Physical limits to membrane curvature sensing by a single protein Indrajit Badvaram1and Brian A. Camley21 1Department of Biophysics Johns Hopkins University Baltimore MD

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Physical limits to membrane curvature sensing by a single protein

Indrajit Badvaram1and Brian A. Camley2, 1

1Department of Biophysics, Johns Hopkins University, Baltimore, MD

2William H. Miller III Department of Physics & Astronomy, Johns Hopkins University, Baltimore, MD

Membrane curvature sensing is essential for a diverse range of biological processes. Recent experiments have

revealed that a single nanometer-sized septin protein can distinguish between membrane-coated glass beads of

one micron and three micron diameters, even though the septin is orders of magnitude smaller than the beads.

This sensing ability is especially surprising since curvature-sensing proteins must deal with persistent thermal

ﬂuctuations of the membrane, leading to discrepancies between the bead’s curvature and the local membrane

curvature sensed instantaneously by a protein. Using continuum models of ﬂuctuating membranes, we investigate

whether it is feasible for a protein acting as a perfect observer of the membrane to sense micron-scale curvature

either by measuring local membrane curvature or by using bilayer lipid densities as a proxy. To do this, we

develop algorithms to simulate lipid density and membrane shape ﬂuctuations. We derive physical limits to

the sensing eﬃcacy of a protein in terms of protein size, membrane thickness, membrane bending modulus,

membrane-substrate adhesion strength, and bead size. To explain the experimental protein-bead association

rates, we develop two classes of predictive models: i) for proteins that maximally associate to a preferred

curvature, and ii) for proteins with enhanced association rates above a threshold curvature. We ﬁnd that the

experimentally observed sensing eﬃcacy is close to the theoretical sensing limits imposed on a septin-sized

protein, and that the variance in membrane curvature ﬂuctuations sensed by a protein determines how sharply

the association rate depends on the curvature of the bead.

I. INTRODUCTION

Membrane curvature is ubiquitous throughout cell biology

[1–3]: proteins that sense membrane curvatures can help lo-

cate the axis of cell division, determine cell polarity, facilitate

membrane remodeling, and serve as a cue for intracellular sig-

naling [4–8]. These proteins often act in tandem by binding

with each other to sense curvature cooperatively. However,

in the case of septin proteins, recent experiments have shown

that in addition to sensing curvatures via cooperative ﬁlament

formation [9,10], even a single septin protein can distinguish

between micron-scale membrane curvatures and preferentially

bind to membranes adhered to glass beads of diﬀerent curva-

tures with diﬀerent association rates [11–13]. How do proteins

only a few nanometers in size eﬀectively sense membrane cur-

vatures that are hundreds of times larger than themselves, on

the order of micrometers? This sensing ability is even more

remarkable considering that biological membranes undergo

persistent thermally-driven undulations [14]. Even if a pro-

tein could perfectly measure the instantaneous shape of the

membrane at the nanometer scale, these undulations drive the

membrane away from its average shape, confounding the pro-

tein’s attempts to measure the membrane’s curvature. How

can a protein reliably make a measurement of micron-scale

curvature in this noisy environment?

Curvatures also induce deviations in the packing of lipids in

the membrane bilayer, and proteins with amphipathic helices

insert themselves into bilayers [15,16]. Proteins that sense

micron-scale curvature may then be, instead of measuring the

shape directly, using a proxy for local curvature related to lipid

packing to sense membrane shape [17]. We conjecture that one

such proxy might be the deviations in lipid densities between

the upper and lower monolayers of the membrane bilayer.

Although there are descriptions of molecular mechanisms

employed by proteins when sensing nanometer-scale curva-

tures [18], curvature sensing at the micron-scale is less well-

average membrane shape

(μm-scale)

protein

(nm-scale)

instantaneous

local membrane shape

(nm-scale)

FIG. 1. Thermal ﬂuctuations of the membrane lead to discrepancies

between the instantaneous local membrane shape present at the pro-

tein’s location and the average membrane shape. Even if the protein

were a perfect detector of curvatures, observing the local curvature

without error, the local shape may not be indicative of the overall

membrane shape. A nanometer-sized protein is challenged by relative

diﬀerences in scale: steep local undulations have radii of curvature

on the scale of nanometers, while the membrane on average may be

much ﬂatter, with a radius of curvature on the scale of micrometers

(not to scale in this illustration).

understood [19]. Previous theoretical studies have modeled

the thermodynamics of curvature sensing [20] and the eﬀects

of helix insertion on the membrane’s energy [21,22]. Here,

we take a diﬀerent approach: we ask how precisely a protein

could measure the micron-scale curvature of a membrane if it

made a perfect measurement of local membrane shape or local

lipid density, subject to the inevitable thermal ﬂuctuations of

the membrane. This gives us the fundamental physical limits

to curvature sensing for an idealized protein, akin to Berg and

Purcell’s classic work on the limits of ligand concentration

sensing for a perfect detector of a ﬁnite size [23] and later

follow-ups [24–27]. Our result builds on the larger literature

of sensing limits in diﬀerent contexts, including gradient sens-

arXiv:2210.12204v1 [physics.bio-ph] 21 Oct 2022

ing [28–32], ﬂow sensing [33], and sensing the mechanical

properties of heterogeneous materials [34,35]. To quantify

curvature sensing in this way, we deﬁne a signal-to-noise ratio

(SNR) to indicate how well a protein is able to extract useful

information about the membrane’s shape despite stochasticity.

To support our analytical models, we develop algorithms to

simulate membranes whose ﬂuctuations in height are coupled

to ﬂuctuations in bilayer lipid densities.

Motivated by the experiments in [12,13], we explore the

theoretical limits to a protein’s ability to distinguish between

membrane-adhered beads of diﬀerent sizes. For a bead of ra-

dius 𝑅, the mean membrane curvature on average is 𝐶=1

𝑅.

However, if we probe the membrane over a region of a pro-

tein size 𝑎, the curvature will diﬀer from this value (Fig. 1).

We derive that in the absence of membrane-substrate adhe-

sion, the curvature ﬂuctuations sensed by a protein of size

𝑎and membrane bending modulus 𝜅are h𝐶2

𝑎i=1

𝑎𝑘𝐵𝑇

16𝜋𝜅 .

Since 𝑎𝑅and 𝑘𝐵𝑇

𝜅≈1/20, the magnitude of these curva-

ture ﬂuctuations is very large relative to the average curvature

1/𝑅—so even with a perfect measurement, a protein could

observe values of curvature far from 1/𝑅, indicating the dif-

ﬁculty in curvature sensing on a freely ﬂuctuating membrane.

Including membrane-bead adhesion suppresses these ﬂuctua-

tions, amplifying the protein’s ability to distinguish between

micron-sized beads. We also study the signal-to-noise ratio as-

sociated with measuring changes in lipid density, and ﬁnd that

measuring lipid densities compares well to an ideal curvature

measurement when the protein is small, or when the mem-

brane is thick and resistant to in-plane compression. Lastly,

to explain the experimentally observed protein-bead associa-

tion rates, we show how ﬂuctuation variances like h𝐶2

𝑎iand

sensing SNR can determine the association rate of a protein to

membrane-adhered beads of diﬀerent curvatures. To do so, we

develop two classes of models: the preferred curvature model,

for proteins that associate maximally to a speciﬁc membrane

curvature, and the curvature threshold model, for proteins that

exhibit enhanced binding to all curvatures above a thresh-

old. Using the preferred curvature model for our estimates

of membrane-substrate adhesion, we ﬁnd that measurements

of curvature by a single septin protein are close to the funda-

mental limit of sensing accuracy. With the curvature threshold

model, we show that the sharpness in the transition to enhanced

association above a threshold is determined by the variance in

membrane curvature ﬂuctuations sensed by the protein.

II. MODELS AND SIMULATION METHODS

A. Modeling membrane, bead, and membrane-bead adhesion

We represent the shape of the membrane in terms of its

height ℎ(r)above a two-dimensional plane as a function of

position r=(𝑥, 𝑦), i.e. using Monge gauge [36]. To induce a

curvature similar to the bead of [12,13], we model a substrate

with a spherical bump on a ﬂat surface (Fig. 2). We assume

that the adhesion energy between the bead and membrane

FIG. 2. Snapshots of thermally-ﬂuctuating simulated membranes: i)

(left) a freely ﬂuctuating ﬂat membrane with no membrane-substrate

adhesion, and ii) (right) a membrane adhered to a bead of radius

𝑅=500 nm with adhesion strength 𝛾=1013 J/m4. System size is

𝐿= 1.6 𝜇m. Note that due to the large diﬀerence in size scales and

strong membrane-substrate adhesion, ﬂuctuations are not apparent in

the plot on the right. Simulation parameters: Table I.

arises from a harmonic potential,

𝐸adh =𝛾

2dr(ℎ(r) − ℎbead (r))2,(1)

where 𝛾is the strength of membrane-substrate adhesion in

units of J/m4,ℎbead (r)traces the height of the bead at each

position in the 𝑥𝑦 plane and serves as the equilibrium height,

and the integral dris over the 𝑥𝑦 plane. This harmonic

potential approximates more detailed potentials, e.g. the Mie

potential of [37] or van der Waals interactions [38]; see Ap-

pendix G. The height ﬁeld ℎbead (r)corresponds to the 𝑧-axis

height of a hemisphere of radius 𝑅, centered in the plane, i.e.

ℎbead (r)=𝑅2−2𝑠2+2𝑠(𝑥+𝑦) − 𝑥2−𝑦2with 𝑠=𝐿/2for

rwithin 𝑅of the bead center (𝑠, 𝑠). We set ℎbead (r)=0out-

side this region: the membrane is adherent to a ﬂat substrate

outside of the bead region.

B. Energy of membrane height and density changes

In addition to the height of the membrane, we also character-

ize the membrane by the lipid densities in each leaﬂet. We use

the Seifert-Langer model [39,40] to represent how the mem-

brane’s height couples to lipid densities. Due to membrane

curvature, the lipid densities measured at diﬀerent depths into

the membrane bilayer will diﬀer. We follow Seifert and Langer

and primarily use the scaled lipid densities of the upper and

lower monolayers at the midsurface, 𝜌+and 𝜌−. These are

deﬁned as 𝜌±≡ (𝜓±/𝜙0−1), where 𝜓±are the densities pro-

jected onto the bilayer midsurface and 𝜙0is the equilibrium

number density of a ﬂat membrane. With this deﬁnition, the

lipid density deviation between the upper and lower monolay-

ers at the midsurface is given by 𝜌≡ (𝜌+−𝜌−)/2, and the

average density is ¯𝜌≡ (𝜌++𝜌−)/2. As shown in Fig. 3,

when the membrane is bent to a positive curvature and the

lipids allowed to laterally relax, the density projected by the

upper leaﬂet at the midsurface is greater than that projected

by the lower leaﬂet. (This is in contrast to the density proﬁle

when momentarily bending the membrane, where the midsur-

face densities are equal and the upper and lower leaﬂets are

stretched and compressed at the neutral surface, respectively

[41].)

Neutral surface

Midsurface

Neutral

surface

FIG. 3. A curved membrane induces deviations in the packing of

lipids in the bilayer. When the membrane is ﬂat, the number densi-

ties of lipids projected by the two monolayers at the midsurface are

equivalent. However, when the membrane is curved and its lipids are

allowed to laterally relax to their minimum energy value, the upper

(+) and lower (−) monolayers project diﬀerent densities at the midsur-

face. The steeper the curvature, the greater the diﬀerence between the

scaled densities 𝜌+and 𝜌−. At steady-state, the neutral surface lipid

number densities 𝜙+=𝜙−. The distance between the midsurface and

either neutral surface is 𝑑.

The membrane’s total free energy 𝐸consists of the sum of

the Helfrich free energy due to bending the membrane [42], the

energy due to lipid density deviations of the upper and lower

membrane monolayers away from their ideal values, and the

adhesion energy in Eq. (1), such that

𝐸=dr𝜅

2(2𝐻)2+𝑘

2[(𝜌+−2𝑑𝐻)2+(𝜌−+2𝑑𝐻)2]+𝐸adh,

(2)

where 𝜅is the membrane bending modulus, 𝐻is the mean

curvature of the membrane, such that 2𝐻=−∇2ℎ(r), and 𝑘is

the monolayer area compressibility modulus. (𝜌+−2𝑑𝐻)and

(𝜌−+2𝑑𝐻)represent the deformations away from the ideal

lipid density in the upper and lower membrane monolayers,

respectively. 𝑑is the distance between the bilayer midsurface

and the neutral surfaces of the monolayers (for brevity, we

refer to 𝑑as the monolayer thickness). The sign conventions

used for the mean curvature 𝐻in Eq. (2) are as in [40]. Since

we do not model asymmetries in lipid composition, we neglect

spontaneous curvature in the curvature dependent contribution

to the membrane energy [43].

The membrane bending energy includes a term ∇2ℎ(r),

which can be more easily dealt with in Fourier space. We

choose our Fourier conventions as representing a ﬁnite system

size of dimensions 𝐿×𝐿, with the Fourier wave-vector q=

2𝜋

𝐿(𝑚, 𝑛)such that −(𝑁−1)/2≤ (𝑚, 𝑛) ≤ (𝑁−1)/2for

𝑁×𝑁modes/lattice points, assuming 𝑁is odd. The Fourier

transform pair for the membrane’s height is then

ℎq=𝐿2drℎ(r)𝑒−𝑖q·r, ℎ(r)=1

𝐿2

ℎq𝑒𝑖q·r,(3)

and similarly for the transform pairs 𝜌q, 𝜌(r)and ¯𝜌q,¯𝜌(r).

Additional comments on the treatment of variables in Fourier

space are included in Appendix A.

The total free energy 𝐸in Eq. (2) is computed by summing

the contributions due to each Fourier mode as

𝐸=1

𝐿2

2(ℎq, 𝜌q,¯𝜌q)E

ℎq

𝜌q

¯𝜌q

∗

+𝐸adh,(4)

E=

˜𝜅𝑞4−2𝑘 𝑑𝑞20

−2𝑘𝑑𝑞22𝑘0

0 0 2𝑘.(5)

In Eq. (5), 𝑞is the magnitude of the Fourier wavevector q,

and ℎq, 𝜌q,¯𝜌qare the membrane height, lipid density deviation

and average density corresponding to a given mode. Here

˜𝜅=𝜅+2𝑑2𝑘is the renormalized bending modulus, which

describes the response of the membrane over short times when

lipids cannot laterally relax. A typical value of 𝜅is about 20

𝑘𝐵𝑇, although this can be a few times larger for particularly

stiﬀ membranes. The strength of membrane substrate adhesion

𝛾can vary over orders of magnitude in diﬀerent contexts. To

best model the experiments in [12,13], we use a fairly strong

𝛾∼1013 J/m4, unless otherwise stated. This is our estimate

of adhesion strengths of supported lipid bilayers (SLBs) on

glass substrates (see Discussion and Appendix G). The other

parameter values used in the model are included in Table I.

C. Dynamics and simulation of ﬂuctuating membranes

A membrane that is deformed away from its equilibrium

state will relax over time. The dynamics of this process are

controlled by the viscosity of the ﬂuid outside the membrane,

the membrane’s own viscosity, and the drag between the two

leaﬂets [39,41,44]. To these relaxation dynamics, we add

a stochastic term obeying a ﬂuctuation-dissipation relation-

ship, which ensures that the system will evolve into thermal

equilibrium. The resulting stochastic dynamical equations for

evolving ℎq,𝜌qand ¯𝜌qin time are (Appendix D)

𝜕

𝜕𝑡 

ℎq

𝜌q

¯𝜌q

=−𝐿2

Ωℎ𝜕𝐸/𝜕ℎ∗

Ω𝜌𝜕𝐸/𝜕𝜌∗

Ω¯𝜌𝜕𝐸/𝜕¯𝜌∗

q+

𝜉q

𝜁q

𝜒q,(6)

where Ωℎ−1=1/4𝜂𝑞,Ω𝜌−1=𝑞2/(4𝑏+4𝜂𝑞 +2𝜇𝑞2), and

Ω¯𝜌−1=𝑞2/(4𝜂𝑞 +2𝜇𝑞2). These Ω−1values play the role of

hydrodynamic mobilities for a membrane with monolayer vis-

cosity 𝜇and intermonolayer friction 𝑏embedded in a ﬂuid of

viscosity 𝜂, setting the time derivative of a ﬁeld 𝜔in terms of

the force-like term −𝐿2𝜕𝐸/𝜕𝜔∗

q(this is the Fourier transform

of −𝛿𝐸/𝛿𝜔(r)in our convention). For instance, Ω−1

ℎ=1/4𝜂𝑞

is the Oseen tensor relating the 𝑧-component force density on

the membrane to the membrane height’s velocity, as used in

[45]. Thermal ﬂuctuations are accounted for with the stochas-

tic terms 𝜉q, 𝜁qand 𝜒q(Appendix D). The deterministic com-

ponents in the equations for 𝜕ℎq

𝜕𝑡 and 𝜕𝜌q

𝜕𝑡 are consistent with

the Seifert-Langer model, and we derive 𝜕¯𝜌q

𝜕𝑡 from the hydro-

dynamic equations in [39] while neglecting inertial eﬀects.

While we present these equations of motion in terms of the

hydrodynamics of the system for generality, our focus is on

the equilibrium properties of the system, which are indepen-

dent of the dynamic parameters 𝜂,𝜇,𝑏, etc. We will use this

dynamical model to sample from the equilibrium thermal dis-

tributions of ℎ(r), 𝜌(r)and ¯𝜌(r). A full understanding of the

dynamics of this problem should also include the eﬀect of the

presence of the substrate near to the surface, which will alter

the hydrodynamic response [44,46].

To simultaneously simulate the ﬂuctuations of membrane

height and lipid density, we numerically integrate Eq. (6).

This essentially extends the Fourier-space Brownian Dynamics

(FSBD) approach [45]—so we will often refer to our simula-

tions as FSBD simulations as well. The simulation algorithms,

their derivations, and guidelines for choosing a manageable

timestep for simulation convergence are included in Appendix

D. To ensure that our approach creates the correct equilibrium

distribution, we compared with an extension of the Fourier

Monte Carlo method [47] (Appendix L).

D. Modeling a protein as a perfect observer

To understand what will limit a protein’s ability to sense

membrane curvature even in ideal circumstances, we treat the

protein as a perfect observer, making a precise measurement

of the membrane curvature at the protein scale. The perfect

observer assumption means that the protein does not aﬀect

the membrane in any way: it is a mere spectator. By a mea-

surement “at the protein scale,” we describe an average over a

region of the membrane of roughly the protein’s size 𝑎. The

local membrane curvature and local lipid density deviation

sensed by the protein are then

𝐶𝑎=𝐿2dr𝐺(r, 𝑎)−∇2ℎ(r)

2,(7)

𝜌𝑎=𝐿2dr𝐺(r, 𝑎)𝜌(r),(8)

where 𝐺(r, 𝑎)is a two-dimensional Gaussian weight centered

at the protein location, such that

𝐺(r, 𝑎)=1

2𝜋𝑎2exp −|r−rprot |2

2𝑎2.(9)

We will always choose the protein to be located at the top of

the spherical bead, rprot =(𝐿/2, 𝐿/2).

The integrals in Eqs. (7)–(8) are evaluated by summing

over discrete membrane lattice points. Membrane curvatures

are computed from ℎqnoting that the Fourier transform of the

curvature is {−1

2∇2ℎ(r)}q=1

2𝑞2ℎq, then using the inverse Fast

Fourier Transform to reconstruct the curvature ﬁeld −1

2∇2ℎ(r).

FIG. 4. Histograms (normalized to probability densities) from FSBD

simulations of local membrane curvatures and local lipid density

deviations sensed by a protein of size 𝑎=16 nm positioned at the top

of membrane-adhered beads of diﬀerent diameters. The strength of

membrane-substrate adhesion is 𝛾=1013 J/m4. When the histogram

distributions associated to diﬀerent beads overlap considerably, there

is more uncertainty about which bead curvature resulted in a particular

local membrane curvature or density deviation sensed by the protein.

The simulation used timesteps of Δ𝑡=3.2ns for total time 𝑡sim =

0.016 seconds. Simulation parameters: Table I.

III. RESULTS

A. Simulations of membrane-adhered beads

We simulate ﬂuctuating membranes adhered to beads of

varying sizes. In Fig. 4, we show the distribution of local

curvature 𝐶𝑎and local lipid density deviation 𝜌𝑎that would

be sensed over a protein scale of 𝑎=16 nm such that 2𝑎

roughly corresponds to the footprint of a yeast septin rod,

which has an end-end length of ∼32 nm [11]. We have chosen

the membrane-substrate adhesion appropriate for a supported

lipid bilayer, which is strongly adherent (Appendix G). These

distributions show the extent to which diﬀerent beads could

be distinguished by a protein: when there is signiﬁcant over-

lap between two distributions, even a perfect detector would

struggle to distinguish between beads of these radii. As the

bead radius is increased, the average curvatures and density

deviations sensed by the protein decrease in magnitude—as

we would expect, because the bead is made locally ﬂatter. The

distributions for larger beads overlap more substantially, so a

protein that measures a particular curvature or density value

in this regime is subjected to more ambiguity as to which bead

the measurement corresponds to.

B. Quantifying sensing eﬃcacy with signal-to-noise ratio

If a particular local curvature or density deviation is sampled

from the distributions in Fig. 4, can the bead size correspond-

ing to the sampled measurement be reliably determined? This

can be challenging due to overlapping distributions, since each

bead size is associated to a multiplicity of instantaneous 𝐶𝑎

and 𝜌𝑎values. We summarize the way that thermal ﬂuctua-

tions of the membrane could confound even a perfect detector

of curvature or lipid density in distinguishing between two

membrane-adhered beads of diﬀerent sizes with a signal-to-

noise ratio (SNR), such that

SNR =(𝜇𝐴−𝜇𝐵)2

𝜎2

𝐴+𝜎2

𝐵

,(10)

where 𝜇𝐴and 𝜇𝐵are the average membrane curvature or lipid

density deviation at steady-state for two beads 𝐴and 𝐵, and

𝜎2

𝐴and 𝜎2

𝐵are the corresponding variances of the membrane

curvature or density deviation sensed by the protein. The mo-

tivation for this deﬁnition is to measure the distance between

the means of two histograms in Fig. 4in terms of their vari-

ance. If we deﬁne a variable 𝑋which is the diﬀerence between

the measured variable on bead 𝐴and the measured variable

on bead 𝐵, the SNR is h𝑋i2/h𝑋2i−h𝑋i2—which gives Eq.

(10) because the variance of two independent variables adds.

The greater this SNR value is, the better a protein can dis-

tinguish between the two beads 𝐴and 𝐵in the presence of

thermal ﬂuctuations of the membrane. We will show in Sec-

tion III E that—with some additional assumptions—this SNR

controls the largest possible ratio of association rates of pro-

teins to beads of radius 𝑅𝐴and 𝑅𝐵, and use this to estimate

experimental SNR.

We use our FSBD simulation to compute the SNR for pairs

of beads in Fig. 5, keeping the diﬀerence 𝑅𝐴−𝑅𝐵to be

200 nm. The smaller the beads, the better a protein can dis-

tinguish between two similarly sized beads, as indicated by

the magnitude of the SNR. For beads on the micron-scale,

the SNR is much smaller than 1when the beads are similarly

sized (such as 1.2 𝜇m and 1.4 𝜇m diameters). This is true

for both the SNR of curvature sensing, SNR𝐶, and the SNR

of density sensing, SNR𝜌. The decrease in SNR is largely

driven by the decreasing signal: large beads have curvatures

1/𝑅that are both increasingly close to zero curvature, so the

term (𝜇𝐴−𝜇𝐵)2will shrink. As we will see later (Fig. 8), for

beads where the radii diﬀer signiﬁcantly, e.g. 1micron vs. 3

micron diameters, the SNR can be appreciable. We also see

that changing the membrane-substrate adhesion energy from

a weakly-adherent membrane value (𝛾=1011 J/m4) to one

appropriate to a SLB (𝛾=1013 J/m4) increases the SNR.

To gain an understanding of how the SNR depends on the

protein size, the mechanical properties of the membrane, the

geometry of the bead, and the membrane-bead adhesion, we

develop a theoretical model for the SNR in the next section,

Sec. III C. We plot this theoretical result against the sim-

ulation result and see good agreement, especially at strong

membrane-substrate adhesion and large bead sizes (Fig. 5).

This deviation at small bead sizes (∼200 −300 nm radii) and

weak adhesion (𝛾∼1011 J/m4) is expected. As we discuss in

the next section, our theoretical model assumes that on aver-

age, the ﬂuctuating membrane perfectly follows the shape of

a spherical bead of any radius 𝑅. However, in the simulated

membrane-bead systems using hemispherical beads with small

𝑅, weak membrane adhesion cannot overcome the substantial

bending forces required at the sharp intersection between the

periphery of a small hemisphere and the horizontal plane from

which it is projected [49]. A stronger adhesion strength such

as 𝛾∼1013 J/m4rectiﬁes this, and allows the membrane to

follow the shape of the bead nearly perfectly. In Appendix

E, we show a comparison of the cross-sectional proﬁles of a

simulated membrane adhered to a small bead for both weak

and strong adhesion.

C. Analytical calculation of the SNR

To ﬁnd an analytical form for the SNR written in Eq. (10),

we need the average values of curvature and density deviation

on a bead as well as their standard deviations. The average

mean curvature and density deviation are reﬂected by the av-

erage shape of the membrane wrapping the bead, and we can

ﬁnd analytical values for them.

If the membrane is strongly adherent to the bead, on average

its shape will just be the bead’s shape, hℎ(r)i =ℎbead (r). The

averaged mean curvature for a membrane adhered to a bead

is then −1

2∇2ℎbead (r). At the top of the bead (r=rprot), the

curvature is then

h𝐶𝑎i ≈ 1/𝑅, (11)

where 𝑅is the bead’s radius. Our assumption that the mem-

brane follows the shape of the bead can be checked with sim-

ulation: we see that it is reasonable at suﬃciently large bead

sizes and strong membrane-substrate adhesion (refer to Fig.

13 in Appendix).

Given that the membrane is deformed to follow the bead, we

can ﬁnd the value of 𝜌(r)that would minimize the energy of

the membrane, solving for 𝜌qsuch that 𝜕𝐸/𝜕 𝜌∗

q=0(using Eq.

(4)). This would be the steady-state 𝜌qholding the membrane

shape ﬁxed. We ﬁnd that this value is

𝜌ss

q=𝑑𝑞2ℎq.(12)

Inverting the Fourier transform, we see that the density at the

protein’s location is

𝜌ss (rprot)=−𝑑∇2ℎ(rprot)=2𝑑

𝑅.(13)

To approximate the standard deviation of observed curvature

and density histograms, we start by noting that in Fig. 4,

the width of the histograms is broadly consistent across many

diﬀerent bead diameters. This suggests that 𝜎𝐴and 𝜎𝐵do not

strongly depend on bead size. In fact, for a large enough bead,

the variances of the observed curvature are essentially those

for a membrane adherent on a ﬂat substrate with the same

adhesion strength. This approximation is warranted for the

same reason that sensing micron-scale curvature is diﬃcult:

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PhysicallimitstomembranecurvaturesensingbyasingleproteinIndrajitBadvaram1andBrianA.Camley2,11DepartmentofBiophysics,JohnsHopkinsUniversity,Baltimore,MD2WilliamH.MillerIIIDepartmentofPhysics&Astronomy,JohnsHopkinsUniversity,Baltimore,MDMembranecurvaturesensingisessentialforadiverserangeofbiologicalpr...

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Physical limits to membrane curvature sensing by a single protein Indrajit Badvaram1and Brian A. Camley21 1Department of Biophysics Johns Hopkins University Baltimore MD.pdf

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Physical limits to membrane curvature sensing by a single protein Indrajit Badvaram1and Brian A. Camley21 1Department of Biophysics Johns Hopkins University Baltimore MD

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