm{p}}=\pi\ell_{1}\ell_{2}/4$. These proteins have an orientation-
dependent excluded-volume interaction and can align on the membrane surface.
When neighboring proteins have a perpendicularly orientation, the excluded
area $A_{\mathrm{exc}}$ between them is larger than that for parallel pairs,
as shown in Fig. 1(c). This area $A_{\mathrm{exc}}$ is approximated as a
function of the angle $\theta_{\mathrm{pp}}$ between the major axes of the two
proteins: Noguchi  _et al._ (2022)
$A_{\mathrm{exc}}=a_{\mathrm{p}}[4-b_{\mathrm{exc}}(\cos^{2}(\theta_{\mathrm{pp}})-1)]$.
The effective excluded area is $A_{\mathrm{eff}}=\lambda A_{\mathrm{exc}}$.
Although $\lambda$ decreases slightly with an increase in the protein density,
we use a constant value $\lambda=1/3$ for simplicity Tozzi  _et al._ (2021);
Roux  _et al._ (2021); Noguchi  _et al._ (2022).

The bending energy of a bound protein is given as follows:

$\displaystyle U_{\mathrm{p}}$ $\displaystyle=$
$\displaystyle\frac{\kappa_{\mathrm{p}}a_{\mathrm{p}}}{2}(C_{\ell
1}-C_{\mathrm{p}})^{2}+\frac{\kappa_{\mathrm{side}}a_{\mathrm{p}}}{2}(C_{\ell
2}-C_{\mathrm{side}})^{2},$ (8) $\displaystyle C_{\ell 1}$ $\displaystyle=$
$\displaystyle
C_{1}\cos^{2}(\theta_{\mathrm{pc}})+C_{2}\sin^{2}(\theta_{\mathrm{pc}}),$ (9)
$\displaystyle C_{\ell 2}$ $\displaystyle=$ $\displaystyle
C_{1}\sin^{2}(\theta_{\mathrm{pc}})+C_{2}\cos^{2}(\theta_{\mathrm{pc}}),$ (10)

where $C_{\ell 1}$ and $C_{\ell 2}$ represent the curvatures along the major
and minor axes of the protein, respectively, and $\theta_{\mathrm{pc}}$
represents the angle between the major protein axis and membrane principal
direction (the azimuthal direction of the cylindrical tube), as shown in Fig.
1(a). The proteins can have different values of bending rigidity and
spontaneous curvature along the major and minor protein axes:
$\kappa_{\mathrm{p}}$ and $C_{\mathrm{p}}$ along the major protein axis and
$\kappa_{\mathrm{side}}$ and $C_{\mathrm{side}}$ are along the minor axis
(side direction).

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

$\displaystyle F_{\mathrm{p}}$ $\displaystyle=$ $\displaystyle\int
f_{\mathrm{p}}\ {\mathrm{d}}A,$ (11) $\displaystyle f_{\mathrm{p}}$
$\displaystyle=$ $\displaystyle\frac{\phi
k_{\mathrm{B}}T}{a_{\mathrm{p}}}\Big{[}\ln(\phi)+\frac{S\Psi}{2}-\ln\Big{(}\int_{-\pi}^{\pi}w(\theta_{\mathrm{ps}})\
{\mathrm{d}}\theta_{\mathrm{ps}}\Big{)}\Big{]},\hskip 14.22636pt$ (12)
$\displaystyle w(\theta_{\mathrm{ps}})$ $\displaystyle=$ $\displaystyle
g\exp\Big{[}\Psi
s_{\mathrm{p}}(\theta_{\mathrm{ps}})+\bar{\Psi}\sin(\theta_{\mathrm{ps}})\cos(\theta_{\mathrm{ps}})$
(13) $\displaystyle-\frac{U_{\mathrm{p}}}{k_{\mathrm{B}}T}\Big{]}\Theta(g),$
$\displaystyle g$ $\displaystyle=$ $\displaystyle
1-\phi(b_{0}-b_{2}Ss_{\mathrm{p}}(\theta_{\rm ps})),$ (14)

where $\Theta(x)$ denotes the unit step function and
$s_{\mathrm{p}}(\theta_{\mathrm{ps}})=\cos^{2}(\theta_{\mathrm{ps}})-1/2$. The
proteins are ordered as $S=2\langle
s_{\mathrm{p}}(\theta_{\mathrm{ps}})\rangle$, where $\theta_{\mathrm{ps}}$
represents the angle between the major protein axis and the ordered direction
and $\langle...\rangle$ denotes the ensemble average (see Fig. 1). Factor $g$
expresses the effect of the orientation-dependent excluded volume, where
$b_{0}=(4+b_{\mathrm{exc}}/2)\lambda$ and $b_{2}=b_{\mathrm{exc}}\lambda$. At
$d_{\rm el}=2$, $3$, $4$, and $6$, $b_{\mathrm{exc}}=0.840$, $1.98$, $3.44$,
and $6.14$, respectively. Non-overlapped states exist at $g>0$. The ensemble
average of a protein quantity $\chi$ is given as

$\displaystyle\langle\chi\rangle=\frac{\int_{-\pi}^{\pi}\chi
w(\theta_{\mathrm{ps}})\
{\mathrm{d}}\theta_{\mathrm{ps}}}{\int_{-\pi}^{\pi}w(\theta_{\mathrm{ps}})\
{\mathrm{d}}\theta_{\mathrm{ps}}}.$ (15)

The quantities $\Psi$ and $\bar{\Psi}$ are the symmetric and asymmetric
components of the nematic tensor, respectively, and are determined using $S$
and $\langle\sin(\theta_{\mathrm{ps}})\cos(\theta_{\mathrm{ps}})\rangle=0$ via
eqn (15). When the nematic order is parallel to one of the directions of the
membrane principal curvatures ($\theta_{\mathrm{sc}}=0$ or $\pi/2$),
$\bar{\Psi}=0$. In this study, the integral is performed in the range of
$-\pi<\theta_{\mathrm{ps}}\leq\pi$. Since the shape is rotationally symmetric,
the range $-\pi/2<\theta_{\mathrm{ps}}\leq\pi/2$ can be used alternatively, in
which the chemical potential is shifted by $\Delta\mu=k_{\mathrm{B}}T\ln(2)$
foo . Note that the separate integrals for the bending energy and other terms
used in Ref. 43 are not applicable, since the orientational fluctuations of
proteins are significantly large Tozzi  _et al._ (2021); Noguchi  _et al._
(2022).

Since an external force $f_{\mathrm{ex}}$ is imposed, the free energy of the
membrane tether is given as
$F=F_{\mathrm{p}}+U_{\mathrm{mb}}-f_{\mathrm{ex}}L_{\mathrm{cy}}$, where the
energy of the bare (unbound) membrane is
$U_{\mathrm{mb}}=\kappa_{\mathrm{d}}A/2{R_{\mathrm{cy}}}^{2}$. This force
$f_{\mathrm{ex}}$ is balanced with the membrane axial force and is obtained
using $\partial F/\partial L_{\mathrm{cy}}|_{\phi}=0$, as follows:

$\displaystyle f_{\mathrm{ex}}=2\pi\frac{\partial
f_{\mathrm{p}}}{\partial(1/R_{\mathrm{cy}})}\bigg{|}_{\phi}+f_{\mathrm{mb}}.$
(16)

Here, the last term $f_{\mathrm{mb}}$ represents the force of the bare
membrane tube: $f_{\mathrm{mb}}=2\pi\kappa_{\mathrm{d}}/R_{\mathrm{cy}}$.

The equilibrium of binding and unbinding is obtained by minimizing $F-\mu
N_{\mathrm{p}}$, where $\mu$ represents the binding chemical potential. Thus,
the protein density is balanced at $\mu=a_{\mathrm{p}}\partial
f_{\mathrm{p}}/\partial\phi$. Details are described in Refs. 26 and 28.

Figure 2:  Fitting of the density–curvature curves for I-BAR-domain binding
based on the isotropic protein model. The density $\phi_{\mathrm{cy}}$ is
normalized by $\phi_{\mathrm{v}}$ for the curvature $1/R_{\mathrm{cy}}$ of the
tethered membrane. Circles, triangles, and squares indicate the experimental
data for $\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$, respectively
(reproduced from Ref. 14). The solid lines are given by eqn (5) with the
fitting parameters $\kappa_{\mathrm{dif}}$ and $C_{\mathrm{s}}$; from top to
bottom, $\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$. (a) Data at
$\phi_{\mathrm{v}}=0.01$ are fitted:
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=45.9$ and $C_{\mathrm{s}}=0.0530$ nm-1.
(b) Data at $\phi_{\mathrm{v}}=0.02$ are fitted:
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=33.6$ and $C_{\mathrm{s}}=0.0578$ nm-1.
(c) Data at $\phi_{\mathrm{v}}=0.05$ are fitted:
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=29.0$ and $C_{\mathrm{s}}=0.0540$ nm-1.
(d) All data are fitted: $\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=27.5$ and
$C_{\mathrm{s}}=0.0629$ nm-1.  Figure 3:  Fit deviation as a function of the
sensing curvature $C_{\mathrm{s}}$ for I-BAR-domain binding based on the
isotropic protein model. From left to right,
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=45.9$, $33.6$, $29$, and $27.5$. (a)
Fit deviation $\Lambda_{1}$ for $\phi_{\mathrm{v}}=0.01$. The minimum
$\Lambda_{1}^{\mathrm{min}}=0.0444$ is obtained at
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=45.9$ and $C_{\mathrm{s}}=0.0530$ nm-1
(corresponding to Fig. 2(a)). (b) Fit deviation $\Lambda_{2}$ for
$\phi_{\mathrm{v}}=0.02$. The minimum $\Lambda_{2}^{\mathrm{min}}=0.0399$ is
obtained at $\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=33.6$ and
$C_{\mathrm{s}}=0.0578$ nm-1 (corresponding to Fig. 2(b)). (c) Fit deviation
$\Lambda_{3}$ for $\phi_{\mathrm{v}}=0.05$. The minimum
$\Lambda_{3}^{\mathrm{min}}=0.0450$ is obtained at
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=29.0$ and $C_{\mathrm{s}}=0.0540$ nm-1
(corresponding to Fig. 2(c)). (d) Sum of fit deviations
$\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$. The minimum
$(\Lambda_{1}+\Lambda_{2}+\Lambda_{3})^{\mathrm{min}}=0.411$ is obtained at
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=27.5$ and $C_{\mathrm{s}}=0.0629$ nm-1
(corresponding to Fig. 2(d)).  Figure 4:  Two-dimensional map of the fit
deviations $\Lambda_{1}$, $\Lambda_{2}$, $\Lambda_{3}$, and
$\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$ for I-BAR-domain binding based on the
isotropic protein model. The solid lines represent the valleys connecting the
values of $C_{\mathrm{s}}$ for the lowest $\Lambda$ values at fixed
$\kappa_{\mathrm{dif}}$. The circles represent the minima and the dashed lines
represent the contours of values exceeding the minima by $10$%.

Since we consider a low density of bound proteins, the proteins in the
spherical vesicle region are randomly oriented, that is,
$S=\Psi=\bar{\Psi}=0$. Hence, the free energy density $f_{\mathrm{p,v}}$ of
the spherical vesicle region is given as

$f_{\mathrm{p,v}}=\frac{\phi_{\mathrm{v}}}{a_{\mathrm{p}}}\Big{\\{}k_{\mathrm{B}}T\big{[}\ln(\phi_{\mathrm{v}})-\ln(1-b_{0}\phi_{\mathrm{v}})-\ln(2\pi)\big{]}+U_{\mathrm{p,v}}\Big{\\}},$
(17)

where
$U_{\mathrm{p,v}}=(\kappa_{\mathrm{p}}{C_{\mathrm{p}}}^{2}+\kappa_{\mathrm{side}}{C_{\mathrm{side}}}^{2})a_{\mathrm{p}}/2$.
Hence, $\mu$ for the density $\phi_{\mathrm{v}}$ is obtained as

$\frac{\mu-
U_{\mathrm{p,v}}}{k_{\mathrm{B}}T}=\ln\bigg{(}\frac{\phi_{\mathrm{v}}}{1-b_{0}\phi_{\mathrm{v}}}\bigg{)}+\frac{b_{0}\phi_{\mathrm{v}}}{1-b_{0}\phi_{\mathrm{v}}}-\ln(2\pi)+1.$
(18)

For the I-BAR and N-BAR domains, we use $d_{\mathrm{el}}=6$, and $3$,
respectively. For both proteins, we use $a_{\mathrm{p}}=50{\mathrm{\,nm}}^{2}$
in accordance with Refs. 14 and 15.

We also calculate the orientational order $S_{z}$ along the tube ($z$) axis,
since it is more easily measured than $S$ in experiments. When the
orientational order is along the azimuthal and axial directions
($\theta_{\mathrm{sc}}=0$ and $\pi/2$), $S_{z}=-S$ and $S_{z}=S$,
respectively. At a high protein density and small tube radius
($1/R_{\mathrm{cy}}>C_{\mathrm{p}}$), the orientational order can deviate from
the azimuthal or axial direction ($0<\theta_{\mathrm{sc}}<\pi/2$). However, in
this study, the fitted results remain in the range of $\theta_{\mathrm{sc}}=0$
and $\pi/2$, since the protein densities are sufficiently low.

###  II.3 Fitting

The experimental data of the bound protein density on the membrane tether are
used for the fitting. We employ a least-squares method and search the
conditions for minimizing the mean squared deviation:

$\Lambda=\frac{1}{{\phi_{\mathrm{m}}}^{2}N}\sum_{i}^{N}(\phi_{i}-\phi_{\mathrm{theory}})^{2},$
(19)

where $N$ represents the number of experimental data, and $\phi_{i}$ and
$\phi_{\mathrm{theory}}$ represent the experimental and theoretical values of
the protein density of the tether region, respectively. This fit deviation is
normalized by the mean value $\phi_{\mathrm{m}}=(1/N)\sum_{i}\phi_{i}$ of the
experimental data. If no normalization is applied, the obtained values of
$\Lambda$ depend on the choice of units ($\phi_{\mathrm{cy}}$ or
$\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$). For the isotropic protein model, two
fitting parameters, $\kappa_{\mathrm{dif}}$ and $C_{\mathrm{0}}$, are used.
For the anisotropic protein model, two fitting parameters,
$\kappa_{\mathrm{p}}$ and $C_{\mathrm{p}}$, or four fitting parameters,
$\kappa_{\mathrm{p}}$, $C_{\mathrm{p}}$, $\kappa_{\mathrm{side}}$, and
$C_{\mathrm{side}}$, are used.

For the I-BAR domain of IRSp53, the experimental data reported in Ref. 14 are
used. The fit deviations for $\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$ are
represented by $\Lambda_{1}$, $\Lambda_{2}$, and $\Lambda_{3}$, respectively.

For the N-BAR domain of amphiphysin 1, the experimental data from Ref. 15 are
used. We assume that the average densities for $n_{\mathrm{v}}<50$ $\mu$m-2,
$50$ $\mu$m${}^{-2}<n_{\mathrm{v}}<120$ $\mu$m-2, and $120$
$\mu$m${}^{-2}<n_{\mathrm{v}}<500$ $\mu$m-2 are $\phi_{\mathrm{v}}=0.0013$,
$0.0043$, and $0.016$, respectively, where $n_{\mathrm{v}}$ represents the
number density of proteins in the spherical vesicle region
($\phi_{\mathrm{v}}=n_{\mathrm{v}}a_{\mathrm{p}}$). The fit deviations for
$\phi_{\mathrm{v}}=0.0013$ and $0.0043$ are represented by $\Lambda_{1}$ and
$\Lambda_{2}$, respectively. The data of higher densities, i.e., $120$
$\mu$m${}^{-2}<n_{\mathrm{v}}<500$ $\mu$m-2, are not used for the fitting,
because they are widely distributed from
$\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}\simeq 3$ to $20$ for a narrow range of
the tether curvature ($0.04{\mathrm{\,nm}}^{-1}\lesssim
1/R_{\mathrm{cy}}\lesssim 0.11{\mathrm{\,nm}}^{-1}$). We compare only the mean
value; when the density ratio at $1/R_{\mathrm{cy}}=0.07{\mathrm{\,nm}}^{-1}$
is in the range $10\lesssim\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}\lesssim 15$,
we consider the fit to be good for $\phi_{\mathrm{v}}=0.016$.

Figure 5:  Fitting of the density–curvature curves for I-BAR-domain-binding
based on the anisotropic protein model. Circles, triangles, and squares
indicate the experimental data of $\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$ for
$\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$, respectively (reproduced from
Ref. 14). The solid lines are given by the theoretical results with the
fitting parameters $\kappa_{\mathrm{p}}$ and $C_{\mathrm{p}}$ at
$\kappa_{\mathrm{side}}=0$; from top to bottom, $\phi_{\mathrm{v}}=0.01$,
$0.02$, and $0.05$. (a) Data at $\phi_{\mathrm{v}}=0.01$ are fitted:
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=100$ and $C_{\mathrm{p}}=0.043$ nm-1. (b)
Data at $\phi_{\mathrm{v}}=0.02$ are fitted:
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=72$ and $C_{\mathrm{p}}=0.0505$ nm-1. (c)
Data at $\phi_{\mathrm{v}}=0.05$ are fitted:
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=60$ and $C_{\mathrm{p}}=0.054$ nm-1.
Figure 6:  Fitting of the density–curvature curves for I-BAR-domain binding
based on the anisotropic protein model. The solid lines are given by the
theoretical results at $\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=82$ and
$C_{\mathrm{p}}=0.047$ nm-1 to minimize $\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$.
(a) Circles, triangles, and squares indicate the experimental data of
$\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$ for $\phi_{\mathrm{v}}=0.01$, $0.02$,
and $0.05$, respectively (reproduced from Ref. 14). (b) Force generated by the
protein binding. (c) Degree $S_{z}$ of protein order along the membrane tube
($z$) axis. (a, c) From top to bottom, $\phi_{\mathrm{v}}=0.01$, $0.02$, and
$0.05$. (b) From top to bottom, $\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$
at $1/R_{\mathrm{cy}}<0.05$ nm-1. The dashed lines in (b) and (c) represent
the data for $\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=60$ and
$C_{\mathrm{p}}=0.054$ nm-1 (the minimum value of $\Lambda_{3}$ corresponding
to Fig. 5(c)).  Figure 7:  Fit deviation as a function of the protein
curvature $C_{\mathrm{p}}$ for I-BAR-domain binding based on the anisotropic
protein model. The solid lines represent the theoretical results for
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=100$, $82$, $72$, and $60$ at
$\kappa_{\mathrm{side}}=0$, from left to right. The dashed lines represent the
data for $\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=86$,
$\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}}=0.5$, and $C_{\mathrm{side}}=0$.
(a) Fit deviation $\Lambda_{1}$ for $\phi_{\mathrm{v}}=0.01$. For
$\kappa_{\mathrm{side}}=0$, the minimum value of
$\Lambda_{1}^{\mathrm{min}}=0.044$ is obtained at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=100$ and $C_{\mathrm{p}}=0.043$ nm-1
(corresponding to Fig. 5(a)). For $\kappa_{\mathrm{side}}\neq 0$, a lower
minimum $\Lambda_{1}^{\mathrm{min}}=0.039$ is obtained at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=86$, $C_{\mathrm{p}}=0.0475$ nm-1, and
$\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}}=0.5$ (corresponding in Fig. 9).
(b) Fit deviation $\Lambda_{2}$ for $\phi_{\mathrm{v}}=0.02$. The minimum
$\Lambda_{2}^{\mathrm{min}}=0.032$ is obtained at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=72$, $C_{\mathrm{p}}=0.0505$ nm-1, and
$\kappa_{\mathrm{side}}=0$ (corresponding to Fig. 5(b)). (c) Fit deviation
$\Lambda_{3}$ for $\phi_{\mathrm{v}}=0.05$. The minimum
$\Lambda_{3}^{\mathrm{min}}=0.045$ is obtained at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=60$, $C_{\mathrm{p}}=0.054$ nm-1, and
$\kappa_{\mathrm{side}}=0$ (corresponding to Fig. 5(c)). (d) Sum of fit
deviations $\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$. The minimum value
$(\Lambda_{1}+\Lambda_{2}+\Lambda_{3})^{\mathrm{min}}=0.140$ is obtained at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=82$, $C_{\mathrm{p}}=0.047$ nm-1, and
$\kappa_{\mathrm{side}}=0$ (corresponding to Fig. 6).  Figure 8:  Two-
dimensional map of the fit deviations of $\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$
for I-BAR-domain binding based on the anisotropic protein model. The circle
represents the minimum point, and the dashed lines represent the contour of
values exceeding the minimum by $10$%. The diamond, triangle, and cross
indicate the minimum points for $\Lambda_{1}$, $\Lambda_{2}$, and
$\Lambda_{3}$, respectively.  Figure 9:  Fitting of the density–curvature
curves for I-BAR-domain binding based on the anisotropic protein model. The
solid lines are given by the theoretical results at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=86$, $C_{\mathrm{p}}=0.0475$ nm-1,
$\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}}=0.5$, and $C_{\mathrm{side}}=0$ to
minimize $\Lambda_{1}$. (a) Circles, triangles, and squares indicate the
experimental data of $\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$ for
$\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$, respectively (reproduced from
Ref. 14). (b) Force generated by the protein binding. (c) Degree $S_{z}$ of
protein order along the membrane tube ($z$) axis. (a, c) From top to bottom,
$\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$. (b) From top to bottom,
$\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$ at $1/R_{\mathrm{cy}}<0.05$ nm-1.
Figure 10:  Fitting of the density–curvature curves for N-BAR-domain binding
based on the isotropic protein model. Circles and triangles indicate the
experimental data of $\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$ for
$\phi_{\mathrm{v}}=0.0013$ and $0.0043$, respectively (reproduced from Ref. 15
with permission from the Royal Society of Chemistry). The solid lines are
given by eqn (5) with the fitting parameters $\kappa_{\mathrm{dif}}$ and
$C_{\mathrm{s}}$; from top to bottom, $\phi_{\mathrm{v}}=0.0013$ and $0.0043$.
(a) Data at $\phi_{\mathrm{v}}=0.0013$ are fitted:
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=17.2$ and $C_{\mathrm{s}}=0.1032$ nm-1.
(b) Data at $\phi_{\mathrm{v}}=0.0043$ are fitted:
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=27.4$ and $C_{\mathrm{s}}=0.0735$ nm-1.
(c) Both datasets are fitted: $\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=23.3$ and
$C_{\mathrm{s}}=0.0820$ nm-1.  Figure 11:  Fit deviation as a function of the
sensing curvature $C_{\mathrm{s}}$ for N-BAR-domain binding based on the
isotropic protein model. From left to right,
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=27.4$, $23.3$, and $17.2$. (a) Fit
deviation $\Lambda_{1}$ for $\phi_{\mathrm{v}}=0.0013$. The minimum
$\Lambda_{1}^{\mathrm{min}}=0.1075$ is obtained at
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=17.2$ and $C_{\mathrm{s}}=0.1032$ nm-1
(corresponding to Fig. 10(a)). (b) Fit deviation $\Lambda_{2}$ for
$\phi_{\mathrm{v}}=0.0043$. The minimum $\Lambda_{2}^{\mathrm{min}}=0.0541$ is
obtained at $\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=27.4$ and
$C_{\mathrm{s}}=0.0735$ nm-1 (corresponding to Fig. 10(b)). (c) Sum of fit
deviations $\Lambda_{1}+\Lambda_{2}$. The minimum
$(\Lambda_{1}+\Lambda_{2})^{\mathrm{min}}=0.498$ is obtained at
$\kappa_{\mathrm{dif}}/k_{\mathrm{B}}T=23.3$ and $C_{\mathrm{s}}=0.0820$ nm-1
(corresponding to Fig. 10(c)).  Figure 12:  Fitting of the density–curvature
curves for N-BAR-domain binding based on the anisotropic protein model. The
lines are given by the theoretical results at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=39$, $C_{\mathrm{p}}=0.072$ nm-1, and
$\kappa_{\mathrm{side}}=0$ for minimizing $\Lambda_{1}+\Lambda_{2}$. (a)
Circles and triangles indicate the experimental data of
$\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$ for $\phi_{\mathrm{v}}=0.0013$ and
$0.0043$, respectively (reproduced from Ref. 15 with permission from the Royal
Society of Chemistry). (b) Force generated by the protein binding. (c) Degree
$S_{z}$ of protein order along the membrane tube ($z$) axis. (a, c) From top
to bottom, $\phi_{\mathrm{v}}=0.0013$, $0.0043$, and $0.016$. (b) From top to
bottom, $\phi_{\mathrm{v}}=0.0013$, $0.0043$, and $0.016$ at
$1/R_{\mathrm{cy}}<0.08$ nm-1.  Figure 13:  Fitting of the density–curvature
curves for N-BAR-domain binding based on the anisotropic protein model.
Circles and triangles indicate the experimental data of
$\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$ for $\phi_{\mathrm{v}}=0.0013$ and
$0.0043$, respectively (reproduced from Ref. 15 with permission from the Royal
Society of Chemistry). The solid lines are given by the theoretical results;
from top to bottom, $\phi_{\mathrm{v}}=0.0013$ and $0.0043$. (a) Data at
$\phi_{\mathrm{v}}=0.0013$ are fitted:
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=28$ and $C_{\mathrm{p}}=0.0895$ nm-1 at
$\kappa_{\mathrm{side}}=0$. (b) Data at $\phi_{\mathrm{v}}=0.0043$ are fitted:
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=55$ and $C_{\mathrm{p}}=0.0585$ nm-1 at
$\kappa_{\mathrm{side}}=0$. (c) Data at $\phi_{\mathrm{v}}=0.0043$ are fitted:
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=46$, $C_{\mathrm{p}}=0.066$ nm-1,
$\kappa_{\mathrm{side}}=0.5\kappa_{\mathrm{p}}$, and
$C_{\mathrm{side}}=-0.0025$ nm-1.  Figure 14:  Fit deviation as a function of
the protein curvature $C_{\mathrm{p}}$ for N-BAR-domain binding based on the
anisotropic protein model at $\kappa_{\mathrm{side}}=0$. From left to right,
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=55$, $39$, and $28$. (a) Fit deviation
$\Lambda_{1}$ for $\phi_{\mathrm{v}}=0.0013$. The minimum
$\Lambda_{1}^{\mathrm{min}}=0.105$ is obtained at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=28$ and $C_{\mathrm{p}}=0.0895$ nm-1
(corresponding to Fig. 13(a)). (b) Fit deviation $\Lambda_{2}$ for
$\phi_{\mathrm{v}}=0.0043$. The minimum $\Lambda_{2}^{\mathrm{min}}=0.061$ is
obtained at $\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=55$ and
$C_{\mathrm{p}}=0.0585$ nm-1 (corresponding to Fig. 13(b)). (c) Sum of fit
deviations $\Lambda_{1}+\Lambda_{2}$. The minimum
$(\Lambda_{1}+\Lambda_{2})^{\mathrm{min}}=0.355$ is obtained at
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=39$ and $C_{\mathrm{p}}=0.072$ nm-1
(corresponding to Fig. 12).  Figure 15:  Fit deviation as a function of the
side curvature $C_{\mathrm{side}}$ for N-BAR-domain binding based on the
anisotropic protein model at $\kappa_{\mathrm{side}}>0$. From left to right,
$(\kappa_{\mathrm{p}}/k_{\mathrm{B}}T,C_{\mathrm{p}}$ (nm-1)$)=(28,0.087)$,
$(46,0.066)$, and $(39,0.068)$ with
$\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}}=0.5$. (a) Fit deviation
$\Lambda_{1}$ for $\phi_{\mathrm{v}}=0.0013$. The minimum
$\Lambda_{1}^{\mathrm{min}}=0.105$ is obtained at
$(\kappa_{\mathrm{p}}/k_{\mathrm{B}}T,C_{\mathrm{p}}$
(nm-1)$,C_{\mathrm{side}}$
(nm-1)$,\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}})=(28,0.087,0.04,0.5)$. (b)
Fit deviation $\Lambda_{2}$ for $\phi_{\mathrm{v}}=0.0043$. The minimum
$\Lambda_{2}^{\mathrm{min}}=0.052$ is obtained at
$(\kappa_{\mathrm{p}}/k_{\mathrm{B}}T,C_{\mathrm{p}}$
(nm-1)$,C_{\mathrm{side}}$
(nm-1)$,\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}})=(46,0.066,-0.0025,0.5)$
(Fig. 13(c)). (c) Sum of fit deviations $\Lambda_{1}+\Lambda_{2}$. The minimum
$(\Lambda_{1}+\Lambda_{2})^{\mathrm{min}}=0.357$ is obtained at
$(\kappa_{\mathrm{p}}/k_{\mathrm{B}}T,C_{\mathrm{p}}$
(nm-1)$,C_{\mathrm{side}}$
(nm-1)$,\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}})=(39,0.068,0.0425,0.5)$.

##  III Binding of I-BAR domains

###  III.1 Isotropic protein model

Before examining the anisotropic protein model, we fit the experimental data
of I-BAR domains using the isotropic protein model with eqn (5). The rigidity
difference $\kappa_{\mathrm{dif}}$ and sensing curvature $C_{\mathrm{s}}$ are
fitted. Figure 2(a), (b), and (c) show the best fits for
$\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$, to minimize $\Lambda_{1}$,
$\Lambda_{2}$, and $\Lambda_{3}$, respectively (see Figs. 4 and 3). The target
density data are fitted very well but the other two datasets exhibit large
deviations. When the sum $\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$ is minimized,
the obtained curves are close to the results of the fit to the middle data,
i.e., $\Lambda_{2}$ (compare Fig. 2(d) and (b)). Therefore, not all of the
data can be reproduced together using the isotropic protein model. Note that
each dataset was fit separately (i.e., different values of the fitting
parameters) in Ref. 14.

As $\kappa_{\mathrm{dif}}$ increases, the lowest value of $\Lambda$ is
obtained at lower $C_{\mathrm{s}}$ (see Fig. 3). This is because
$\kappa_{\mathrm{dif}}C_{\mathrm{s}}$ ($=\kappa_{\mathrm{pi}}C_{0}$) is a
factor of the major term in the exponent of eqn (5). Thus, $\Lambda$ values
close to the minimum are obtained in the long narrow regions along the solid
curves in Fig. 4. When a statistical error is considered to be $10$% of
$\Lambda$, $\kappa_{\mathrm{dif}}$ and $C_{\mathrm{s}}$ are in the regions of
the long ellipses in Fig. 4. The anisotropic model exhibits a similar
dependence for $\kappa_{\mathrm{p}}$ and $C_{\mathrm{p}}$ as described later.

Equation (7) for $\phi_{\mathrm{cy}}\ll 1$ has been used instead of eqn (5) in
the previous studies Tsai  _et al._ (2021); Prévost  _et al._ (2015); Rosholm
_et al._ (2017); Yang  _et al._ (2022). We compared the results of these two
equations for $\Lambda_{1}$, $\Lambda_{2}$, and $\Lambda_{3}$ using the best-
fit parameters at $\phi_{\mathrm{v}}=0.01$, $0.02$, and $0.05$, respectively.
We found that the low-density limit approximation overestimates the values of
$\phi_{\mathrm{cy}}$ by approximately $20$%, $30$%, and $40$%, respectively. A
bound protein prevents the binding of other proteins to the same position (the
binding rate is proportional to $(1-\phi)$) Goutaland  _et al._ (2021).
Although this effect is negligible in the limit $\phi_{\mathrm{cy}}\ll 1$, it
becomes recognizable at $\phi_{\mathrm{cy}}\simeq 0.1$ (compare functions
$\exp(-x)$ and $1/[1+\exp(x)]$). Therefore, eqn (5) should be used in
$\phi_{\mathrm{cy}}\gtrsim 0.1$.

###  III.2 Anisotropic protein model

The experimental data of the I-BAR domains are fitted using the anisotropic
protein model. First, we use the protein rigidity $\kappa_{\mathrm{p}}$ and
protein curvature $C_{\mathrm{p}}$ as the fitting parameters with
$\kappa_{\mathrm{side}}=0$, as shown in Figs. 5–8. To determine the minimum
for each fit, the parameters are varied discretely with
$\Delta\kappa_{\mathrm{p}}=2k_{\mathrm{B}}T$ and $\Delta
C_{\mathrm{p}}=0.0005$ nm-1. Very good agreement is obtained for not only the
target density–curvature curve but also the other two curves (see Fig. 5). The
minimum $\Lambda$ values for the target curves are almost identical to those
of the isotropic protein model (compare Fig. 7(a)–(c) and Fig. 3(a)–(c)).
However, the sum, $\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$, is significantly
reduced; thus, the total fitness is improved (compare Fig. 7(d) and Fig.
3(d)).

Although the density–curvature curves are sufficiently reproduced using
$\kappa_{\mathrm{side}}=0$, a deviation is recognized in
$1/R_{\mathrm{cy}}\gtrsim 0.07$ nm-1 for $\phi_{\mathrm{v}}=0.01$. In the
experimental data, the protein density decreases by a larger amount as
$1/R_{\mathrm{cy}}$ increases. The other two curves do not exhibit such a
deviation, possibly owing to limited data for the narrow tubes (one and no
data points for $1/R_{\mathrm{cy}}>0.07$ nm-1 at $\phi_{\mathrm{v}}=0.02$ and
$0.05$, respectively). This deviation can be eliminated by using a finite
value of the bending rigidity $\kappa_{\mathrm{side}}$ in the side direction.
In addition, $\kappa_{\mathrm{side}}$ is varied discretely with
$\Delta\kappa_{\mathrm{side}}=0.5\kappa_{\mathrm{p}}$ at $C_{\mathrm{side}}=0$
for $0\leq\kappa_{\mathrm{side}}\leq\kappa_{\mathrm{p}}$. A better fit is
obtained for $\phi_{\mathrm{v}}=0.01$, as shown in Figs. 7(a) and 9. Although
$C_{\mathrm{side}}$ does not vary together here, $\Lambda_{1}$ has a minimum
at $C_{\mathrm{side}}=0$ for the variation in $C_{\mathrm{side}}$ with $\Delta
C_{\mathrm{side}}=0.0025$ nm-1 when the other parameters are fixed. Thus, the
zero side curvature is reasonable. In contrast to $\Lambda_{1}$, lower values
of the others ($\Lambda_{2}$, $\Lambda_{3}$, and
$\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$) are not obtained using
$\kappa_{\mathrm{side}}\neq 0$. Thus, the best fit for the total data
($(\Lambda_{1}+\Lambda_{2}+\Lambda_{3})^{\mathrm{min}}=0.140$) is given at
$\kappa_{\mathrm{side}}=0$, as shown in Fig. 6.

When a $10$% larger value of $\Lambda_{1}+\Lambda_{2}+\Lambda_{3}$ is allowed
as a statistical error, the elliptical region surrounded by dashed lines in
Fig. 8 is the expected range of $\kappa_{\mathrm{p}}$ and $C_{\mathrm{p}}$.
The minimum points of $\Lambda_{1}$ and $\Lambda_{2}$ are also included in
this range. Hence, we concluded that the I-BAR domain has
$\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=82\pm 20$ and
$C_{\mathrm{p}}(\mathrm{nm}^{-1})=0.047-0.0003(\kappa_{\mathrm{p}}/k_{\mathrm{B}}T-82)\pm
0.001$.

Although the experimental data are well-fitted without the side rigidity, the
existence of the side rigidity is not excluded. The measurement of other
quantities can increase the estimation accuracy for the mechanical properties
of proteins. Here, we propose the force $f_{\mathrm{ex}}$ and orientational
degree $S_{z}$ along the tube axis as candidates. The force $f_{\mathrm{ex}}$
has been experimentally measured from the position of optically trapped beads
Heinrich and Waugh (1996); Dimova (2014). Force modification due to the
binding of BAR domains has been reported Prévost  _et al._ (2015); Sorre  _et
al._ (2012). Although $S_{z}$ has not been experimentally measured, it is
measurable using polarizers in principle. As the tube curvature
$1/R_{\mathrm{cy}}$ increases, the two quantities vary in different manners
with respect to the protein density, as shown in Figs. 6 and 9. Each has a
minimum value at a tube curvature lower than the sensing curvature. Since the
densities are sufficiently low, $S_{z}$ exhibits only a small dependence on
$\phi_{\mathrm{v}}$. Importantly, they vary with changes in the bending
parameters, although the density curves do not vary significantly (compare the
solid and dashed lines in Fig. 6(b) and (c), and also see Fig. 9(b) and (c)).
This suggests that the bending rigidity and curvature of proteins can be more
accurately estimated through additional fitting of $f_{\mathrm{ex}}$ and
$S_{z}$ to the experimental data.

##  IV Binding of N-BAR domains

###  IV.1 Isotropic protein model

In this section, we consider the binding of N-BAR domains reported in Refs. 13
and 15. First, we examine the isotropic protein model, as for the I-BAR
domains considered in Sec. III. Figures 10 and 11 show the fitted
density–curvature curves and fit deviations, respectively, using eqn (5). The
target curve is well-fitted, whereas the other is not, similar to the case of
the I-BAR domain.

###  IV.2 Anisotropic protein model

The experimental data of the N-BAR domains are fitted using the anisotropic
protein model. First, the fitting is performed using the fitting parameters
$\kappa_{\mathrm{p}}$ and $C_{\mathrm{p}}$ at $\kappa_{\mathrm{side}}=0$. The
parameters are varied discretely with
$\Delta\kappa_{\mathrm{p}}=k_{\mathrm{B}}T$ and $\Delta C_{\mathrm{p}}=0.0005$
nm-1. Figure 12(a) shows the fitted density–curvature curves for minimizing
$\Lambda_{1}+\Lambda_{2}$. The two curves exhibit far better agreements than
those obtained using the isotropic protein model (compare Figs. 12(a) and
10(c)). Nonetheless, the deviations from the curves fitted to the data for
$\phi_{\mathrm{v}}=0.0013$ or $0.0043$ are large. Additionally, the curve
obtained for $\phi_{\mathrm{v}}=0.016$ is slightly exceeded the expected
values ($\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}=18$ at $1/R_{\mathrm{cy}}=0.07$
nm-1, as indicated by the dashed line in Fig. 12(a)).

The fit deviation of the curve minimizing $\Lambda_{1}$ is identical to that
of the isotropic model, but that for $\Lambda_{2}$ is slightly worse (see
Figs. 13(a), (b) and 14). Similar to the case of the I-BAR domain with
$\phi_{\mathrm{v}}=0.01$, this is due to the smaller reduction in
$\phi_{\mathrm{cy}}/\phi_{\mathrm{v}}$ at high tube curvatures (see Fig.
13(b)). Thus, we perform the fit with a finite side rigidity for
$0<\kappa_{\mathrm{side}}\leq\kappa_{\mathrm{p}}$, as shown in Figs. 13(c) and
15. The parameters are varied discretely with
$\Delta\kappa_{\mathrm{p}}=k_{\mathrm{B}}T$, $\Delta C_{\mathrm{p}}=0.001$
nm-1, $\Delta\kappa_{\mathrm{side}}=0.5\kappa_{\mathrm{p}}$, and $\Delta
C_{\mathrm{side}}=0.0025$ nm-1. A better fit to the data at
$\phi_{\mathrm{v}}=0.0043$ is obtained ($\Lambda_{2}^{\mathrm{min}}$ is 4%
smaller than that of the isotropic model). For the other fit deviations
($\Lambda_{1}+\Lambda_{2}$ and $\Lambda_{1}$), the minimum values are almost
identical to those at $\kappa_{\mathrm{side}}=0$. Interestingly, in all cases,
better fits are obtained at $\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}}=0.5$
than at $\kappa_{\mathrm{side}}/\kappa_{\mathrm{p}}=1$.

Since the density $\phi_{\mathrm{v}}$ has a wide distribution in the
experimental data, we additionally performed the fitting at
$\phi_{\mathrm{v}}=0.001$ and $0.0015$ to examine the effect of the choice of
$\phi_{\mathrm{v}}$ values for $n_{\mathrm{v}}<50$ $\mu$m-2. The minimum value
of the sum $\Lambda_{1}+\Lambda_{2}$ decreases by 5% and increases by 4% for
$\phi_{\mathrm{v}}=0.001$ and $0.0015$,
($(\Lambda_{1}+\Lambda_{2})^{\mathrm{min}}=0.336$ and $0.368$), respectively.
The corresponding values of $\kappa_{\mathrm{p}}/k_{\mathrm{B}}T$ and
$C_{\mathrm{p}}$ (nm-1) are shifted by only $1$ and $0.001$, respectively. For
$\Lambda_{1}$, $\kappa_{\mathrm{p}}/k_{\mathrm{B}}T$ and $C_{\mathrm{p}}$
(nm-1) are shifted by only $2$ and $0.0025$, respectively, and the minimum
value does not change. Therefore, a slight variation in $\phi_{\mathrm{v}}$
does not result in a significant change.

Next, we consider the effects of variations in the other fixed parameters. An
aspect ratio of $d_{\mathrm{el}}=3$ is used for the N-BAR domains. Since the
protein densities are low, the excluded-volume effect is weak. We examined the
cases of $d_{\mathrm{el}}=2$ and $4$ with the best-fit conditions, and only
obtained a 5% decrease in $(\Lambda_{1}+\Lambda_{2})^{\mathrm{min}}$ at
$d_{\mathrm{el}}=4$. Thus, we concluded that the N-BAR domains have
$30\lesssim\kappa_{\mathrm{p}}/k_{\mathrm{B}}T\lesssim 60$ and $0.06\lesssim
C_{\mathrm{p}}(\mathrm{nm}^{-1})\lesssim 0.09$.

Note that the protein shape on the membrane surface might be modified by the
membrane curvature. However, as discussed above, their effects are considered
small for N-BAR domains. For other proteins that exhibit large structural
changes on a membrane surface as observed in deformable colloids Midya  _et
al._ (2023), their deformations can have significant effects.

The protein area $a_{\mathrm{p}}$ can be slightly varied by area definition.
In our previous studies Tozzi  _et al._ (2021); Roux  _et al._ (2021); Noguchi
_et al._ (2022), we used $a_{\mathrm{p}}=60$ nm2, since the protein was
approximated as elliptical with $\ell_{1}=15$ nm and $\ell_{2}=5$ nm, and the
protein area partially included bare membrane regions. To fit a
density–curvature curve, the area $a_{\mathrm{p}}$ only appears as pairs with
bending rigidities in the theoretical models
($a_{\mathrm{p}}\kappa_{\mathrm{p}}$ and
$a_{\mathrm{p}}\kappa_{\mathrm{side}}$ in the anisotropic model and
$a_{\mathrm{p}}\kappa_{\mathrm{dif}}$ in the isotropic model). Thus, the area
variation influences the bending rigidity in accordance with
$\kappa^{\prime}=\kappa a_{\mathrm{p}}/a_{\mathrm{p}}^{\prime}$ (e.g., a $20$%
decrease in $a_{\mathrm{p}}$ results in a $25$% increase in the bending
rigidities). The force, $f_{\mathrm{ex}}-f_{\mathrm{mb}}$, is also modified by
the factor $a_{\mathrm{p}}/a_{\mathrm{p}}^{\prime}$. When the protein density
is measured as the number density $n_{\mathrm{v}}$, the area fraction is
slightly modified according to the area definition
($\phi_{\mathrm{v}}=n_{\mathrm{v}}a_{\mathrm{p}}$). However, this results in
only slight changes, as discussed above.

In our previous study Noguchi  _et al._ (2022), we compared the results of the
anisotropic model and a coarse-grained membrane simulation. When the proteins
are distributed homogeneously in the membrane, the results agree very well. In
contrast, deviations are obtained, when the proteins form small clusters.
Since the clusters have a larger area than the individual proteins, they are
more oriented in the preferred direction. The ratio of the clusters increases
with the protein density. Thus, we consider that the deviation in this study
suggests non-negligible cluster formation of the N-BAR domains. Note that the
clustering can be induced by direct attractive interaction between proteins
and membrane-mediated interactions. Large clusters can deform a membrane tube
into polygonal shapes Noguchi (2022a, 2015).

##  V Summary

We have developed an estimation method for the mechanical properties of bound
proteins based on the experiments of tethered vesicles and applied it to the
I-BAR and N-BAR domains. When the anisotropy of the proteins is taken into
account, the experimental data are reproduced far better. When the classical
isotropic model is used, each density–curvature curve is well reproduced but
the other curves largely deviate. When the recently developed anisotropic
model is used, this deviation is significantly reduced. For the I-BAR domains,
all three curves are well-fitted by a single parameter set, and the bending
rigidity $\kappa_{\mathrm{p}}/k_{\mathrm{B}}T=82$ and spontaneous curvature
$C_{\mathrm{p}}(\mathrm{nm}^{-1})=0.047$ along the protein axis are
determined. The estimation errors are small along
$\kappa_{\mathrm{p}}C_{\mathrm{p}}$ as
$\delta_{\mathrm{er}}(\kappa_{\mathrm{p}}/k_{\mathrm{B}}T)=\pm 20$ and
$\delta_{\mathrm{er}}(C_{\mathrm{p}}(\mathrm{nm}^{-1}))=-0.0003(\kappa_{\mathrm{p}}/k_{\mathrm{B}}T-82)\pm
0.001$. For the N-BAR domains, the two density–curvature curves are not
completely fitted simultaneously, even when the anisotropic model is used.
This deviation is likely caused by a small cluster formation, and
$30\lesssim\kappa_{\mathrm{p}}/k_{\mathrm{B}}T\lesssim 60$ and $0.06\lesssim
C_{\mathrm{p}}(\mathrm{nm}^{-1})\lesssim 0.09$ are estimated. If the
definition of the protein area is modified, the bending rigidity is changed as
$\kappa_{\mathrm{p}}^{\prime}=\kappa_{\mathrm{p}}a_{\mathrm{p}}/a_{\mathrm{p}}^{\prime}$.

The experimental data were well-fitted without the side bending rigidity.
Including them, the fitness was improved for some of the conditions but the
others were not changed significantly. Since positive and negative side
curvatures can promote and suppress the tubulation, respectively Noguchi
(2016), the estimation of the side rigidity and side curvature is important.
Recent experiments Zeno  _et al._ (2019); Stachowiak  _et al._ (2012); Busch
_et al._ (2015); Snead  _et al._ (2019) revealed that the intrinsically
disordered domains of curvature-inducing proteins play a significant role in
membrane remodeling. The disordered domains can be modeled by excluded-volume
chains. At a low protein density, the membrane–chain interaction slightly
increases the bending rigidity and spontaneous curvature isotropically (i.e.,
in both the axial and side directions of proteins) Hiergeist and Lipowsky
(1996); Bickel  _et al._ (2001); Auth and Gompper (2003); Wu  _et al._ (2013).
At a high density, the inter-chain interactions have strong effects in protein
clusters Hiergeist and Lipowsky (1996); Marsh  _et al._ (2003); Evans  _et
al._ (2003); Noguchi (2022b) and also between the clusters Wu  _et al._
(2013); Noguchi (2022b). These effects should be further examined by the
comparison of tether-vesicle experiments.

In this study, we fitted only the density–curvature curves. The estimation
quality can be further improved by additional fitting for other quantities.
For this purpose, we have proposed two parameters, the axial force and the
orientational degree. They exhibit different behaviors from the
density–curvature curve; thus, comparison with the experimental results can
facilitate the determination of the mechanical properties.

###### Acknowledgements.

This work was supported by JSPS KAKENHI Grant Number JP21K03481.

## References

  * McMahon and Gallop (2005) H. T. McMahon and J. L. Gallop, Nature 438, 590 (2005).
  * Suetsugu  _et al._ (2014) S. Suetsugu, S. Kurisu, and T. Takenawa, Physiol. Rev. 94, 1219 (2014).
  * Johannes  _et al._ (2015) L. Johannes, R. G. Parton, P. Bassereau, and S. Mayor, Nat. Rev. Mol. Cell Biol. 16, 311 (2015).
  * Brandizzi and Barlowe (2013) F. Brandizzi and C. Barlowe, Nat. Rev. Mol. Cell Biol. 14, 382 (2013).
  * Hurley  _et al._ (2010) J. H. Hurley, E. Boura, L.-A. Carlson, and B. Różycki, Cell 143, 875 (2010).
  * McMahon and Boucrot (2011) H. T. McMahon and E. Boucrot, Nat. Rev. Mol. Cell Biol. 12, 517 (2011).
  * Baumgart  _et al._ (2011) T. Baumgart, B. R. Capraro, C. Zhu, and S. L. Das, Annu. Rev. Phys. Chem. 62, 483 (2011).
  * Has and Das (2021) C. Has and S. L. Das, Biochim. Biophys. Acta 1865, 129971 (2021).
  * Itoh and De Camilli (2006) T. Itoh and P. De Camilli, Biochim. Biophys. Acta 1761, 897 (2006).
  * Masuda and Mochizuki (2010) M. Masuda and N. Mochizuki, Semin. Cell Dev. Biol. 21, 391 (2010).
  * Mim and Unger (2012) C. Mim and V. M. Unger, Trends Biochem. Sci. 37, 526 (2012).
  * Frost  _et al._ (2008) A. Frost, R. Perera, A. Roux, K. Spasov, O. Destaing, E. H. Egelman, P. De Camilli, and V. M. Unger, Cell 132, 807 (2008).
  * Sorre  _et al._ (2012) B. Sorre, A. Callan-Jones, J. Manzi, B. Goud, J. Prost, P. Bassereau, and A. Roux, Proc. Natl. Acad. Sci. USA 109, 173 (2012).
  * Prévost  _et al._ (2015) C. Prévost, H. Zhao, J. Manzi, E. Lemichez, P. Lappalainen, A. Callan-Jones, and P. Bassereau, Nat. Commun. 6, 8529 (2015).
  * Tsai  _et al._ (2021) F.-C. Tsai, M. Simunovic, B. Sorre, A. Bertin, J. Manzi, A. Callan-Jones, and P. Bassereau, Soft Matter 17, 4254 (2021).
  * Rosholm  _et al._ (2017) K. R. Rosholm, N. Leijnse, A. Mantsiou, V. Tkach, S. L. Pedersen, V. F. Wirth, L. B. Oddershede, K. J. Jensen, K. L. Martinez, N. S. Hatzakis, P. M. Bendix, A. Callan-Jones, and D. Stamou, Nat. Chem. Biol. 13, 724 (2017).
  * Aimon  _et al._ (2014) S. Aimon, A. Callan-Jones, A. Berthaud, M. Pinot, G. E. Toombes, and P. Bassereau, Dev. Cell 28, 212 (2014).
  * Yang  _et al._ (2022) S. Yang, X. Miao, S. Arnold, B. Li, A. T. Ly, H. Wang, M. Wang, X. Guo, M. Pathak, W. Zhao, C. D. Cox, and Z. Shi, Nat. Commun. 13, 7467 (2022).
  * Roux  _et al._ (2010) A. Roux, G. Koster, M. Lenz, B. Sorre, J.-B. Manneville, P. Nassoy, and P. Bassereau, Proc. Natl. Acad. Sci. USA 107, 4141 (2010).
  * Moreno-Pescador  _et al._ (2019) G. Moreno-Pescador, C. D. Florentsen, H. Østbye, S. L. Sønder, T. L. Boye, E. L. Veje, A. K. Sonne, S. Semsey, J. Nylandsted, R. Daniels, and P. M. Bendix, ACS Nano 13, 6689 (2019).
  * Larsen  _et al._ (2020) J. B. Larsen, K. R. Rosholm, C. Kennard, S. L. Pedersen, H. K. Munch, V. Tkach, J. J. Sakon, T. Bjørnholm, K. R. Weninger, P. M. Bendix, K. J. Jensen, N. S. Hatzakis, M. J. Uline, and D. Stamou, ACS Cent. Sci. 6, 1159 (2020).
  * Hatzakis  _et al._ (2009) N. S. Hatzakis, V. K. Bhatia, J. Larsen, K. L. Madsen, P.-Y. Bolinger, A. H. Kunding, J. Castillo, U. Gether, P. Hedegård, and D. Stamou, Nat. Chem. Biol. 5, 835 (2009).
  * Zeno  _et al._ (2019) W. F. Zeno, W. T. Snead, A. S. Thatte, and J. C. Stachowiak, Soft Matter 15, 8706 (2019).
  * Canham (1970) P. B. Canham, J. Theor. Biol. 26, 61 (1970).
  * Helfrich (1973) W. Helfrich, Z. Naturforsch 28c, 693 (1973).
  * Tozzi  _et al._ (2021) C. Tozzi, N. Walani, A.-L. L. Roux, P. Roca-Cusachs, and M. Arroyo, Soft Matter 17, 3367 (2021).
  * Roux  _et al._ (2021) A.-L. L. Roux, C. Tozzi, N. Walani, X. Quiroga, D. Zalvidea, X. Trepat, M. Staykova, M. Arroyo, and P. Roca-Cusachs, Nat. Commun. 12, 6550 (2021).
  * Noguchi  _et al._ (2022) H. Noguchi, C. Tozzi, and M. Arroyo, Soft Matter 18, 3384 (2022).
  * Nascimento  _et al._ (2017) E. S. Nascimento, P. Palffy-Muhoray, J. M. Taylor, E. G. Virga, and X. Zheng, Phys. Rev. E 96, 022704 (2017).
  * Dommersnes and Fournier (1999) P. G. Dommersnes and J.-B. Fournier, Eur. Phys. J. B 12, 9 (1999).
  * Dommersnes and Fournier (2002) P. G. Dommersnes and J.-B. Fournier, Biophys. J. 83, 2898 (2002).
  * Schweitzer and Kozlov (2015) Y. Schweitzer and M. M. Kozlov, PLoS Comput. Biol. 11, e1004054 (2015).
  * Kohyama (2019) T. Kohyama, J. Phys. Soc. Jpn. 88, 024008 (2019).
  * Noguchi and Fournier (2017) H. Noguchi and J.-B. Fournier, Soft Matter 13, 4099 (2017).
  * Smith  _et al._ (2004) A.-S. Smith, E. Sackmann, and U. Seifert, Phys. Rev. Lett. 92, 208101 (2004).
  * Noguchi (2021a) H. Noguchi, Soft Matter 17, 10469 (2021a).
  * Noguchi (2021b) H. Noguchi, Phys. Rev. E 104, 014410 (2021b).
  * Noguchi (2022a) H. Noguchi, Int. J. Mod. Phys. B 36, 2230002 (2022a).
  * Safran (1994) S. A. Safran,  _Statistical Thermodynamics of Surfaces, Interfaces, and Membranes_ (Addison-Wesley, Reading, MA, 1994).
  * Ramaswamy  _et al._ (2000) S. Ramaswamy, J. Toner, and J. Prost, Phys. Rev. Lett. 84, 3494 (2000).
  * Tozzi  _et al._ (2019) C. Tozzi, N. Walani, and M. Arroyo, New J. Phys. 21, 093004 (2019).
  * (42) In Ref. 28, the formula is written for $-\pi/2<\theta_{\mathrm{ps}}\leq\pi/2$, but the chemical potential and axial forces are inconsistently calculated.
  * Vyas  _et al._ (2022) P. Vyas, P. B. S. Kumar, and S. L. Das, Soft Matter 18, 1653 (2022).
  * Goutaland  _et al._ (2021) Q. Goutaland, F. van Wijland, J.-B. Fournier, and H. Noguchi, Soft Matter 17, 5560 (2021).
  * Heinrich and Waugh (1996) V. Heinrich and R. E. Waugh, Ann. Biomed. Eng. 24, 595 (1996).
  * Dimova (2014) R. Dimova, Adv. Colloid Interface Sci. 208, 225 (2014).
  * Midya  _et al._ (2023) J. Midya, T. Auth, and G. Gompper, ACS Nano 17, 1935 (2023).
  * Noguchi (2015) H. Noguchi, J. Chem. Phys. 143, 243109 (2015).
  * Noguchi (2016) H. Noguchi, Sci. Rep. 6, 20935 (2016).
  * Stachowiak  _et al._ (2012) J. C. Stachowiak, E. M. Schmid, C. J. Ryan, H. S. Ann, D. Y. Sasaki, M. B. Sherman, P. L. Geissler, D. A. Fletcher, and C. C. Hayden, Nat. Cell Biol. 14, 944 (2012).
  * Busch  _et al._ (2015) D. J. Busch, J. R. Houser, C. C. Hayden, M. B. Sherman, E. M. Lafer, and J. C. Stachowiak, Nat. Commun. 6, 7875 (2015).
  * Snead  _et al._ (2019) W. T. Snead, W. F. Zeno, G. Kago, R. W. Perkins, J. B. Richter, C. Zhao, E. M. Lafer, and J. C. Stachowiak, J. Cell Biol. 218, 664 (2019).
  * Hiergeist and Lipowsky (1996) C. Hiergeist and R. Lipowsky, J. Phys. II France 6, 1465 (1996).
  * Bickel  _et al._ (2001) T. Bickel, C. Jeppesen, and C. M. Marques, Eur. Phys. J. E 4, 33 (2001).
  * Auth and Gompper (2003) T. Auth and G. Gompper, Phys. Rev. E 68, 051801 (2003).
  * Wu  _et al._ (2013) H. Wu, H. Shiba, and H. Noguchi, Soft Matter 9, 9907 (2013).
  * Marsh  _et al._ (2003) D. Marsh, R. Bartucci, and L. Sportelli, Biochim. Biophys. Acta 1615, 33 (2003).
  * Evans  _et al._ (2003) A. R. Evans, M. S. Turner, and P. Sens, Phys. Rev. E 67, 041907 (2003).
  * Noguchi (2022b) H. Noguchi, J. Chem. Phys. 157, 034901 (2022b).


11institutetext: Dept. of Computer Science and Engineering, The Chinese
University of Hong Kong 22institutetext: Dept. of Psychology, The University
of Hong Kong 33institutetext: Division of Social Science, The Hong Kong
University of Science and Technology

# Weakly-supervised Medical Image Segmentation with Gaze Annotations

Yuan Zhong1 Chenhui Tang1 Yumeng Yang2 Ruoxi Qi2 Kang Zhou1
Yuqi Gong1 Pheng Ann Heng1 Janet H. Hsiao${}^{3(\textrm{\scriptsize{\char
0\relax}})}$ Qi Dou${}^{1(\textrm{\scriptsize{\char 0\relax}})}$

###### Abstract

Eye gaze that reveals human observational patterns has increasingly been
incorporated into solutions for vision tasks. Despite recent explorations on
leveraging gaze to aid deep networks, few studies exploit gaze as an efficient
annotation approach for medical image segmentation which typically entails
heavy annotating costs. In this paper, we propose to collect dense weak
supervision for medical image segmentation with a gaze annotation scheme. To
train with gaze, we propose a multi-level framework that trains multiple
networks from discriminative human attention, simulated with a set of pseudo-
masks derived by applying hierarchical thresholds on gaze heatmaps.
Furthermore, to mitigate gaze noise, a cross-level consistency is exploited to
regularize overfitting noisy labels, steering models toward clean patterns
learned by peer networks. The proposed method is validated on two public
medical datasets of polyp and prostate segmentation tasks. We contribute a
high-quality gaze dataset entitled GazeMedSeg as an extension to the popular
medical segmentation datasets. To the best of our knowledge, this is the first
gaze dataset for medical image segmentation. Our experiments demonstrate that
gaze annotation outperforms previous label-efficient annotation schemes in
terms of both performance and annotation time. Our collected gaze data and
code are available at: https://github.com/med-air/GazeMedSeg.

###### Keywords:

Gaze Annotation Weakly-supervised Image Segmentation

##  1 Introduction

Recent studies have witnessed increasing interest in incorporating human
factors into deep learning applications [13, 24]. Eye tracking data, serving
as a popular tool reflecting the underlying cognitive processes [27], has
stood out as a promising and accessible media for human-AI interaction.
Previous works commonly utilize gaze as auxiliary information to guide deep
networks [9, 15, 22, 23], with recent explorations of employing gaze as the
sole supervision signal for label-efficient classification [19]. However,
leveraging gaze for supervising image segmentation models remains under-
explored yet valuable, since it alleviates annotators’ workload by alleviating
the need for labor-intensive pixel-wise annotation. Unlike existing label-
efficient annotation schemes that provide sparse supervision with bounding
boxes [6, 20], points [4] or scribbles [25], gaze data yield dense pixel-wise
supervision signals, which is crucial for medical images featuring ambiguous
boundaries and low contrast. These motivate us to investigate the potential of
gaze-supervised medical image segmentation.

A straightforward way to gaze supervision is training with pseudo-masks
generated by binarizing gaze heatmaps with a fixed threshold, where the dense
gaze heatmaps contain continuous values indicating the degree of observational
attention. A similar approach is widely employed in image-level semantic
segmentation [3, 26] for binarizing class activation maps (CAMs) [29]. In
practice, this approach yields suboptimal performance for gaze supervision due
to the distinct characteristics of gaze data as discriminative and noisy.
Firstly, the quality of pseudo-masks is sensitive to the selection of
threshold (Fig. 1). And it is unreasonable to rive precise object boundaries
with a global threshold over all images since human annotators usually pay
discriminative attention even on different parts of a single object. Secondly,
the error in eye tracking and human subjectivity makes gaze data a noisy
supervision signal for segmentation. For example, the annotator may check
every suspicious area when annotating the targets, thus some noisy gaze will
be left. Current noise-robust approaches are based on the symmetric or
asymmetric assumptions of simulated noise and design robust loss functions [8,
28] or regularization [14]. For correlated real-world gaze noise, however, the
assumptions on simulated noise do not necessarily hold.

Figure 1: Illustrations of full and different label-efficient annotation
schemes. Dense binarized gaze pseudo-masks are generated with various
thresholds $t$, which trade off the activation of the foreground and
background.

The key to robust gaze supervision lies in the unity and consideration of the
aforementioned two characteristics of gaze data. Inspired by multi-expert
models [17] benefiting from comprehensively integrating knowledge from
multiple experts, we propose to fuse multiple diverged networks learning from
multi-level human attention, simulated by applying a set of hierarchical
thresholds on gaze heatmaps. These networks are designed to learn
heterogeneous knowledge from discriminative human attention. Moreover, to
mitigate gaze noise, we exploit the clean knowledge learned by peer networks
of different levels to compensate for overfitting gaze noise with analysis on
the memorization effect of deep networks.

In this paper, we propose a new gaze annotation scheme that collects dense
annotation in an annotator-friendly and efficient manner for segmentation
tasks. Utilizing the scheme, we introduce the gaze dataset GazeMedSeg, which
extends the Kvasir-SEG [10] and NCI-ISBI [2] datasets with gaze data of
multiple annotators. To train with gaze, we propose a multi-level approach
that trains multiple divergent deep networks to ensemble information from
different levels of human discriminative attention. In addition, a cross-level
consistency regularization term over predictions smoothed by a local pixel
propagation module is exploited to compensate for overfitting on noisy gaze
labels. The advantage of the proposed neat approach is in its ability to
seamlessly fit into standard training pipelines with no changes to model
architectures. In experiments, we validate gaze annotation on polyp and
prostate segmentation tasks using our GazeMedSeg dataset. Compared to the
existing label-efficient annotation schemes, gaze supervision narrows the gaps
with full supervision and consistently boosts performance by over 2.0% in Dice
while being 15.4% faster to annotate, striking a sweet trade-off between
performance and annotation time.

##  2 Gaze Annotation Collection

####  2.0.1 Gaze annotation scheme.

We develop the eye-tracking program utilizing SR Research Experiment Builder
platform. At the beginning of gaze annotation, each annotator goes through a
9-point gaze calibration process. Our gaze annotation collection consists of
two stages. When presented with an image, the annotator (with eye-tracker)
first roughly scans the image and locates the target objects. Following that,
the annotator is requested to scan the objects thoroughly. Typically,
participants start from central areas and then move on to the boundaries,
ensuring that all parts of the target are covered. This step avoids partial
activation by encouraging annotators to pay more attention to the target
objects. Therefore, the noise will be relatively weakened when normalizing the
heatmap. After finishing the annotating, a key is pressed to switch to the
next image. More details on the eye-tracking settings can be found in the
Appendix.

GazeMedSeg dataset. Our collected GazeMedSeg dataset includes gaze annotations
for two public medical segmentation datasets. We use the Kvasir-SEG [10]
dataset for polyp segmentation from gastrointestinal images, and the NCI-ISBI
[2] dataset for prostate segmentation from T2-weighted MR images. The Kvasir-
SEG dataset includes 900 training and 100 testing images and the NCI-ISBI
dataset includes 60 training and 10 testing volumes, where we retain slices
containing prostate and obtain 789 training and 117 testing images. One
annotator finishes the annotation of all images in the datasets, and we use it
in our major experiments. We also invite two additional annotators to annotate
a subset for sensitivity studies (Sec. 4.2). All annotators are experienced in
medical imaging and are well-trained for eye-tracking trials.

##  3 Methodology

###  3.1 Multi-level Learning from Discriminative Attention

The original gaze data is a series of gaze positions collected at a certain
frequency. Given the gaze positions, the attention heatmaps are obtained by
convolving an isotropic Gaussian over them. We further apply dense conditional
random fields (CRF) [11] to enhance the initial heatmaps and generate pseudo-
masks. However, the quality of pseudo-masks varies significantly with distinct
thresholds (Fig. 1). It is hard to balance the over-activation and under-
activation of foregrounds via a fixed global threshold, because human
subjectively pay discriminative attention to target objects. For instance,
annotators may focus on the most discriminative part while only roughly
scanning ambiguous parts of an object, which may be neglected with a fixed
global threshold.

Figure 2: (a) Overview of the proposed method. For simplicity, we present the
case with two levels and $\mathcal{L}_{cons}$ of network 1. The consistency
loss is applied to all networks in the implementation. (b) We visualize the
dynamics of early-learning (the Dice of output and ground-truth on wrongly
annotated pixels) and overfitting (the Dice of output and noisy gaze pseudo-
mask on wrongly annotated pixels) with and without the proposed consistency
regularization on Kvasir-SEG [10] training data. The proposed consistency
prevents overfitting on the noisy labels. We use two levels and plot the
average of all levels in this experiment. (c) We visualize the gradients of
cross-entropy and consistency terms in the training process. The gradients
that encourage dilation and erosion of the predicted target are scaled for
visualization in different colors. The cross-entropy term gives noisy
supervision of erosion on the top of the target object, which is compensated
by consistency with clean patterns of dilation learned by other networks.

Our idea is to train $m$ deep networks simultaneously supervised by pseudo-
masks generated from $m$ different thresholds (Fig. 2 (a)), simulating multi-
level human attention. Each network is independently initialized, resulting in
varying learning capabilities. Being supervised by pseudo-masks with different
activation degrees, the networks evolve to learn various representations of
human attention.

In practice, we select a pair of thresholds based on annotators’ feedback to
generate diverse heatmaps that closely resemble ground truth and complement
each other, with one tending to erode (under-activate) and the other dilate
(over-activate) targets. Additional thresholds are linearly interpolated from
the pair. Empirical results indicate that two levels are sufficient for decent
performance (Sec. 4.2). The final segmentation prediction is obtained by
ensembling the predictions of these networks. We maintain the multi-level
structure throughout both the training and inference stages.

###  3.2 Cross-level Consistency for Noise Compensation

Another essential aspect of gaze-supervised segmentation is the inevitable
noise in the pseudo-masks. The noises can be introduced in the process of
human interpretation, gaze estimation by eye-trackers, heatmap generation,
etc. We observe a memorization effect [1] of deep networks when training with
gaze data on the medical image segmentation task. As shown in Fig. 2 (b), the
model captures clean patterns on incorrectly annotated pixels of pseudo-masks
at the beginning of training, but eventually overfits on noisy labels.

In the multi-level framework, though the networks are diverged with distinct
supervisions, they share the same input and various pseudo-masks supervise
different representations of the shared gaze data. Based on the assumption
that networks learn clean patterns at the beginning of the memorization, for
each network, we propose to exploit the knowledge learned by peer networks of
other levels to compensate for the noisy label via a consistency term.

To ensure noise-robust consistency, we first use a non-parametric local pixel
propagation (LPP) module to filter the feature of each pixel by propagating
the features of surrounding pixels in local regions inspired by recent works
[5, 7] proving noise-robust feature correspondence distillation. Given the
feature map $\phi$, for each pixel feature $\phi_{p}$, the LPP module computes
the transform $\widehat{\phi}_{p}$ as:

$\widehat{\phi}_{p}:=\sum_{q\in\mathcal{P}_{p}}\mathrm{softmax}\left(\max\\{\mathrm{cos}(\phi_{p},\phi_{q}),0\\}\right)\cdot\phi_{q},$
(1)

where $\mathrm{cos}$ denotes the cosine similarity, $\mathcal{P}_{p}$ denotes
the set of neighboring pixels (e.g., a $3\times 3$ region with dilation 1) of
pixel $p$. This refinement has a feature denoising/smoothing effect that
reduces the outliers and enhances the features with local context by
introducing spatial smoothness which encourages spatially close pixels to be
similar. Given an arbitrary pair of levels $i,j\in\mathbb{Z}^{+}$, where
$i,j\leq m$ and $i\neq j$, the consistency loss applied on the $i$-th level
maximizes dot-product between the propagated prediction $\hat{p}^{i}$ of
$i$-th level and non-propagated prediction $p^{j}$ of $j$-th level, i.e.,
$\mathcal{L}_{\mathrm{cons}}^{(i,j)}:=-\hat{p}^{i}\cdot p^{j}$. Notably, we
have $\hat{p}^{i}=g^{i}(\widehat{\phi}^{i})$ for the propagated prediciton and
$p^{i}=g^{i}(\phi^{i})$ for the non-propagated one, where $g$ denotes the
shallow classifier.

###  3.3 Overall Optimization of Gaze Supervision

The overall loss for the $i$-th level is the combination of the supervision
term and the consistency term over all other peer-level $j$:

$\mathcal{L}^{i}=\mathcal{L}_{\mathrm{CE}}^{i}+\frac{\lambda}{m-1}\sum_{j=1,j\neq
i}^{m}\mathcal{L}_{\mathrm{cons}}^{(i,j)},$ (2)

where $\mathcal{L}_{\mathrm{CE}}$ is the cross-entropy loss and can be
replaced by any other segmentation loss such as dice loss, and $\lambda$ is
empirically set to 3. Note that when optimizing the network of the $i$-th
level, the parameters of all other networks are frozen.

The key to understanding the overall optimization process lies in the trade-
off of cross-entropy and consistency terms. Intuitively,
$\mathcal{L}_{\mathrm{CE}}$ trains a set of divergent networks utilizing
multi-level pseudo-masks. However, this term tends to vanish after the early
learning stage and each network starts to overfit on the respective noisy
label. The consistency term $\mathcal{L}_{\mathrm{cons}}$ compensates for it,
implicitly forcing networks to continue learning from clean patterns learned
by networks of other levels. The mechanism can be viewed as pushing networks
to struggle to find a balance between divergence and consistency, in which the
hyper-parameter $\lambda$ controls this balance. It is worth noting that both
divergence and consistency are equally essential for optimization. The
consistency term ensures robustness to noise and the cross-entropy term
expands and diversifies clean knowledge learned and prevents collapsing into a
single network.

##  4 Experiments

We validate the proposed gaze annotation scheme on the aforementioned Kvasir-
SEG [10] dataset for polyp segmentation and NCI-ISBI [2] dataset for prostate
segmentation. For all datasets, we only utilize weak annotations for training
and report performance on the testing set. We train a 2D UNet [18] from
scratch for 15k iterations with a NVIDIA A40 GPU. We use Adam optimizer with
batch size 8 and learning rate $4e^{-4}$. More details on the training recipe
can be found in the Appendix.

###  4.1 Comparison Among Label-efficient Annotation Schemes

####  4.1.1 Comparison with state-of-the-art weakly-supervised methods.

We compare the new gaze-based annotation scheme with full mask supervision and
other state-of-the-art weakly-supervised methods using different sparse
annotations including box, point, and scribble on two datasets. To compare
annotation time, we also invite the same annotator to annotate a randomly
sampled subset of the Kvasir-SEG dataset containing 100 images using other
annotation schemes and report the annotation time in Table 3. Note that we
annotate the bounding box using extreme points clicking [16], and annotate
points using the scheme suggested by [4]. The estimated time is close to that
reported in the literature [4, 12, 16, 21], where the narrow gaps may come
from the difference in the complexity of different target objects to be
annotated. For all datasets, we simulate weak annotations based on the ground
truth. Note that we randomly sampled 10 pixels and 10 background pixels inside
and outside the bounding box respectively as suggested by [4] for point
annotations, and we follow [21] to simulate scribble annotations. In Table 3,
our results show that gaze supervision outperforms previous weakly-supervised
methods trained with other sparse annotation schemes and achieves over 95% of
the fully-supervised performance.

Table 1: Comparison with full mask supervision and SOTA weakly-supervised
methods using different annotation schemes. We report the mean and standard
deviation of three runs with different seeds. Dice is used as the evaluation
metric. The reported annotation time is estimated to annotate 900 images in
Kvasir-SEG [10] training set.

Method | Sup. | Polyp | Prostate
---|---|---|---
‘ |  Anno. Time | Dice | Dice
Vanilla | Full | 18.7 hrs | $82.12_{\pm 1.11}$ | $80.58_{\pm 0.48}$
BoxInst [20] | Box | 3.1 hrs | $65.72_{\pm 2.97}$ | $73.78_{\pm 1.15}$
BoxTeacher [6] | Box | 3.1 hrs | $73.33_{\pm 1.30}$ | $75.60_{\pm 1.15}$
PointSup [4] | Point | 4.8 hrs | $73.05_{\pm 1.64}$ | $73.46_{\pm 4.71}$
AGMM [25] | Point | 4.8 hrs | $75.57_{\pm 0.84}$ | $73.86_{\pm 1.26}$
AGMM [25] | Scribble | 2.6 hrs | $67.23_{\pm 1.02}$ | $72.70_{\pm 1.03}$
Ours | Gaze | 2.2 hrs | $\mathbf{77.80_{\pm 1.02}}$ | $\mathbf{77.64_{\pm 0.57}}$

Figure 3: Performance versus annotation time for different annotation schemes.
To match annotation times among annotatíion forms, we train a 2D UNet model
using from 10% to 100% of the Kvasir-SEG training set.

Figure 4: Visualization of gaze data and predictions. The model without
consistency term ensemble the noise of different levels. Instead, the model
regularized by consistency learns clean patterns of pseudo-masks and
demonstrates robustness to noises.

Trade-off between performance and annotation time. We compare the proposed
gaze annotation scheme with other label-efficient sparse annotation schemes
for image segmentation under the same annotation budget, i.e., the time
required to annotate training data. Fig. 3 presents our results on Kvasir-SEG
[10], proving that gaze annotation boosts weakly-supervised segmentation by
striking a sweet performance/annotation time trade-off by maximizing
performance with the least annotation time among existing weak annotation
schemes.

Figure 5: Ablation studies on polyp segmentation. (a) Effects of proposed
components. (b) Effects of the hyper-parameter $m$ ($m=2$ by default) and
$\lambda$ ($\lambda=3$ by default). The orange and blue lines deficit the
results of varying $m$ and $\lambda$, respectively. (c) Effects of three
different annotators (major annotator A and additional annotators B and C) and
comparison with other annotation schemes.

###  4.2 Ablation Studies

####  4.2.1 Modules and hyper-parameters.

To evaluate the impact of critical components of the proposed method, we study
the effectiveness of the proposed components for gaze supervision in Fig. 5
(a). We further present the result of different choices of hyper-parameter $m$
(number of levels) and $\lambda$ in Fig. 5 (b). Our experiments show that
$m=2$ is optimal for gaze training while increasing $m$ gives limited benefits
but leads to greater training and inference complexity. The results on
$\lambda$ echo the intuition of it in Sec. 3.3. We also observe that having
$\lambda$ greater than 7 results in a degenerated model that collapses to
consistency with performance worse than simply ensembling without consistency
regularization. We further visualize gaze-supervised predictions in Fig. 4,
where the model with consistency regularization demonstrates resistence to
gaze noise.

Sensitivity to annotator. We invite two additional annotators to annotate a
subset of the Kvasir-SEG training set containing 500 images and train a UNet
on this subset using different annotation schemes. Note that all annotators
receive the same training for gaze annotating. The results presented in Fig. 5
(c) show that though eye-tracking is subjective, different annotators
demonstrate comparable annotation time and supervision quality, consistently
outperforming other annotation schemes.

##  5 Conclusion and Future Work

In this paper, we propose to train deep networks with gaze annotations
efficiently collected using a new gaze annotation scheme for medical image
segmentation. The proposed method can be seamlessly integrated into standard
training pipelines. The results show that gaze annotation achieves a sweet
performance and annotation time trade-off compared to other annotation forms.

Our explorations give rise to several potential directions for future work:
(1) While we expect gaze supervision to be broadly applicable to medical
applications, the multiplied complexities in the training and especially the
inference stage hamper real-time scenarios since multiple networks are
maintained even though we have shown that $m=2$ is sufficient. One potential
direction is to aggregate networks at a certain frequency in the training and
only keep an aggregated model for inference. (2) Though specialized hardware
is required to collect gaze data, we foresee that eye-tracking will not pose a
bottleneck for clinical practicality even at present with the advance of
commercial VR/XR headsets with precise and affordable eye-tracking
capabilities. (3) We focus on binary segmentation in this paper, and the
extension to multiple cases is straightforward via annotating each class
separately and deciding the label for each pixel as the class with the highest
value in gaze heatmaps of different classes.

Acknowledgments. This work was supported by Hong Kong Research Grants Council
(Project No. T45-401/22-N), and Science, Technology and Innovation Commission
of Shenzhen Municipality (Project No. SGDX20220530111201008).

Disclosure of Interests. The authors have no competing interests to declare
that are relevant to the content of this article.

## References

  * [1] Arpit, D., Jastrzębski, S., Ballas, N., Krueger, D., Bengio, E., Kanwal, M.S., Maharaj, T., Fischer, A., Courville, A., Bengio, Y., et al.: A closer look at memorization in deep networks. In: International conference on machine learning. pp. 233–242. PMLR (2017)
  * [2] Bloch, B.N., Madabhushi, A., Huisman, H., Freymann, J., Kirby, J., Grauer, M., Enquobahrie, A., Jaffe, C., Clarke, L., Farahani, K.: Nci-isbi 2013 challenge: Automated segmentation of prostate structures (isbi-mr-prostate-2013) (2015). https://doi.org/10.7937/K9/TCIA.2015.ZF0VLOPV
  * [3] Chang, Y.T., Wang, Q., Hung, W.C., Piramuthu, R., Tsai, Y.H., Yang, M.H.: Mixup-cam: Weakly-supervised semantic segmentation via uncertainty regularization. British Machine Vision Conference (2020)
  * [4] Cheng, B., Parkhi, O., Kirillov, A.: Pointly-supervised instance segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 2617–2626 (2022)
  * [5] Cheng, H., Zhu, Z., Sun, X., Liu, Y.: Mitigating memorization of noisy labels via regularization between representations. International Conference on Learning Representations (2022)
  * [6] Cheng, T., Wang, X., Chen, S., Zhang, Q., Liu, W.: Boxteacher: Exploring high-quality pseudo labels for weakly supervised instance segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 3145–3154 (2023)
  * [7] Hamilton, M., Zhang, Z., Hariharan, B., Snavely, N., Freeman, W.T.: Unsupervised semantic segmentation by distilling feature correspondences. International Conference on Learning Representations (2022)
  * [8] Han, B., Yao, Q., Yu, X., Niu, G., Xu, M., Hu, W., Tsang, I., Sugiyama, M.: Co-teaching: Robust training of deep neural networks with extremely noisy labels. Advances in neural information processing systems 31 (2018)
  * [9] Huang, Y., Li, X., Yang, L., Gu, L., Zhu, Y., Seo, H., Meng, Q., Harada, T., Sato, Y.: Leveraging human selective attention for medical image analysis with limited training data. British Machine Vision Conference (2021)
  * [10] Jha, D., Smedsrud, P.H., Riegler, M.A., Halvorsen, P., de Lange, T., Johansen, D., Johansen, H.D.: Kvasir-seg: A segmented polyp dataset. In: International Conference on Multimedia Modeling. pp. 451–462. Springer (2020)
  * [11] Krähenbühl, P., Koltun, V.: Efficient inference in fully connected crfs with gaussian edge potentials. Advances in neural information processing systems 24 (2011)
  * [12] Kuznetsova, A., Rom, H., Alldrin, N., Uijlings, J., Krasin, I., Pont-Tuset, J., Kamali, S., Popov, S., Malloci, M., Kolesnikov, A., et al.: The open images dataset v4: Unified image classification, object detection, and visual relationship detection at scale. International Journal of Computer Vision 128(7), 1956–1981 (2020)
  * [13] Li, Q., Peng, Z., Zhou, B.: Efficient learning of safe driving policy via human-ai copilot optimization. International Conference on Learning Representations (2022)
  * [14] Liu, S., Niles-Weed, J., Razavian, N., Fernandez-Granda, C.: Early-learning regularization prevents memorization of noisy labels. Advances in neural information processing systems 33, 20331–20342 (2020)
  * [15] Liu, Y., Zhou, L., Zhang, P., Bai, X., Gu, L., Yu, X., Zhou, J., Hancock, E.R.: Where to focus: Investigating hierarchical attention relationship for fine-grained visual classification. In: European Conference on Computer Vision. pp. 57–73. Springer (2022)
  * [16] Papadopoulos, D.P., Uijlings, J.R., Keller, F., Ferrari, V.: Extreme clicking for efficient object annotation. In: Proceedings of the IEEE international conference on computer vision. pp. 4930–4939 (2017)
  * [17] Pavlitskaya, S., Hubschneider, C., Weber, M., Moritz, R., Huger, F., Schlicht, P., Zollner, M.: Using mixture of expert models to gain insights into semantic segmentation. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition Workshops. pp. 342–343 (2020)
  * [18] Ronneberger, O., Fischer, P., Brox, T.: U-net: Convolutional networks for biomedical image segmentation. In: Medical Image Computing and Computer-Assisted Intervention–MICCAI 2015: 18th International Conference, Munich, Germany, October 5-9, 2015, Proceedings, Part III 18. pp. 234–241. Springer (2015)
  * [19] Saab, K., Hooper, S.M., Sohoni, N.S., Parmar, J., Pogatchnik, B., Wu, S., Dunnmon, J.A., Zhang, H.R., Rubin, D., Ré, C.: Observational supervision for medical image classification using gaze data. In: Medical Image Computing and Computer Assisted Intervention–MICCAI 2021: 24th International Conference, Strasbourg, France, September 27–October 1, 2021, Proceedings, Part II 24. pp. 603–614. Springer (2021)
  * [20] Tian, Z., Shen, C., Wang, X., Chen, H.: Boxinst: High-performance instance segmentation with box annotations. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 5443–5452 (2021)
  * [21] Valvano, G., Leo, A., Tsaftaris, S.A.: Learning to segment from scribbles using multi-scale adversarial attention gates. IEEE Transactions on Medical Imaging 40(8), 1990–2001 (2021)
  * [22] Wang, C., Zhang, D., Ge, R.: Eye-guided dual-path network for multi-organ segmentation of abdomen. In: International Conference on Medical Image Computing and Computer-Assisted Intervention. pp. 23–32. Springer (2023)
  * [23] Wang, S., Ouyang, X., Liu, T., Wang, Q., Shen, D.: Follow my eye: Using gaze to supervise computer-aided diagnosis. IEEE Transactions on Medical Imaging 41(7), 1688–1698 (2022)
  * [24] Wang, S., Zhao, Z., Ouyang, X., Wang, Q., Shen, D.: Chatcad: Interactive computer-aided diagnosis on medical image using large language models. arXiv preprint arXiv:2302.07257 (2023)
  * [25] Wu, L., Zhong, Z., Fang, L., He, X., Liu, Q., Ma, J., Chen, H.: Sparsely annotated semantic segmentation with adaptive gaussian mixtures. In: Proceedings of the IEEE/CVF Conference on Computer Vision and Pattern Recognition. pp. 15454–15464 (2023)
  * [26] Wu, T., Gao, G., Huang, J., Wei, X., Wei, X., Liu, C.H.: Adaptive spatial-bce loss for weakly supervised semantic segmentation. In: European Conference on Computer Vision. pp. 199–216. Springer (2022)
  * [27] Yun, K., Peng, Y., Samaras, D., Zelinsky, G.J., Berg, T.L.: Exploring the role of gaze behavior and object detection in scene understanding. Frontiers in psychology 4, 917 (2013)
  * [28] Zhang, Z., Sabuncu, M.: Generalized cross entropy loss for training deep neural networks with noisy labels. Advances in neural information processing systems 31 (2018)
  * [29] Zhou, B., Khosla, A., Lapedriza, A., Oliva, A., Torralba, A.: Learning deep features for discriminative localization. In: Proceedings of the IEEE conference on computer vision and pattern recognition. pp. 2921–2929 (2016)

## A. Gaze Data Collection

Table 2: Eye tracking settings to build the GazeMedSeg dataset using SR Research Experiment Builder. Major settings for the program building are reported. Eye Tracker | SR Research EyeLink 1000 Plus
---|---
Tracked Eye | Monocular (ocular dominance)
Sampling Rate | 1000 Hz
Tracking Error |  $\leq 0.5^{\circ}$ visual angle
Screen-Eye Distance |  $46-55$ cm
Screen Size | 21.5 inch
Screen Resolution | $1024\times 768$
Screen Refresh Rate | 60 Hz
Displayed Image Size | $768\times 768$
Displayed Image Position | Centered
Displayed Image Extent | Vertical & Horizontal: $\approx 54$ cm
Using Chin Rest | Yes
Wearing Glasses | No (contact lenses are allowed)
Tracking Program | SR Research Experiment Builder
Host PC OS | Microsoft Windows 10
Figure 6: Eye-tracking equipment and setup. We use a chin rest to stabilize
the eye-tracking process. During gaze collection, only one image is displayed
on the screen. Annotators use their eyes to annotate and their hands to press
a key to switch to the next image once they have finished.

## B. Implementation Details

Table 3: Hyper-parameters used in experiments on two datasets. Backbone | 2D UNet
---|---
Supervision Loss | Cross-entropy loss
Training Iterations | 15000
Batch Size | 8
Optimizer | SGD
SGD Momentum $\mu$ | 0.99
Scheduler | CosineAnnealingLR
Base Learning Rate | $1e^{-2}$
Minimum Learning Rate | $1e^{-4}$
Resolution | $224\times 224$
Data Augmentation | Random Flip
Data Split | Kvasir-SEG: 900 training and 100 testing images
NCI-ISBI: 789 training and 117 testing images