# Excess area dependent scaling behavior of nano-sized membrane tethers

N. Ramakrishnan  Department of Bioengineering, University of Pennsylvania,
Philadelphia, PA, 19104, USA,  Arpita Roychoudhury  Department of Physics,
Indian Institute of Science Education and Research, Pune, 411008, India,
David M. Eckmann  Department of Bioengineering, University of Pennsylvania,
Philadelphia, PA, 19104, USA,  Department of Anesthesiology and Critical Care,
University of Pennsylvania, Philadelphia, PA, 19104, USA,  Portnovo S.
Ayyaswamy  Department of Mechanical engineering and Applied Mechanics,
University of Pennsylvania, Philadelphia, PA, 19104, USA,  Tobias Baumgart
Department of Chemistry, University of Pennsylvania, Philadelphia, PA, 19104,
USA,  Thomas Pucadyil  Department of Biology, Indian Institute of Science
Education and Research, Pune, 411008, India,  Shivprasad Patil  Department of
Physics, Indian Institute of Science Education and Research, Pune, 411008,
India,  Valerie M. Weaver  Department of Surgery and Anatomy, University of
California San Francisco, San Francisco, CA, 94143, USA,  Ravi Radhakrishnan
Department of Chemical and Biomolecular engineering, University of
Pennsylvania, Philadelphia, PA, 19104, USA,  Department of Biochemistry and
Biophysics, University of Pennsylvania, Philadelphia, PA, 19104, USA
<EMAIL_ADDRESS>
###### Abstract

Thermal fluctuations in cell membranes manifest as an excess area (${\cal
A}_{\rm ex}$) which governs a multitude of physical process at the sub-micron
scale. We present a theoretical framework, based on an in silico tether
pulling method, which may be used to reliably estimate ${\cal A}_{\rm ex}$ in
live cells. The tether forces estimated from our simulations compare well with
our experimental measurements for tethers extracted from ruptured GUVs and
HeLa cells. We demonstrate the significance and validity of our method by
showing that all our calculations along with experiments of tether extraction
in 15 different cell types collapse onto two unified scaling relationships
mapping tether force, tether radius, bending stiffness $\kappa$, and membrane
tension $\sigma$. We show that ${\cal R}_{\rm bead}$, the size of the wetting
region, is an important determinant of the radius of the extracted tether,
which is equal to $\xi=\sqrt{\kappa/2\sigma}$ (a characteristic length scale
of the membrane) for ${\cal R}_{\rm bead}{}<\xi$, and is equal to ${\cal
R}_{\rm bead}$ for ${\cal R}_{\rm bead}>\xi$. We also find that the estimated
excess area follows a linear scaling behavior that only depends on the true
value of ${\cal A}_{\rm ex}$ for the membrane, based on which we propose a
self-consistent technique to estimate the range of excess membrane areas in a
cell.

Keywords : _mechanotype, excess area, membrane tether, tether pulling,
umbrella sampling, dynamically triangulated Monte Carlo_

The mechanical properties of a cell can be used as a surrogate marker to
identify cellular phenotypes. Mechanical characterization (or mechanotyping)
has been particularly useful in identifying a number of pathophysiologies —
well known examples include the stiffening of malaria infected erythrocytes
and hepatocytes, the softening of metastatic cancer cells, and the sickle
shape of an erythrocyte laden with hemoglobin S [1, 2, 3]. Several works in
biomechanics have aimed to characterize cells based on mechanical measurements
using a wide range of techniques such as flow and optical cytometry,
manipulation using micropipette aspiration, optical tweezers and laser traps,
and microfluidic devices (see [1, 4, 5] for comprehensive reviews). These
studies have focused on whole cell measurements and hence have investigated
the relationship between the mechanotype and pathophysiology at the cellular
and tissue scales. In many cases, the changes in mechanical properties are
primarily caused by variations in the structure and organization of the
cellular cytoskeleton [6] and the extracellular matrix [7]. Such subcellular
scale rearrangements can significantly impact the mechanical properties of the
cell membrane at length-scales smaller than cellular dimensions (i.e., tens of
nanometers to less than one micron), a range which also corresponds to the
scale at which the cell membrane is effective as an organizer and a host of
functional signaling complexes.

The sub-cellular scale relevant to the above discussion corresponds to the
dimensions primarily set by the cortical cytoskeletal mesh, which has been
estimated to be between $l_{c}=150-500$ nm [8, 9]. The mechanical properties
of a patch of the cell membrane that spans the region between multiple
cytoskeletal pinning points, with typical dimensions $l_{c}$, can differ from
the bulk because the nature of the thermal undulations (and the associated
conformational entropy of the membrane) depends directly on $l_{c}$, and in
turn influences the system’s free energy. The total area of the membrane
(denoted by ${\cal A}$) is in general larger than the projected area of the
cytoskeletal mesh (denoted by ${\cal A}_{\rm patch}{}$). The characteristics
of the membrane deformations and undulations can be described by a
dimensionless scalar quantity called the membrane excess area given as ${\cal
A}_{\rm ex}=100*({\cal A}-{\cal A}_{\rm patch}{})/{\cal A}_{\rm patch}{}$ and
the membrane is taken to be flat when ${\cal A}_{\rm ex}$=0 and curved/ruffled
if ${\cal A}_{\rm ex}{}>0$. The presence of excess area (and curvature
gradients) can alter the local signaling microenvironment for a number of
biophysical processes whose downstream components include curvature sensing
proteins like BAR, Exo70, and ENTH domains [10, 11, 12]. Notable processes
where modulations in the membrane excess area at the sub-cellular scale can
significantly impact common cellular functions including intracellular
transport of cargo or viral/bacterial internalization through
exo-/endo-/phago-cytosis [13, 14], cell polarization [15, 16], and cell
motility [17]. Hence it is logical to posit that the primary mechanisms
linking the cell-microenvironment to cell fate can revolve around the physical
factors impacting the membrane at length-scales below $l_{c}$ [6, 18, 19, 20,
21].

We note that a number of experimental studies have focused on how membranous
reservoirs respond to perturbations in the physical environment of the cell.
The estimates for excess membrane area determined using conventional
morphometric measurements, involving osmotic shock assays and cryo-EM [22] do
not delineate thermally undulating excess areas, which causes a mis-estimation
of the area. Moreover, such methods, by averaging over the entire cell (or
even 100s of cells), ignore the heterogeneity on the scale of $l_{c}$ at a
single cell level or the asymmetry in membrane response that could exist in a
polarized cell (where the basal and apical surfaces may sustain very different
membrane properties). In this article, we propose a theoretical
framework/computational model applicable to tether pulling assays (reviewed in
[18]) to obtain reliable estimates for the membrane excess area. Unique to our
modelling approach is a new methodology that allows incorporation of large
deformations as well as thermal membrane undulations in the estimate.

##  1 Computational model

We consider a square frame with a lateral size ${\cal L}_{\rm patch}{}=510$
nm, which houses the membrane surface. As noted in the introduction ${\cal
A}$, ${\cal A}_{\rm patch}$, and ${\cal A}_{\rm ex}$ are respectively the
curvilinear, projected, and excess areas of the membrane. We discretize the
membrane surface into a triangulated surface that contains $M$ triangles
intersecting at $N$ vertices and forming $L$ links [23, 24] and the
statistical weights of the membrane conformations are governed by the discrete
form of the Canham-Helfrich Hamiltonian [25, 26]:

${\cal
H}=\sum\limits_{i=1}^{N}\left\\{\frac{\kappa}{2}\left(c_{1,i}+c_{2,i}\right)^{2}+\sigma{}\right\\}{\cal
A}_{v}.$ (1)

$\kappa$ and $\sigma$ are respectively the bending rigidity and the bare
surface tension of the membrane and ${\cal A}_{v}$ is the curvilinear area per
vertex on the surface. $c_{1,i}$ and $c_{2,i}$ are the principal curvatures at
a given vertex $i$ computed as in our earlier work [27]. In our studies we
hold ${\cal A}_{\rm patch}$ to be a constant and take $\sigma=0$. However when
thermal undulations are taken into account, the effective surface tension in
the membrane will be non-zero due to renormalization effects and a mapping
between the renormalized tension and excess area has been quantified in our
earlier work [28]. All our simulations have been performed in a constant
$N$-${\cal A}_{\rm patch}$-$T$ ensemble, where $T$ is the absolute
temperature.

The conformational states of the triangulated surface are evolved using the
dynamically triangulated Monte Carlo (MC) technique which consists of two
independent MC moves: (i) a vertex move that simulates thermal fluctuations
and (ii) a link flip that captures the fluid nature of biological membranes
(see supplementary information Sec. S1 for details). A MC step consists of $N$
vertex moves and $L$ link flips that are performed at random and all the moves
are accepted using the Metropolis scheme [29]. All the simulations reported
here have been performed using a membrane patch with $N=2601$ vertices and the
statistics are collected over 1.5 million MC steps.

###  1.1 Analytical model for the membrane excess area

The excess area of a planar membrane in the small deformation limit ($|\nabla
h|\ll 1$) can be analytically estimated to be [30, 31];

${\cal G}=\dfrac{100}{2{\cal L}_{\rm patch}^{2}}\sum\limits_{q=q_{\rm
min}}^{q=q_{\rm max}}\dfrac{{k}_{\rm B}T{}}{\kappa q^{2}+\sigma},$ (2)

where $q$ denotes the wavenumber of all possible undulation modes in the
membrane and $k_{\rm B}$ the Boltzmann constant. The maximum value of the
wavenumber $q_{\rm max}=2\pi a_{0}^{-1}$ is set by the size of the
triangulated vertices $a_{0}$ and its minimum value $q_{\rm min}=2\pi
l_{p}^{-1}$ is set by the length scale $l_{p}$ such that $l_{p}\gg a_{0}$ and
$l_{p}\leq{\cal L}_{\rm patch}{}$. We have performed all our analysis using
three values of $l_{p}=150$, $250$, and $510$ nm that represent the variations
in the cytoskeletal length-scales. We note that this model only has
applicability in the regime of small ${\cal A}_{\rm ex}$ when $|\nabla h|\ll
1$ is satisfied and is expected to fail in regimes where the ${\cal A}_{\rm
ex}$ of the cell is not small (see supplementary information Sec. S3) .

###  1.2 In silico tether pulling assay

If ${\cal F}_{\rm t}$ be the force required to extract a tether of radius
${\cal R}_{\rm t}$ and length ${l}_{\rm t}$ from the membrane patch, as
illustrated in Fig. 1, the total energy ${\cal H}_{\rm tot}$, which has a
contribution due to membrane deformations (eqn. (1)) and an additional part
from the work done to extract the tether (assuming that the tether is a
perfect cylinder and ignoring thermal undulations), is given by [32]:

${\cal H}_{\rm tot}=\dfrac{\kappa\pi{l}_{\rm t}{}}{{\cal R}_{\rm
t}{}}+2\pi\sigma{}{l}_{\rm t}{}{\cal R}_{\rm t}{}-{\cal F}_{\rm t}{}{l}_{\rm
t}{}.$ (3)

Minimization of the total energy with respect to ${l}_{\rm t}$ and ${\cal
R}_{\rm t}$ yields: (i) $\kappa={\cal F}_{\rm t}{}{\cal R}_{\rm t}{}/(2\pi)$
and (ii) $\sigma={\cal F}_{\rm t}{}/(4\pi{\cal R}_{\rm t}{})$. These
relationships allow one to determine the elastic properties of the cell
membrane through tether pulling experiments; however, the non-trivial geometry
of a tether (which in general is not a perfect cylinder) and the underlying
membrane patch (which is not a perfect planar entity but rather a ruffled
surface subject to undulations, especially under high ${\cal A}_{\rm ex}$)
limits the applicability of eqn. 3. To overcome these limitations, we have
extended the umbrella sampling technique [33] to extract tethers of a
specified length ${\cal L}_{\rm t}$ from a membrane in the $N$-${\cal A}_{\rm
patch}$-$T$ ensemble. This is analogous to tether extraction in experiments
where a constant outward force is applied on a selected region of the cell
membrane through an AFM or an optical tweezer. In our model, we use an
additional harmonic biasing potential of the form ${\cal H}_{\rm bias}=k_{\rm
bias}({l}_{\rm t}-{\cal L}_{\rm t})^{2}/2$ in place of the force employed in
experiments. Here $k_{\rm bias}$ is the spring constant of the biasing
potential and ${\cal L}_{\rm t}$ is a reaction coordinate that denotes the
prescribed length of the extruded tether. In our calculations we take $k_{\rm
bias}=0.5\,{k}_{\rm B}T{}/{\rm nm}^{2}$ and this value is chosen such that the
undulation modes of the membrane remains unaltered. It should be noted that
the addition of the biasing potential does not alters the equilibrium
characteristics of the membrane since its contribution will be removed in the
WHAM analysis.

Figure 1:  (a) Representative equilibrium conformation of a membrane with
$\kappa=20\,k_{\rm B}T$ and ${\cal A}_{\rm ex}\sim 40\%$. The set of biased
vertices at the tip ($\\{{\bf X}_{T}\\}$) and at the base ($\\{{\bf
X}_{B}\\}$) along with the position of their respective centers of mass ${\bf
R}_{T}$ and ${\bf R}_{B}$ (shown as crosses) are also shown. $\\{{\bf
X}_{T}\\}$ is the set of all vertices within a region of size ${\cal R}_{\rm
bead}$. (b) Conformation of the membrane in panel (a) with a fully developed
tether, obtained for ${\cal L}_{\rm t}{}=600$ nm. The tether force and radius,
${l}_{\rm t}$ and ${\cal R}_{\rm t}$ and the membrane dimension ${\cal L}_{\rm
patch}$ are also marked.

The length of the tether ${l}_{\rm t}$ is defined using a macroscopic order
parameter, determined from two different sets of vertices $\\{{\bf X}_{T}\\}$
and $\\{{\bf X}_{B}\\}$, that are shown in Fig. 1(a). ${\bf R}_{T}$ and ${\bf
R}_{B}$, which are also shown in Fig. 1(a), represent the centers of mass of
the chosen vertices that define the two macroscopic variables from which the
instantaneous tether length is calculated as $l_{t}=|{\bf R}_{T}-{\bf
R}_{B}|$. While $\\{{\bf X}_{T}\\}$ is predetermined at the start of the
simulation, $\\{{\bf X}_{B}\\}$ is computed at runtime and taken to be the set
of all vertices at the boundary of the membrane patch (also see supplementary
information Movie M1).

In a typical tether pulling assay, the bead used to extract the tether is only
partially wetted by the membrane surface and in general the wetting area is
unknown. Also, due to the non-specific nature of these adhesions the wetting
area may vary in different experiments, even for the same cell. In order to
investigate the role of the wetting area on the properties of the extracted
tether, we choose the biased vertices in the tip to be a circular region of
radius ${\cal R}_{\rm bead}$. This is illustrated in the lower panel of Fig.
1(a).

###  1.3 Potential of mean force

For a given membrane patch, independent simulations are performed to extract
tethers within a given umbrella sampling window. For all simulations reported
in this article, we use at least $64$ windows each of width $5$ nm — the
number of windows required to extract fully developed tethers increases with
increasing ${\cal A}_{\rm ex}$. Histograms of the instantaneous tether length
in each of the windows are recorded for $1.5$ million Monte Carlo steps and
these statistics are converted to a potential of mean force (PMF) using the
Weighted Histogram Analysis method [34]. The typical runtime for an umbrella-
sampling window to sample $1.5$ million MCS is around $36$ hours on a $2.6$
GHz processor.

###  1.4 Computing the radius and length of membrane tethers

The radius and length of the membrane tether ${\cal R}_{\rm t}$ and ${l}_{\rm
t}$, respectively, can be determined exactly in the simulations, as shown in
Fig. 1(b). Let ${\bf[r]}$ be the set of all $N_{c}$ vertices on the tubular
region and ${\bf r}_{CM}=(N_{c})^{-1}\sum_{i}{\bf r}_{i}$ their center of
mass: here ${\bf r}_{i}$ is the three-dimensional position vector of vertex
$i$ in the Cartesian coordinates. The center of mass can be used to construct
the gyration tensor as, ${\bf G}=(N_{c})^{-1}\sum_{i=1}^{N_{c}}({\bf
r}_{i}-{\bf r}_{CM})\otimes({\bf r}_{i}-{\bf r}_{CM})$ whose eigenvalues are
$\lambda_{1}$, ${\lambda_{2}}$, and ${\lambda_{3}}$. Since the tethers formed
are axi-symmetric we identify $\lambda_{2}$ and $\lambda_{3}$ using the
relation $\lambda_{2}\approx\lambda_{3}$. Of the three eigenvalues,
$\lambda_{1}$ represents the length of the tether, with ${l}_{\rm t}{}\approx
2\sqrt{\lambda_{1}}$, and $\sqrt{\lambda_{2}}$ and $\sqrt{\lambda_{3}}$
represent its two principal radii. We estimate the average tether radius as
${\cal R}_{\rm t}{}=(\sqrt{\lambda_{2}}+\sqrt{\lambda_{3}})/2$.

##  2 Experimental Methods

###  2.1 Cell culture

HeLa cells were placed in $35$ mm petridishes at $37$° C in $5$% CO2 in DMEM
(Dulbecco’s Modified Eagle’s medium, Lonza) containing $10$% FBS (Fetal Bovine
Serum, Gibco) and $0.02$% Penicillin/Streptomycin for $48$ hours before
commencing the experiment. A confluent culture of HeLa cells was treated with
$0.25$% Trypsin-EDTA (Gibco), detrypsinised in DMEM containing $10$% FBS and
seeded at a density of $80,000$ cells/coverslip (Ted Pella Inc., Redding), so
that a single monolayer of cells are obtained on the coverslip.

###  2.2 Giant Unilamellar Vesicles (GUVs)

For the preparation of vesicles, $1,2$-dioleolyl-sn-glycero-$3$-phosphocholine
(DOPC), $1,2$-dioleolyl-sn-glycero-$3$-phospho-L-serine (DOPS) (Avanti Polar,
Alabaster, AL) and $1,2$-dioleolyl-sn-
glycero-$3$-phosphoethanolamine-N-(lissamine rhodamine B sulfonyl)(RhPE)
(Invitrogen) stock solutions in chloroform, at room temperature were used. The
lipid mix was aliquoted in a glass vial to a total lipid concentration of 1 mM
at a ratio of DOPC:DOPS:RhPE ($84$:$15$:$1$ mol%).

Gel-assisted formation of GUVs were carried out using polyvinyl alcohol (PVA)
as described earlier [35], with a few modifications as per the requirements of
the experiments. In this method of GUV formation, a drop of $5$% w/v degassed
PVA (MW $145,000$, Sigma) in deionized water is added to a clean glass
coverslip placed on a hot plate set at $75$° C. The water gets evaporated in
about $10$ minutes leaving a dry thin film of PVA on the coverslip. To this,
around $3$ $\mu$L of the $1$ mM lipid stock solution in chloroform was added
to dry PVA while on the hot plate to let the chloroform evaporate. The thin
film was peeled off and immersed in eppendorfs containing $20$ mM HEPES, $150$
mM NaCl, pH $7.4$ with $100$ mM sucrose. This immersed film was left
undisturbed for around one hour followed by gentle tapping to release the GUVs
from the PVA film to the buffer solution. The buffer containing large free
floating GUVs ($10$-$15$ $\mu{\rm m}$) was pipetted out and used for tether
pulling experiments.

###  2.3 AFM Experiments

AFM-based force spectroscopic experiments were performed using Nanowizard II
atomic force microscope (JPK Instruments). The AFM liquid cell was assembled
with freshly cleaved mica discs prior to adding the GUV solution. The liquid
cell was then mounted on the AFM stage and left undisturbed for $20$ minutes
to allow the vesicles to settle on the mica surface. Using a fluorescence
microscope attached with the AFM set up, we could confirm that the GUVs
settled on the surface and the floating ones were washed away by exchanging
buffer solution with HBS. Subsequently, the GUVs got ruptured on the mica
surface and they were imaged using AFM. The images obtained using AFM revealed
the location and height of the ruptured GUV patches which matched with that of
the height of a single bilayer membrane ($5$-$6$ nm). Force spectroscopy was
then performed on these particular patches to pull membrane tethers. Silicon
nitride cantilevers (MikroMasch CSC$38$/AlBS) were used for pulling the
tethers. Cantilevers were calibrated before each experiment and its spring
constant was determined using equipartition theorem [36]. The measured spring
constant of the cantilevers used for most experiments was found to be range of
$20$-$80$ mN/m. Constant speed mode was used for approaching the tip to the
sample surface followed by retraction at the same speed. The approach-retract
cycle was repeated at various points on the membrane patch using force mapping
tool built in Nanowizard II software and force-displacement curves were
recorded. Force curves showing step profiles were selected and analyzed using
JPK data processing software by fitting the curves with the in-built functions
to measure the force minimum corresponding to the tether force and step
heights in retraction force curves.

##  3 Results

###  3.1 Extraction of membrane tether proceeds through three distinct regimes

We first demonstrate the characteristics of a tether extracted from a model
membrane with $\kappa=20$ ${k}_{\rm B}T$ and ${\cal A}_{\rm ex}\sim 40\%$,
using a bead size of ${\cal R}_{\rm bead}=50$ nm in the $N$-${\cal A}_{\rm
patch}$-$T$ ensemble. The tether is extracted using the umbrella sampling
technique described in the methods section, for reaction coordinate (imposed
tether length) values in the range $0<{\cal L}_{\rm t}<500$ nm, with a window
size of $5$ nm. The top panel in Fig. 2 shows representative snapshots of the
membrane stabilized at four different values of ${\cal L}_{\rm t}$ = $0$,
$200$, $300$, and $450$ nm. At small values of ${\cal L}_{\rm t}$, the
membrane conformations show large undulations whose magnitudes are set by the
value of ${\cal A}_{\rm ex}$. However, at large values of ${\cal L}_{\rm t}$,
the membrane undulations are absorbed into the large out of plane protrusions
that resemble a tether extracted from a planar membrane. It is noted that the
shape of a fully developed tether (i.e., when the undulations in the planar
region becomes very small) is consistent with that predicted for nearly planar
membranes, using analytical methods [37].

Figure 2:  (a) Representative conformations of a membrane with
$\kappa=20\,k_{\rm B}T$ and ${\cal A}_{\rm ex}\sim 40\%$ as a function of
${\cal L}_{\rm t}$. Panels (b) and (c) show the computed values of the tether
length ${l}_{\rm t}$, and radius ${\cal R}_{\rm t}$, respectively, as a
function of ${\cal L}_{\rm t}$. These quantities are computed as described in
Sec. 1.4. The shaded regions mark the three regimes for tether extraction
namely, regime 1: suppression of undulations, regime 2: formation of tethers,
and regime 3: extrusion of tethers at a constant radius. The boxed numbers in
the top panel denote the regimes to which the configurations correspond to.

The instantaneous length and radius of the tether region, denoted by ${l}_{\rm
t}$ and ${\cal R}_{\rm t}$, as a function of the reaction coordinate ${\cal
L}_{\rm t}$, are shown in the middle and lower panels of Fig. 2, respectively.
Both ${l}_{\rm t}$ and ${\cal R}_{\rm t}$ show non-monotonic behaviors with
respect to ${\cal L}_{\rm t}$, which are solely attributable to the non-zero
excess area of the membrane. For membrane with thermal undulations, and hence
non-zero excess areas, we identify three characteristic regimes for tether
growth which are marked as shaded regions in the figure. These regions are
characterized as follows:

  * •

Regime 1 (${l}_{\rm t}{}\approx{\cal R}_{\rm t}{}$): for ${\cal L}_{\rm
t}$$<75$ nm, where the tether radius and length are similar, the applied
biasing potential only serves to suppress the short wavelength undulations in
the membrane. This is reflected in the fact that the membrane conformations in
this regime are not distinguishable from their equilibrium counterparts.

  * •

Regime 2 (${l}_{\rm t}{}\approx$ constant and ${\cal R}_{\rm t}{}\propto{\cal
L}_{\rm t}{}^{-1}$): for $75<{\cal L}_{\rm t}<300$ nm a pronounced protrusion
is seen in the vicinity of the region where the biasing potential is applied.
The radius of this protrusion decreases with increasing ${\cal L}_{\rm t}$,
while its length remains unchanged.

  * •

Regime 3 (${\cal R}_{\rm t}{}\approx$ constant and ${l}_{\rm t}{}\propto{\cal
L}_{\rm t}{}$): for ${\cal L}_{\rm t}$$>300$ nm in Fig. 2, the tether radius
remains constant while its length increases linearly with ${\cal L}_{\rm t}$,
marking a region of tether growth. The linear increase in ${l}_{\rm t}$ fails
to hold when all excess area in the membrane is drawn into the tether region.

The extent of the three regimes, depend on the values of $\kappa$ and ${\cal
A}_{\rm ex}$. This is shown in the supplementary information, where we have
displayed the effects of ${\cal A}_{\rm ex}$ and $\kappa$ on the radius of the
extracted tether.

The characteristic length scale for a membrane, given by
$\xi=\sqrt{\kappa/2\sigma}$ [38, 39], sets the limit below which curvature
contributions are dominant. In our model, $\xi$ is an increasing function of
$\kappa$ and ${\cal A}_{\rm ex}$ — the latter may be deduced from the inverse
relationship between $\sigma$ and ${\cal A}_{\rm ex}$ in eqn. (2). In a tether
pulling experiment performed in the $N$-${\cal A}_{\rm patch}$-$T$ ensemble,
the radius of the extracted tether depends either on $\xi$ or on the size of
the biased region ${\cal R}_{\rm bead}$ used for tether extraction. This is
shown in Fig. 3 where we display the values of ${\cal R}_{\rm t}$ as a
function of ${\cal R}_{\rm bead}$, for $\kappa=20,\,40,$ and $160$ ${k}_{\rm
B}T$ and ${\cal A}_{\rm ex}{}=10$ and $40\%$. The conformations shown in panel
(a) for a membrane with $\kappa=20\,k_{\rm B}T$ and ${\cal A}_{\rm ex}\sim
10\%$, for ${\cal L}_{\rm t}{=300}$ nm, clearly illustrates the interplay
between the characteristic length $\xi$ and the imposed length ${\cal R}_{\rm
bead}$. While we observe fully grown and geometrically identical tethers for
${\cal R}_{\rm bead}\leq 75$ nm, we find the tether extracted with ${\cal
R}_{\rm bead}=100$ nm to be significantly different. This feature is also
quantified in Fig. 3(b) where we find the nearly constant tether radius
(${\cal R}_{\rm t}{}\sim 80$ nm) for ${\cal R}_{\rm bead}\leq 75$ nm to show a
marked increase to ${\cal R}_{\rm t}{}\sim 110$ nm when ${\cal R}_{\rm
bead}=100$ nm.

In panels (b) and (c) of Fig. 3 two key features are worth noting: (i) as
expected, the value of ${\cal R}_{\rm t}$ is an increasing function of
$\kappa$ for all values of ${\cal R}_{\rm bead}$, and (ii) the dependence of
${\cal R}_{\rm t}$ on ${\cal R}_{\rm bead}$ is minimal for large values of
$\kappa$ and also when ${\cal A}_{\rm ex}$ is large.

Figure 3:  Dependence of the tether radius on the size of the biasing region.
(a) Representative conformations of tethers extracted using beads with ${\cal
R}_{\rm bead}=25$, $50$, $75$, and $100$ nm, from a membrane with
$\kappa=20\,k_{\rm B}T$ and ${\cal A}_{\rm ex}\sim 10\%$. Panels (b) and (c)
show the computed values of ${\cal R}_{\rm t}$, as a function of ${\cal
R}_{\rm bead}$, for $\kappa=20,\,40,$ and $160$ ${k}_{\rm B}T$ for ${\cal
A}_{\rm ex}{}=10$ and $40\%$, respectively.

###  3.2 PMF and tether force

The PMF (${\cal W}_{\rm t}$) to extract a tether of length ${l}_{\rm t}$ from
a membrane patch of fixed ${\cal A}_{\rm ex}$ is computed from the umbrella
sampling data using the WHAM technique (see methods section). ${\cal W}_{\rm
t}$ for a membrane with $\kappa=20\,k_{\rm B}T$ and ${\cal A}_{\rm ex}\sim
40\%$ is shown in the top panel of Fig. 4(a). The three characteristic regimes
seen for ${\cal R}_{\rm t}$ (see Sec. 3.1) are also reflected in the form of
${\cal W}_{\rm t}$. Here, we again observe three scaling regimes : (i) an
initial linear regime given by ${\cal F}_{1}{l}_{\rm t}{}$, (ii) a second non-
linear regime, $\propto{l}_{\rm t}{}^{2}$, and (iii) a final linear regime,
$\propto{\cal F}_{2}{l}_{\rm t}{}$. Both the linear regimes are shown as solid
lines in panel (a) of Fig. 4 and the latter is attributable to tether
extrusion at a constant radius, for which the elastic energy is expected to
scale as ${\cal H}_{\rm tot}\propto{l}_{\rm t}{}$ (eqn. (3)). On the other
hand, the source of the non-linear scaling is attributed to ${\cal R}_{\rm t}$
being a decreasing function of ${l}_{\rm t}$. We note that the scaling
behavior is universal and is observed for all systems investigated.

Figure 4:  (a) The potential of mean force ${\cal W}_{\rm t}$ and the tether
force ${\cal F}_{\rm t}$, as a function of the tether length ${l}_{\rm t}$,
for a membrane with $\kappa=20\,k_{\rm B}T$ and ${\cal A}_{\rm ex}\sim 40\%$.
In the top panel, ${\cal W}_{\rm t}$ shows a linear scaling in regimes 1 and
3, which are represented by the functions ${\cal F}_{1}{l}_{\rm t}{}$ and
${\cal F}_{2}{l}_{\rm t}{}$, respectively. The lower panel compares values of
${\cal F}_{\rm t}$ estimated from direct numerical differentiation of ${\cal
W}_{\rm t}$ (symbols) to that obtained from the scaling relations (lines). (b)
Force displacement curves for experimental tether pulling assay using ruptured
GUVs (top panel) and HeLa cells (lower panel) – the inset shows a transition
between regions of constant force. The illustration in the top panel shows the
state of the membrane tether at various stages of the experiment. The vertical
deflection of the AFM tip is measure of the tether force ${\cal F}_{\rm t}$
and its separation from the sample is a measure of the tether length ${l}_{\rm
t}$.

The force required to extract the tether may be computed as ${\cal F}_{\rm
t}{}=|\nabla_{{l}_{\rm t}{}}{\cal W}_{\rm t}{}|$, where $\nabla_{{l}_{\rm
t}{}}$ denotes a gradient with respect to ${l}_{\rm t}$. ${\cal F}_{\rm t}$
can be estimated either from direct numerical differentiation of ${\cal
W}_{\rm t}$ or from the scaling relations — for the latter, ${\cal F}_{\rm
t}{}={\cal F}_{1}$ in regime 1 and ${\cal F}_{\rm t}{}={\cal F}_{2}$ in regime
3. The tether forces computed using the two methods for ${\cal W}_{\rm t}$ in
Fig. 4(a) are shown in the lower panel — symbols and lines correspond to
${\cal F}_{\rm t}$ obtained using numerical differentiation and using the
scaling relations, respectively. We find the estimates from both the methods
to be in excellent agreement. Since direct numerical differentiation is
subject to a large noise to signal ratio, we primarily rely on the scaling
relation based method to estimate ${\cal F}_{\rm t}$. As in experiments, we
report the value of the force in the second regime as the tether force, i.e.,
${\cal F}_{\rm t}{}\sim{\cal F}_{2}$.

The tether force shown in Fig. 4(a) has the same qualitative and quantitative
behavior as that normally observed in experiments. The top and bottom panels
in Fig. 4(b) show forces required to extrude a tether from ruptured GUVs on
mica and from the HeLa cells, respectively. The pulling speeds in both the
experimental assays are taken to be 1 $\mu$m/s, which satisfies the assumption
of quasi-equilibrium tether extraction employed in our simulations.
Measurements at speeds less than that reported here are not possible due to
the noise arising from cantilever thermal drift. Though there are no known
techniques to calculate the precise value of ${\cal A}_{\rm ex}$ for both
systems, it is reasonable to assume that it is finite. While the force-
displacement curves for both the systems depend on the properties of their
respective bilayer membrane, in the case of HeLa cells there may be additional
contributions due to the underlying cytoskeletal mesh. Though we would expect
ruptured GUVs on a mica surface to be free of any pinning contacts, there
could be a finite number of pinning sites due to the chemical heterogeneity on
the surface in spite of the surface being atomically smooth. The salt
concentration in the buffer may screen the interactions between the membrane
and the mica surface leading to a sparse contact between the two and the
effect of these non-specific contacts on the force-displacement curves are
minimal. The forces measured in experiments match very well with the
numerically computed values of ${\cal F}_{\rm t}$. The measured tether force
is about 20 pN for tethers pulled from both the ruptured GUVs and the HeLa
cells. For the case of ruptured GUVs, the tether length at which we observe a
transition to the tether extrusion regime is consistent with that seen in our
simulations, while that for the cells is considerably higher extending into
few microns. We attribute this deviation to the lack of a suitable reference
frame for cellular measurements.

Figure 5:  The potential of mean force ${\cal W}_{\rm t}$ as a function of the
tether length ${l}_{\rm t}$, extracted with ${\cal R}_{\rm bead}{}=50$ nm,
from membranes with ${\cal L}_{\rm patch}{}=0.51$ $\mu$m and $1.02$ $\mu$m,
and excess areas ${\cal A}_{\rm ex}=10\%$ and $40\%$. Data for
$\kappa=20\,k_{\rm B}T$ are shown in panel (a) and that for $\kappa=40\,k_{\rm
B}T$ is shown in panel (b).

As noted in the introduction, the size of the cytoskeletal mesh ($l_{c}$)
bounding the cell membrane significantly influences the characteristics of the
extracted tether. The current theoretical model only considers tethers from a
homogeneous membrane with constant $\kappa$ and ${\cal A}_{\rm ex}$. However,
to zeroth order, the role of the cytoskeleton in suppressing long wavelength
undulations beyond $l_{c}$ can be taken into account in our model by examining
the dependence on the membrane patch size ${\cal L}_{\rm patch}$. In Fig. 5,
we investigate this effect by extracting tethers from two planar patches with
${\cal L}_{\rm patch}{}=510$ nm and ${\cal L}_{\rm patch}{}=1.02$ $\mu$m,
which are representative of cell membranes scaffolded by dense and sparse
cytoskeletal meshes, respectively. Panels (a) and (b) show data for membranes
with $\kappa=20$ and $40$ ${k}_{\rm B}T$, respectively, for excess areas
${\cal A}_{\rm ex}{}=10$ and $40\%$. It is evident from these figures that the
PMF, and hence ${\cal F}_{\rm t}$ and ${\cal R}_{\rm t}$, in addition to the
elastic parameters $\kappa$ and ${\cal A}_{\rm ex}$, are also functions of
${\cal L}_{\rm patch}$. This points to the fact the cell may have a
heterogeneous mechanical microenvironment depending on the cytoskeletal mesh
size and may provide varied response to biochemical processes, such as
nanocarrier or viral binding, depending of the characteristic value of $l_{c}$
at the site of the process [40]. Hence, characterizing the mechanical
properties of the cell membrane at the scale of $l_{c}$ would be extremely
important. In the following, we will only focus on membrane patches with
${\cal L}_{\rm patch}{}=510$ nm to establish how the excess area of the
membrane can be inferred from tether pulling experiments.

###  3.3 Tether radii and forces measured in silico compare well with range of
values measured in in vivo experiments

Figure 6: (a) Six model membrane systems, denoted M1–M6, with specified values
of ${\cal A}_{\rm ex}$ and $\kappa$. For any system Mi ($i=1\cdots 6$), Mi1,
Mi2, and Mi3 correspond to tethers extracted with ${\cal R}_{\rm bead}=25$,
$50$, and $75$ nm, respectively. The values of ${\cal W}_{\rm t}$, ${\cal
F}_{\rm t}$, and ${\cal R}_{\rm t}$ for all the systems are shown in panels
(b), (c), and (d), respectively.

Pontes et. al. [41] have recently reported results for in vivo tether pulling
assays studies of 15 different cell types in the central nervous system (CNS)
— the data is also shown in the supplementary information. Based on this
study, we classify cells in the CNS into four distinct categories: (i) small
$\kappa$ ($20-60$${k}_{\rm B}T$) & small $\sigma$, (ii) small $\kappa$ & large
$\sigma$, (iii) large $\kappa$ ($\sim 160$ ${k}_{\rm B}T$) & small $\sigma$,
and (iv) large $\kappa$ & large $\sigma$. In order to establish the
quantitative accuracy of our model, we compute the values of ${\cal R}_{\rm
t}$ and ${\cal F}_{\rm t}$ for six model systems which are representative of
the cells in the CNS. They are denoted by M1 ($\kappa=20\,k_{\rm B}T$, ${\cal
A}_{\rm ex}\sim 10\%$), M2 ($\kappa=20\,k_{\rm B}T$, ${\cal A}_{\rm ex}\sim
44\%$), M3 ($\kappa=40\,k_{\rm B}T$, ${\cal A}_{\rm ex}\sim 9\%$), M4
($\kappa=40\,k_{\rm B}T$, ${\cal A}_{\rm ex}\sim 43\%$), M5
($\kappa=160\,k_{\rm B}T$, ${\cal A}_{\rm ex}\sim 13\%$), and M6
($\kappa=160\,k_{\rm B}T$, ${\cal A}_{\rm ex}\sim 38\%$). These model systems
are also depicted in Fig. 6(a).

We extract tethers from all the six model system (Mi, with $i=1\cdots 6$),
using bead sizes ${\cal R}_{\rm bead}=25,\,50$, and $75$ nm — the
corresponding data are denoted by Mij, where $j=1$, $2$, and $3$,
respectively. The PMFs for these systems are displayed in Fig. 6(b) and the
presence of the three characteristic regimes for ${\cal W}_{\rm t}$, discussed
earlier, are evident. Despite a similarity in the scaling behavior, the values
of ${\cal W}_{\rm t}$ are highly sensitive to changes in both ${\cal R}_{\rm
bead}$ and the elastic parameters $\kappa$ and ${\cal A}_{\rm ex}$,
predominantly so for the latter. The average values of ${\cal R}_{\rm t}$ and
${\cal F}_{\rm t}$ for the model systems are displayed in Figs. 6(c) and (d)
respectively. ${\cal R}_{\rm t}$ is found to be independent of ${\cal R}_{\rm
bead}$ and, as expected, we find: (i) for a given $\kappa$, ${\cal R}_{\rm t}$
is a decreasing function of ${\cal A}_{\rm ex}$ (e.g. M1$>$M2), and (ii) for a
fixed ${\cal A}_{\rm ex}$, ${\cal R}_{\rm t}$ is an increasing function of
$\kappa$ (e.g. M5$>$M3$>$M1). The tether force also shows a similar behavior,
with ${\cal F}_{\rm t}$ being larger for systems with smaller ${\cal A}_{\rm
ex}$ and larger $\kappa$. The range of values for the tether force ($10<{\cal
F}_{\rm t}{}<50$ pN) and radius ($60<{\cal R}_{\rm t}{}<110$ nm) measured in
our simulations compare very well with the experiments of Pontes et. al. [41],
where they report values in the range $15<{\cal F}_{\rm t}{}<70$ pN and
$43<{\cal R}_{\rm t}{}<158$ nm. This establishes the validity of our present
model as a tool for interpreting tether pulling assays that aim to probe
tethers in the nanoscopic scale.

Figure 7:  Validity of the scaling relations for $\kappa$ and $\sigma$ for
data from simulations (M1–M6, shown as open symbols) and experiments (C1–C15,
shown as filled symbols). Panel (a) shows the relation $\kappa/\alpha=1/2\pi$
and panel (b) shows the scaling relation $\sigma/\Gamma=1/4\pi$, and the
corresponding correlation coefficients for systems $M1-M6$ are found to be
$r^{2}=0.846$ and $r^{2}=0.952$, respectively. The dotted lines in panels (a)
and (b) correspond to $1/2\pi$ and $1/4\pi$ respectively.

Our results in Figs. 7(a) and (b), depict the adherence to the constitutive
relations derived by minimizing eqn. (3). Briefly, the effective bending
rigidity and the surface tension are expected as follow the relations
$\kappa/\alpha=(2\pi)^{-1}$ and $\sigma/\Gamma=(4\pi)^{-1}$, respectively.
Here the scaling parameters are $\alpha={\cal F}_{\rm t}{}{\cal R}_{\rm
t}{}/{k}_{\rm B}T{}$ and $\Gamma={\cal F}_{\rm t}{}/{\cal R}_{\rm t}{}$. As
can be seen from the figures, data from both our simulations (marked M1–M6 and
shown as open symbols) and from the experiments of Pontes et. al. [41] (marked
C1–C15 and shown as filled symbols) show a good collapse, with correlation
coefficients of $r^{2}=0.846$ for $\kappa$ and $r^{2}=0.952$ for $\sigma$,
which further establishes the agreement of our calculations and the referred
experiments with known scaling relationships. The dotted lines in Figs. 7(a)
and (b) correspond to $(2\pi)^{-1}$ and $(4\pi)^{-1}$, respectively.

###  3.4 Data from tether pulling experiments may be classified according to
${\cal A}_{\rm ex}$

Using a suitable choice of scaling parameters, data from various tether
pulling assays may be classified according to the excess area in the membrane.
We demonstrate this feature in Fig. 8(a) where we show a plot of $\alpha$ vs
$\Gamma$ for the six model systems we have chosen. Each system is represented
by a set of four data points which correspond to tethers extracted with ${\cal
R}_{\rm bead}=25$, $50$, $75$, and $100$ nm. The entire set of data clusters
into groups, that are primarily dependent on the value of ${\cal A}_{\rm ex}$
in the model membrane. It may be seen that systems M1, M3, and M5 (with ${\cal
A}_{\rm ex}\sim 10\%$) are clustered in the top right while M2, M4, and M6
(with ${\cal A}_{\rm ex}\sim 40\%$) are clustered in the bottom left, and
these two clusters are marked as shaded regions. Such a clustering analysis
provides a useful route to experimentally classify cells. However, it does not
yield any information about the value of ${\cal A}_{\rm ex}$.

Figure 8:  (a) A plot of $\alpha$ vs $\Gamma$ for M1–M6, for different values
of ${\cal R}_{\rm bead}$, show data clustering in an excess area dependent
fashion. (b) ${\cal G}(\alpha)$, the analytical estimates for the membrane
excess area for M1–M6, computed using eqn. (2). The dotted line denotes a
scaling of the form $G/\alpha$, with $G\sim 1107$.

Based on eqn. (2), we recognize that ${\cal G}(\alpha)$ shows a scaling of the
form $G/\alpha$ (dotted line in Fig. 8(b)). The data from our calculations are
consistent with this scaling as depicted in Fig. 8(b). Given the potential for
clustering of our data in Fig. 8(a) on the basis of ${\cal A}_{\rm ex}$, and
the scaling shown in ${\cal G}(\alpha)$ in Fig. 8(b), we define a
dimensionless variable $\eta={\cal A}_{\rm ex}{}/{\cal G}$.

A plot of $\eta$ as a function of $\alpha$ for systems M1–M6, for four
different values of ${\cal R}_{\rm bead}$, are shown in Fig. 9(a).
Intriguingly, the data collapse into a linear scaling behavior when $\eta$ is
plotted against $\alpha$ (see Fig. 8(a)) where the slope of the scaling line
depends only on ${\cal A}_{\rm ex}$. The scaling is represented as:

$\eta_{i}=m_{i}\alpha+1,$ (4)

with $i=1\cdots 6$. The intercept is taken to be $1$ since $m_{i}\rightarrow
0$ as $\eta_{i}\rightarrow 1$, i.e., when ${\cal G}\rightarrow{\cal A}_{\rm
ex}{}$. We estimate the values of $m_{i}$ for each system by fitting the
corresponding data to a linear function. The three representative dotted lines
in Fig. 8(a), corresponding to the small, intermediate, and large excess area
regimes, show the clustering of data that only depends on the value of ${\cal
A}_{\rm ex}$ in the membrane. The values of $m_{i}$ computed for each set of
data in M1–M6 (Fig. 9(a)) are shown as a function of ${\cal A}_{\rm ex}$ in
Fig. 9(b). In general, the dependence of $m_{i}$ on ${\cal A}_{\rm ex}$ may be
expressed as:

$m_{i}=f({\cal A}_{\rm ex}{}_{,i}),$ (5)

where $f$ is an unknown function. As a first approximation, we find $m_{i}$ to
be a linear function of ${\cal A}_{\rm ex}$ and hence $f({\cal A}_{\rm
ex}{}_{,i})=K{\cal A}_{\rm ex}{}_{,i}$ with $K$ being the slope of the best
fit linear function, shown as a dotted line in Fig. 9(b).

Figure 9:  (a) Scaling plot of $\eta$ vs $\alpha$ for systems M1–M6 for four
different values of ${\cal R}_{\rm bead}$. The dotted lines, show
representative scaling relations of the form $\eta_{i}=m_{i}\alpha+1$, for
small, intermediate, and large ${\cal A}_{\rm ex}$ regimes. (b) A plot of the
slope $m_{i}$ as a function of ${\cal A}_{\rm ex}$ and the dotted lines denote
the best linear fit to the data. Fitting $f({\cal A}_{\rm ex}{}_{,i})=K{\cal
A}_{\rm ex}{}_{,i}$ we find the value of $K=0.00085/({k}_{\rm B}T{})$.

The presence of an excess area dependent scaling described by the slope $m$ in
Fig. 9(b) can allow one to devise strategies to estimate the range of ${\cal
A}_{\rm ex}$ in cells directly from tether pulling experiments. One possible
approach is to use eqn. (5) in eqn. (4) and self consistently solve for ${\cal
A}_{\rm ex}$ using the relationship:

${\cal A}_{\rm ex}{}=\left(f({\cal A}_{\rm ex})\alpha+1\right){\cal G}.$ (6)

Here, the variables $\alpha={\cal F}_{\rm t}{}{\cal R}_{\rm t}{}/{k}_{\rm
B}T{}$ and ${\cal G}$ are directly computed from the tether force and radius
measured in tether pulling experiments. The form of the unknown function
$f({\cal A}_{\rm ex})$ is in turn obtained from simulations of model systems,
that correctly accounts for the size of the cytoskeletal mesh in the target
cell. The excess membrane area may then be estimated by self consistently
solving eqn. (6).

##  4 Discussion

We have presented a computational approach based on umbrella sampling and the
weighted histogram analysis technique to compute the free energy landscape and
the force-extension relationship for the pulling of membrane tethers from
membrane patches of different excess membrane areas, ${\cal A}_{\rm ex}$. The
tether forces measured in our simulations agree very well with in vitro tether
pulling experiments on ruptured GUVs on substrate and on HeLa cells. Unlike
existing models, we are able to account for both mechanical work as well as
entropic work in tether extraction by performing finite temperature
calculations, delineation of the Helmholtz free energy, and performing the
analysis in an ensemble with non-zero ${\cal A}_{\rm ex}$. Based on the
computed values of the force required for tether extraction and the tether
radius, we established scaling relationships involving the ${\cal F}_{\rm t}$,
${\cal R}_{\rm t}$, and ${\cal A}_{\rm ex}$. We demonstrated the relevance of
the calculations by showing the scaling of $\kappa$ with $\alpha$ and $\sigma$
with $\Gamma$ from the model and those obtained from 15 different cell
experiments collapse on to a single curve. These scaling curves can be used to
construct new schemes for estimating the excess membrane area, which alleviate
the limitations of previous methods by being valid for large curvatures, and
by taking into account the thermal membrane undulations in the high curvature
limit. We have shown that our results successfully recapitulate the results of
the previous model in the small-curvature limit. However, in the large-
curvature limit, when the domain of applicability of the previous model is
limited, we predict the values of the excess membrane areas that are
substantially larger than the estimates from the small-curvature model. In
light of the discussion above, there is a profound biomedical ramification of
the excess membrane area distribution as revealed by our analyses of the
tether pulling experiments using the fully non-linear model of the membrane
patch subject to finite temperature undulations.

Our model while directly relevant to tether extraction in well behaved in
vitro setups, such as GUVs or supported bilayers, does not include the full
complexity required to recapitulate the cellular experiments. The complexities
arise due to: (i) the dynamic nature of the cytoskeletal reorganization, (ii)
changes in ${\cal A}_{\rm ex}$ due to cellular trafficking mechanisms; the
latter poses an important constraint regarding the ensemble. While in in vitro
experiments or in our model, we have the ability to either select/design a
constant ${\cal A}_{\rm ex}$ or a constant $\sigma$ ensemble, it is not
obvious what the correct cellular condition would be. For example, at early
timescales (i.e. too short for changes in $l_{c}$) the cell membrane patch may
be under a state of tension but at later times both $\sigma$ and ${\cal
A}_{\rm ex}$ can change due to signaling and trafficking. Notwithstanding
these considerations, our model can still be applicable under certain cellular
conditions, namely (i) the timescale of the tether extraction is faster than
that for cytoskeletal reorganization and trafficking ($\sim 10$-$100$ s [42]);
(ii) the dimensions of the extracted tethers are smaller than $l_{c}$. When
these conditions are met, one can treat the tether extraction as a quasi-
equilibrium process where the cytoskeleton merely serves as a pinning boundary
condition for the membrane. This is further justified because the membrane
tension equilibrates at a much faster time scale of $\tau_{\rm
tension}=\eta_{s}\textfractionsolidus\sigma{}\sim 1$-$100$ $\mu{\rm s}$,
(where $\eta_{s}$ is the surface dilational viscosity of the bilayer $\approx
0.35$ Ns/m [43]). Under these assumptions, ${\cal L}_{\rm patch}$ can serve as
an approximate surrogate to include cytoskeletal pinning effects. These
considerations and caveats must be taken into consideration in developing
experimental methods for determining ${\cal A}_{\rm ex}$ in cells based on the
model we have described here.

A bi-directional coupling can be established between the cell exterior and
cell interior in a “mechano-sensitive” fashion through the control of membrane
excess area [19], because ${\cal A}_{\rm ex}$ is the conjugate variable for
membrane tension as well as membrane curvature. Several signaling mechanic
events can therefore be transduced via the regulation in ${\cal A}_{\rm ex}$ :
they include cell-ECM interactions, which can tune acto-myosin tension and
influence cell-proliferation through integrin-mediated signaling pathways [44,
45, 46]. Glycocalyx remodeling can influence membrane-curvature distribution
on the cell surface and initiate a proliferative cell-response, funneling
through integrin-mediating signals [20]. Cellular recycling pathways
responsible for cargo transport from the endosome to the plasma membrane can
also induce and nucleate cell-membrane protrusions providing dominant
mechanisms for cell migration and motility [47, 12]. These examples serve to
reiterate how membrane excess area, in response to the tuning of tension, and
by influencing the curvature distribution of the cell membrane, can transduce
signals impacting cell-fate decisions in ECM-specific, and mechano-sensitive
fashion.

Mechanotyping cells to characterize the state of the cell membrane is,
therefore, expected to be crucial in circumstances where the underlying
heterogeneity is intrinsic such as in a tumor microenvironment and influences
cell fate through outside-in mechanisms relayed via membrane
mechanotransduction to intracellular signaling. Mechanotyping will be equally
important in circumstances where the membrane plays a dominant role such as in
the viral invasion of host cells in virology, formation of the immunological
synapse in adaptive immunity, or targeted delivery of nanocarriers in
pharmacology.

## Acknowledgements

This work was supported in part by Grants NSF-CBET-1236514 (R.R),
NIH/U01EB016027 (R.R), NIH/1U54CA193417 (R.R and T.B), and NIH/R01GM097552
(T.B). T.P and S.P acknowledge support from the Wellcome Trust-DBT India
alliance. Computational resources were provided in part by the Grant MCB060006
from XSEDE and NSF/DMR-1120901.

## Author contributions statement

R.R. and N.R. designed and performed the simulations. A.R, T.P and S.P
designed and performed the experiments. All authors were involved in data
analysis and interpretation and in writing of the manuscript.

## Competing financial interests

The authors declare that they have no competing financial interests.

Supplementary Information

##  S1 Dynamical Triangulated Monte Carlo

The dynamical triangulation Monte Carlo technique consists of two independent
moves to alter the degrees of freedom that define the triangulated surface
which is taken as a model for the fluid membrane [24, 27]:

1) Vertex Move: A randomly chosen vertex is randomly displaced to a new
position within a cube of size $\epsilon$, centered around the vertex. The
move is performed by the holding the connectivity fixed as shown in Fig. S1(a)
and accepted using the Metropolis scheme [29].

2) Link Flip: A randomly chosen tether shared between two triangles on the
surface is removed and reconnected between the two previously unconnected
vertices as shown in Fig. S1(b), by holding the vertex positions fixed.

Both moves are accepted using the standard Metropolis scheme with a
probability given by the Boltzmann constant of the energy change ($\Delta{\cal
H}_{\rm tot})$ due to the move. In the case of tether pulling simulations the
total energy of the membrane is given by ${\cal H}_{\rm tot}={\cal H}+{\cal
H}_{\rm bias}$, where ${\cal H}$ denotes the elastic Hamiltonian and ${\cal
H}_{\rm bias}$ is the harmonic biasing potential as defined in the main
manuscript. Here, ${k}_{\rm B}T$ = 1 is the inverse temperature, with $k_{\rm
B}$ the Boltzmann constant and $T$ the absolute temperature.

Figure S1:  Dynamical triangulated Monte Carlo scheme to independently modify
the position (a) and the connectivity (b) of the vertices in the triangulated
surface model.

The state of the membrane can be affected by variations either in the bending
stiffness or in the self-avoidance parameter, leading to membranes with
different excess areas ${\cal A}_{\rm ex}$. Snapshots of the membrane
conformations in the parameter space of bending rigidity and excess area are
shown in Fig. S2.

##  S2 Membrane conformations in various limits

The conformations of a planar membrane, when ${\cal H}_{\rm bias}=0$, for two
different bending rigidities ($\kappa=10$ and $40$ ${k}_{\rm B}T$) for two
different values of ${\cal A}_{\rm ex}$ ($=4\%$ and $40\%$) are shown in Fig.
S2. The surface is colored with respect to the $z$ position of the vertices.

Figure S2:  Conformations of membranes with different bending stiffness and
excess area. Shown are shapes for two values of the excess area ${\cal A}_{\rm
ex}{}=4$ and $40\%$.

##  S3 Undulation spectrum for the planar membrane

In the continuum limit, a planar membrane can be parameterized based on its
height with respect to a reference plane and such a parameterization is called
the Monge gauge. If the reference plane is taken to be the plane, then the
height of the membrane at a chosen point on the plane, with coordinates $x$
and $y$, is given by $h(x,y)$. The height of the membrane can also be
expressed in terms of its Fourier modes as [39]

$h({\bf X})=\frac{1}{{\cal L}_{\rm patch}^{2}}\int d{\bf q}\,\,h_{\bf
q}\exp(-i{\bf q}\cdot{\bf X})$ (S7)

Figure S3:  Validation of the small deformation limit. The power spectrum, for
each of the Fourier modes, scales as $q^{-4}$ when the membranes have small
excess area or large bending stiffness.

Here we have used the short hand notations ${\bf X}=[x,y]$ and ${\bf
q}=[q_{x},q_{y}]$ to denote two dimensional real and Fourier spaces and the
Fourier amplitude also has two components given by $h_{\bf
q}=[h_{q_{x}},h_{q_{y}}]$. When the elastic Hamiltonian ${\cal H}$ (see eqn. 1
of the main manuscript) is expressed in terms of its Fourier modes, the power
spectrum for each of the modes can be shown to obey the relation,

${\cal A}_{\rm patch}{}\left\langle h_{q}h_{-q}\right\rangle=\dfrac{{k}_{\rm
B}T{}}{\kappa q^{4}+\sigma q^{2}}$ (S8)

This result is derived for nearly planar membranes (where $|\nabla h\ll 1|$)
and hence should be reproducible in the simulations for membranes with either
large bending stiffnesses or small excess areas or both. The power spectrum
for planar membranes with small excess area and for a range of values of is
shown in Fig. S3. The observed undulation modes scale as $q^{-4}$, which is in
good agreement with the theoretical expression given above. However, it should
be remembered that membranes with large excess area would not adhere to this
scaling behavior, since the excess area manifests as large amplitude
undulations, which takes the systems beyond the small deformation limit (as
$|\nabla h\sim 1|$).

##  S4 Properties of the tether as a function of $\kappa$ and ${\cal A}_{\rm
ex}$

In this section, we display the effect of the membrane excess area and bending
rigidity on the length and radius of a tether extracted from a cell membrane.
In Fig. S4 we show ${l}_{\rm t}$ and ${\cal R}_{\rm t}$, along with the
membrane conformations, as a function of the imposed tether length ${\cal
L}_{\rm t}$ for a membrane with $\kappa=20\,k_{\rm B}T$ and ${\cal A}_{\rm
ex}\sim 10\%$.

Figure S4:  The length and radius of the tether extracted from a membrane with
$\kappa=20\,k_{\rm B}T$ and ${\cal A}_{\rm ex}\sim 10\%$ as a function of the
imposed tether length ${\cal L}_{\rm t}$.

Similarly, in Fig. S5 we show the effect of $\kappa$ on ${l}_{\rm t}$ and
${\cal R}_{\rm t}$ for membranes with similar excess areas, chosen to be
${\cal A}_{\rm ex}\sim 10\%$. The tether pulling data is displayed for
$\kappa=20$, and $160$ ${k}_{\rm B}T$.

Figure S5:  Effect of $\kappa$ on the length and radius of the extracted
tether as a function of the imposed tether length ${\cal L}_{\rm t}$, for
membranes with similar excess areas, taken to be ${\cal A}_{\rm ex}\sim 10\%$.

As noted in the discussions on Fig.2 in the main manuscript, we find both the
systems to exhibit the three distinct scaling regimes previously identified
for the tether radius. However, for the membranes with low excess area
considered here we find the third regime to occur at a smaller value of ${\cal
L}_{\rm t}$ compared to that seen for membranes with large excess areas.
Similarly, the value of ${\cal R}_{\rm t}$ in the final regime is an
increasing function of $\kappa$, as is evident from Fig. S5.

##  S5 Tether pulling experiments

A typical tether pulling experiment proceeds through many stages as
illustrated in Fig. S6. In the first stage, the tip of an atomic force
microscope (AFM), attached to a cantilever, is indented into the cell surface
and held fixed until the tip makes a contact with the cell membrane; these
stages are illustrated in Figs. S6(a) and (b). Stage (b) in the experiments is
analogous to the initial configurations used in our simulations. After the
formation of a stable contact the AFM tip is retracted at a constant velocity
until it returns to its undeflected state, as shown in Figs. S6(c) and (d). In
the course of retraction the adherence between the tip and the membrane leads
to formation of a tether followed by its extrusion and these process are
identical to those observed in our simulations and described in Sec.4 of the
main manuscript.

Figure S6:  Various stages of a tether pulling experiment.

##  S6 Mechanical properties of the 15 different cells in the CNS

Here we show data from Pontes et. al. [41] for the mechanical properties of 15
different cells in the central nervous system (CNS). The tether force ${\cal
F}_{\rm t}$ and radius ${\cal R}_{\rm t}$ for each of these cells (marked
C1–C15) satisfies the scaling relation ${\cal F}_{\rm t}{\cal R}_{\rm
t}/(2\kappa)=\pi$ and this is shown in Fig. S7(a). The values of $\kappa$ and
$\sigma$ are shown in Fig. S7(b) and the spread of the data show three
characteristic mechanical regimes namely: (i)low $\kappa$ and low $\sigma$,
(ii)low $\kappa$ and high $\sigma$, and (iii) high $\kappa$ and high $\sigma$.

Figure S7:  (a) The scaling relation ${\cal F}_{\rm t}{\cal R}_{\rm
t}/2\kappa$ and (b) the values of $\kappa$ and $\sigma$ for 15 different cells
(marked C1–C15) in the CNS. Data from Pontes et. al. [41].

##  S7 Movie M1

The movie shows the conformations of a tether extracted from a planar membrane
as a function of the reaction coordinate ${\cal L}_{\rm t}$ – data shown for a
membrane with ${\cal L}_{\rm patch}{}=510$ nm, $\kappa=40$ ${k}_{\rm B}T$, and
${\cal A}_{\rm ex}\sim 40\%$. The histogram shown alongside corresponds to the
distribution of the mean curvature of the membrane surface

Figure S8: Movie showing the evolution of tether as a function of the reaction
coordinate ${\cal L}_{\rm t}$.

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