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Dynamical Control of the Shape and Size of Stereocilia and Microvilli

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Dynamical Control of the Shape and Size of Stereocilia and Microvilli
Jacques Prost, Camilla Barbetta, and Jean-Franc
x
ois Joanny
Physico Chimie Curie, Institut Curie, Paris, France
ABSTRACT We discuss theoretically the shape of actin-based protrusions such as stereocilia or microvilli that have importantfunctions in many biological systems. These linear protrusions are dynamical structures continuously renewed by treadmilling:actinpolymerizesatthetipoftheciliumanddepolymerizesinitsbulk.Theyalsooftenhaveawell-controlledlengthsuchasinthehairbundlesoftheinnerearcellswheretheyappearinagradedstaircasestructure.Recentexperimentalresultsbyanothergroupofresearchersshowthatthetreadmillingvelocityofthehaircellstereociliaisproportionaltotheirlength.Weusegenericargumentsto describe the physics of stereocilia taking into account the effect of many individual proteins at a coarse-grained level by a fewphenomenologicalparameters.Atthetipofthecilium,weﬁndthatactinpolymerizationinducesaneffectivepressure.Belowthetip,the shape of the cilium is determined by depolymerization: Agreement with the observed shape requires that depolymerizationoccurs at least in two steps. Under these conditions, we calculate the cilium shape and provide physical grounds for theproportionality between treadmilling velocity and cilium length. We also calculate the penetration of the stereocilium in the actincortical layer.
INTRODUCTION
Actin-based protrusions play an important role in cell bio-logy (1,2). They extend either in a planarlike fashion as inlamellipodia or linearly as in ﬁlopodia (3), microvilli (4), or stereocilia (5–7). In both cases, the actin polymerization/ depolymerization process controls the protrusion dynamics.Assembly takes place at or very near the plasma membraneand disassembly occurs deeper in the cell.Whereas the actin network in lamellipodia is signiﬁcantlybranched and can be remodeled under the action of molec-ular motors such as myosin II (8), it is composed of tightlybundled parallel ﬁlaments in the linear protrusions (3). Cili-ated structures are ubiquitous in epithelial tissues and their physiological function can vary considerably from cell typeto cell type (1,2). Examples are the brush-border ciliatedcells of the intestine or the stereocilia of inner ear cells.Stereocilia present two remarkable features (5–7): theycan be as long as 100
m
m and they are assembled in groupsof 50–100, called a hair-bundle. In a given hair-bundle, a stereocilium length can range from a few microns to 100
m
m.One can thus compare stereocilia of different lengths in thesame cell. Furthermore, the scale is large enough that accu-rate dynamical studies can be performed. It has been clearlyshown that:The polymerization process takes place at the tip of thecilia (9).In a given cell, the steady-state cilium length is propor-tional to the actin polymerization rate, i.e., to the tread-milling velocity; in other words, irrespective of their length, cilia are renewed in a given time, typically twodays (10). Note that, between different cells such ascells of the organ of Corti and cells of the vestibule,this time can be different.In a given cell, thick cilia are longer than thin ones (11,12).Biopolymer cables, called tip-links, connect adjacent cilia and deform the tip of the smallest cilia in a character-istic way: tallest cilia which are not deformed by tip-links are oblate or quasi-spherical, whereas smaller ones which are deformed are prolate or pointed, the tipbeing tilted in the direction of the cable (13,14).The base of the cilia, which is tapered, extends signif-icantly into the cell cortex (15).To avoid any confusion, in the following, we call fasciclethe parallel network of actin ﬁlaments and bundle the groupof cilia in hair cells.These experimental observations raise the followingquestions:Whyisthetreadmillingvelocityproportionaltothelength?Whatistherelationshipbetweenciliumdiameterandlength?What does determine the root and emerging part of thecilium?Can the observation of the shape of a cilium provide in-formation on the nature and intensity of the forces in-volved in the process?It would be ideal to answer those questions starting from a molecular level. This is, however, too complex a task: thenumber of proteins present in a cilium is by far too large andtheir structures by far too complex to allow for a detaileddescription.Forinstance,atthetipofaciliumanelectrondenseregion has been observed, called the tip-complex, in which allproteins have not yet been identiﬁed (10). Even the outer membrane embracing a cilium cannot be described in all of itsdetails. We thus have to adopt some level of coarse-graining.
Submitted September 28, 2006, and accepted for publication April 19, 2007.
Address reprint requests to J. Prost, Tel.: 33-40-79-45-00; E-mail: jacques.prost@curie.fr.Editor: Marileen Dogterom.
2007 by the Biophysical Society0006-3495/07/08/1124/10 $2.00 doi: 10.1529/biophysj.106.098038
1124 Biophysical Journal Volume 93 August 2007 1124–1133
We describe the membrane as a ﬂexible surface under tension, a procedure that is well accepted now (16). The actinﬁlaments are cross-linked in stereocilia by proteins such asespin (11). We consider here the actinfascicle as a solidbodymade of close-packed ﬁlaments that all move at the sametreadmillingvelocity.Theﬁlamentspolymerizeattheirbarbedend located at the cilium tip and depolymerize statistically inthe bulk of the fascicle (1,10). The depolymerization is hin-dered by binding proteins such as espin or possibly by spe-ciﬁc minus end capping proteins. On the sides of the cilium,the fascicle/membrane interaction can be characterized by a standard interaction energy per unit area.At the tip, the actin-growing front is interacting with themembrane: we write generic dynamical equations showingthat the polymerization process generates an effective pres-sure, which depends on the difference between the ‘‘would-be’’ polymerization rate in a stress-free situation and theactual polymerization rate. This formulation is model-free inthe sense that the parameters involved could be measured inindependent experiments. Alternatively the parameters couldbe extracted from molecular theories such as the polymer-ization-ratchet model (17,18), or a model describing the roleof formins (19). For stereocilia, the polymerization process isknown to depend on the presence of myosin XV (20,21), via a mechanism yet to be understood and the role of the tipcomplex is essentially unknown. For these reasons, weconcentrate in this manuscript on those results, which do not depend on molecular details. This is in particular the case for the cilia shape and the forces involved. Recently, a theo-retical description of ﬁlopodia has been proposed in whichthe interaction of the polymerizing ﬁlaments with the mem-brane is described by a ratchet model (22). This description isuseful as it stresses the necessity of a certain degree of bun-dling to obtain long enough ﬁlopodia. The obtained shapesare, however, valid only if the fascicle radius is much smaller than the natural radius of membrane tubes pulled by a point force in the absence of actin. Furthermore, neither the lateralfascicle-membrane interaction nor depolymerization is con-sidered. In a cilium, the diameter is signiﬁcantly larger thanthe natural tube radius; the fascicle-membrane interaction anddepolymerization areessential featuresof the cilium dynamics.Our article is organized as follows. In Shape of the Tip, wediscusstheshapeofthetipofaciliumbothintheabsenceandin the presence of tip links. In particular, we obtain a generalrelation between polymerization rate and cilium radius. InShape of the Stem, we describe the depolymerization in thefascicle and deduce the shape of the associated stem. Mem-brane Shape is devoted to the shape of the membrane sur-rounding the cilium. In Partition between Emerged andImmersed Parts of the Fascicle, we study the interaction of the cilium with the cortical actin layer and the partition of thecilium between an emerged and an immersed part. In theDiscussion, we extract from the experimentally observedshapes the typical forces involved in cilia and discuss the roleof speciﬁc proteins such as espin, myosin XV, or myosin VI.
SHAPE OF THE TIPOblate shape of the tip in the absence of tip-link
The polymerization process at the tip of the cilia is complex,and involves, in general, regulation by several proteins. Agood example is given by proteins of the formin family(23,24), which build complexes with the barbed ends of theactin ﬁlaments: they regulate the actin critical concentrationand are very sensitive to forces of approximately picoNew-tons; their activity requires ATP hydrolysis. In the case of stereocilia, the role of myosin XV has been evidenced (21):this motor could simply push the membrane to help providespace for the addition of new monomers (20). PicoNewtonforces are again large enough to severely modify the poly-merization rate. More generally, other proteins such as ezrinare known to regulate the polymerization process (25,26).In view of the complexity of the phenomena involved at a molecular level, it is useful to develop a model free for-mulation. We ﬁrst write generic equations valid for ‘‘weakforces,’’ and then extend our results to account for strongnonlinearities in Appendix A.The physics of the tip growth is governed by the existenceof two physically coupled surfaces: the plasma membraneand the actin polymerization front. The plasma membranerepresented in Fig. 1 is characterized by its shape and normalvelocity, which depend on its effective tension, its curvaturemodulus, and the forces that the polymerizing front exertson the membrane. The polymerizing front in turn is alsocharacterized by its shape and velocity along the actin axis
FIGURE 1 Geometry of the tip of the cilium: the actin ﬁlaments are par-allel to the vertical axis and their barbed end is in the direction of the arrow.The polymerization front is represented by a dashed line at a distance
d
fromthe membrane (
continuous line
). The inset shows the crossover to the stemregion with an edge of radius of curvature
r
e
.
Shape of Stereocilia 1125Biophysical Journal 93(4) 1124–1133
and the force that the membrane exerts on the front. Ac-cording to Newton’s third law, the force exerted by the mem-brane on the front is equal and opposite to the force exertedby the front on the membrane.We parameterize the membrane shape by a vector
r
(
x
)(where
x
is a two-dimensional vector giving the coordinatesin the plane perpendicular to the stereocilium) and the front by the local height along the
ˆz
direction parallel to the ciliumaxis,
h
(
x
) as deﬁned on Fig. 1.At linear order in the force, the coupled dynamical equa-tions for the motion of the two fronts read
v
n
¼
l
m
d
F
d
r
n
@
h
@
t
1
v
T
¼
l
a
d
F
d
h
1
v
0p
;
(1)where
v
n
is the velocity normal to the membrane,
r
n
the localcoordinate along the normal
n
to the membrane, and
@
h
=@
t
the velocity of the actin front in the
z
direction; both ve-locities are deﬁned in the cell rest frame.Theseequationsdescribetheaverageshapeofthemembrane-actin front interface. Both the polymerization front and themembrane are ﬂuctuating. Equation 1 should thus includegeneralized ﬂuctuating forces. In this work, we omit themsince we only consider the average behavior.The dynamical equations involve two positive dissipativecoefﬁcients
l
m
and
l
a
. A more general formulation wouldalso include cross-terms, but this would lead to the same re-sults with a slight difference in the deﬁnition of the pressuredeﬁned below. Both
l
m
and
l
a
can, in principle, be measuredexperimentally. Indeed,
l
m
is a mobility coefﬁcient relatingthe membrane velocity to an applied force and
l
a
describesthe force dependence of the polymerization rate due to themembrane. The dynamic equations automatically satisfy thelocal force balance.The treadmilling velocity
v
T
is the velocity of the actinmonomers counted as positive toward the base of the cilium;the polymerization velocity
v
p0
is the polymerization velocityfor a ﬂat unperturbed actin front at the equilibrium distance
d
0
from the membrane (
d
0
is deﬁned more precisely below).The total free energy of the system
F
is a functional of both the membrane position
r
(
x
) and the front height
h
(
x
).We write it as
F ¼F
m
1
F
i
:
The membrane free energy
F
m
at a distance
d
0
from the fascicle front involves a tensioncontribution, a curvature contribution, and a pressure term:
F
m
¼
Z
ds
m
s
1
12
k
ð
H
2
c
0
Þ
2
Z
Pdv
:
(2)Here,
ds
m
is the surface element parallel to the membrane.The energy per unit area
s
of the membrane in interactionwith the front at a distance
d
0
depends both on the number of phospholipids per unit area, and on the interaction energy of that membrane with the actin fascicle, in particular via link-ing proteins. As a result,
s
depends, in principle, on the angle
u
between the membrane normal and the fascicle axis. Thevalue
k
is the membrane curvature modulus,
H
the total localcurvature, and
c
0
a spontaneous curvature that exists whenthe membrane is asymmetrical.
P
is the hydrostatic pressuredifference between the inside and the outside of the cilium.Note that the only difference with the standard Helfrich freeenergy of membranes is in the
u
dependence of the tension
s
.The interaction energy between the membrane and thefascicle front
F
i
can be expanded in powers of the perpen-dicular distance
d
between the tip of the actin ﬁlaments andthe membrane. To lowest order, we write:
F
i
¼
Z
ds
m
12
k
ð
d
d
0
Þ
2
:
(3)Here,
k
is a spring constant per unit area, and
d
0
themembrane-front average distance at thermal equilibrium.Both
k
and
d
0
depend on the membrane characteristics andon the nature of the linking proteins between the membraneand the fascicle front. They provide a coarse-grained descrip-tion of the tip complex and its interaction with the mem-brane. The equilibrium distance
d
0
might, in general, dependon the angle
u
between the membrane normal
n
and thecilium axis
ˆz
; in a ﬁrst approximation, we ignore this orien-tational dependence. The membrane-front distance is
d
¼
n
ð
r
h
ˆz
Þ
:
The detailed calculation of the derivatives of theinteraction free energy is given in Appendix A and leads to
d
F
i
d
r
n
¼
k
ð
d
d
0
Þ
12
k
ð
d
d
0
Þ
2
H
d
F
i
d
h
¼
k
ð
d
d
0
Þ
:
(4)At steady state (
v
n
¼ð
@
h
=@
t
Þ¼
0), the shape equation of the membrane is then obtained from Eq. 1:
d
F
m
d
r
n
¼
v
0p
v
T
l
a
1
12
H
ð
v
T
v
0p
Þ
2
k
l
2a
:
(5)This equation is formally equivalent to the equilibriumequation of a membrane submitted to an effective hydrostaticpressure difference and effective tension:
P
eff
¼
P
1
v
0p
v
T
l
a
s
eff
¼
s
1
12
ð
v
T
v
0p
Þ
2
k
l
2a
:
(6)We show in Appendix A that this result is general and that even for the full nonlinear problem the effect of treadmillingcan be recast in the form of an effective pressure and aneffective membrane tension.Note that the shape Eq. 5 implies that, for any treadmillingvelocity
v
T
, there is, in general, a unique solution for themembrane shape (16). Equation 5 implies also a global forcebalance
P
eff
p
r
20
¼
2
p
r
0
s
eff
1
ð
k
=
2r
20
Þ
;
where
r
0
is the fas-cicle radius at the edge of the tip where
u
¼
p
/2 and theeffective tension
s
eff
is calculated for an angle
u
¼
p
/2. Theglobal force balance determines the effective pressure, i.e.,the difference between the treadmilling velocity and the bare
1126 Prost et al.Biophysical Journal 93(4) 1124–1133
polymerization velocity for a given fascicle radius
r
0
. In thelimit where there is no pressure difference
P
¼
0, we ﬁnd, ina linear approximation,
v
T
¼
v
0p
l
a
2
s
eff
r
0
1
k
r
30
:
(7)We have neglected here the treadmilling contribution tothe tension, which is of higher order. This relation imposesthat thicker cilia have a larger treadmilling velocity at con-stant polymerization velocity
v
p0
.The shape of the tip may be described by three regimes:Large tips (
r
0
ð
k
=
2
s
Þ
1
=
2
). One can distinguish tworegions as shown on Fig. 1: the central region essentiallysphericalwitharadius
R
¼
2
s
eff
(
u
¼
0)/
P
eff
andtheedgeswith two principal radii of curvature, a small radius of curvature
r
e
’ð
k
=
2
s
Þ
1
=
2
;
and a large one equal to
r
0
.Small tips (
r
0
’ð
k
=
2
s
Þ
1
=
2
). The central region disappearsand the tip is almost spherical with a radius (
k
/2
s
)
1/2
.Small fascicles (
r
0
ð
k
=
2
s
Þ
1
=
2
). This case is discussedin Atilgan et al. (22) and has been observed with mi-crotubules bundles in vesicles (27). The fascicle acts asa point force on the membrane, and the results of Dere´nyiet al. (28) hold. This regime only exists if the mem-brane does not adhere to the ﬁlaments which is pos-sible if the energy of the nonadhering membrane issmaller than that of the adhering one,
r
0
ð
s
eff
1
ð
k
=
2r
20
ÞÞ
.
ð
2
s
0
k
Þ
1
=
2
;
where
s
0
is the tension of the non-adhering membrane.
Prolate shape of the tip in the presence of tip-link
As already mentioned, in a hair cell, the hair-bundle has a graded array of stereocilia. The stereocilia in one row allhave the same size and are connected to larger stereocilia inthe next row by tip-links. It has been shown that the larger stereocilium exerts a point force on the smaller ones via thetip-link. A point force
f
in the direction of the tip must therefore be included in Eq. 2. In the presence of this forcethe shape of the stereocilium is no longer rotationally sym-metric and one must rely on numerical methods to ﬁnd it. Weused the Surface Evolver program (29) in the same way as inDere´nyi et al. (28). We give examples of the obtained shapeon Fig. 2. Fig. 2
a
corresponds to no force and no poly-merization pressure; Fig. 2
b
corresponds to no force andﬁnite polymerization pressure; and Fig. 2
c
corresponds to a ﬁnite polymerization pressure and a ﬁnite force.The set of dynamical expressions in Eq. 1 describes the tipshape due to the polymerization kinetics. It can no longer beused below the location
z
¼
0, where the membrane tangent is parallel to the cilium axis. From there on, no polymeri-zation occurs, but instead a statistical depolymerization takesplace. We describe this process, which controls the shape of what we call the stem, in the following section.
SHAPE OF THE STEM
The actin fascicle being solidly held together by bindingproteins such as ﬁlamin and espin, it can be considered asincompressible to a very good approximation. This allows usto discuss the shape of the fascicle independently of that of the membrane in this part of the cilium.For the discussion of the stem shape of a long cilium, wecan ignore the tip curvature and suppose that all ﬁlamentshave their barbed ends in the same plane. The radius of thecilium in this plane is the radius
r
0
at the base of the tip. If the surface density of actin barbed ends perpendicular to thefascicle axis is
c
s
, the number of actin ﬁlaments is
n
f
¼
c
s
p
r
02
. Actin ﬁlaments polymerize at their barbed ends witha polymerization rate
k
p
and depolymerize at their pointedend located inside the fascicle at a rate
k
d
. The treadmill-ing velocity is
v
T
¼
k
p
a
, where
a
is the size of an actinmonomer.In physiological conditions, it is likely that actin ﬁlamentspointed ends are capped by various capping proteins whenthey are formed. We consider here one of these capping pro-teins and assume that it controls the depolymerization, whichis possible only if the ﬁlament is uncapped. The uncappingrate of the pointed ends is
k
u
. For simplicity, we ignore hererecapping events and we suppose that once a ﬁlament is un-capped it depolymerizes at its pointed end. A full calculationincluding recapping events leads to similar results. Bundlingproteins can also prevent depolymerization and can be treatedin essentially the same way.We ﬁrst study the distribution of the pointed ends in thefascicle. There are capped pointed ends and uncapped pointedends. We call
p
c
(
n
) the probability to ﬁnd a capped pointedend of an actin ﬁlament containing
n
monomers, and
p
u
(
n
)the probability to ﬁnd an uncapped pointed end of a ﬁlament containing
n
monomers. The master equation for these twoprobabilities (30):
FIGURE 2 Shape of the tip of a stereocilium. (
a
) Shape with no poly-merization pressure and no force; membrane tension
s
¼
5 10
5
N/m,bendingmodulus
k
¼
10
kT
. Notethe ﬂat regionin the center,correspondingto the absence of polymerization pressure. (
b
) Shape with a polymerizationpressure
P
eff
¼
9 10
3
Pa and no force; membrane tension
s
¼
5 10
5
N/m,bending modulus
k
¼
10
kT
. Note the curvature difference between thecentral region and the edges as described in the text. (
c
) Shape with a polymerization pressure
P
eff
¼
6 10
3
Pa with an external force of
;
20 pN;
s
¼
5 10
5
N/m, bending modulus
k
¼
45
kT
. Note the prolate shape dueto the tip-link pulling force.
Shape of Stereocilia 1127Biophysical Journal 93(4) 1124–1133
@
p
c
@
t
¼
k
u
p
c
ð
n
Þ
1
k
p
p
c
ð
n
1
Þ
k
p
p
c
ð
n
Þ
@
p
u
@
t
¼
k
u
p
c
ð
n
Þ
1
k
p
p
u
ð
n
1
Þ
k
p
p
u
ð
n
Þ
1
k
d
p
u
ð
n
1
1
Þ
k
d
p
u
ð
n
Þ
:
(8)When an uncapped pointed end depolymerizes, the size of the corresponding ﬁlament decreases and the ﬁlament dis-appears when all monomers have depolymerized. Therefore
p
u
(
n
¼
0)
¼
0. In a steady state, statistically, the loss of ﬁlaments is balanced by the appearance of new ﬁlaments.The formation of new ﬁlaments occurs by a nucleationprocess, the minimal seed for growth being of the order of three or four monomers. There is a ﬁnite ﬂux of new actinﬁlaments in the fascicle and the probability
p
c
is ﬁnite closeto the polymerization plane. This probability is deﬁned for a number of monomers larger than the seed size. A steady stateof the fascicle then exists with ﬁlaments of different lengthshaving been created at different times.The steady-state solution of the master equation with theseboundary conditions is
p
c
ð
n
Þ¼
a
k
p
k
p
1
k
u
n
p
u
ð
n
Þ¼
a
k
u
1
k
p
k
d
k
u
k
p
k
p
k
p
1
k
u
n
k
p
k
d
n
:
(9)The constant
a
is ﬁxed by the normalization below. Notethat the distribution of endpoints decreases to zero when
n
islarge only if
k
d
.
k
p
, which we assume below.The distance
z
between a fascicle section and the tip of thefascicle can be labeled by the number of monomers
m
¼
z
/
a
between this section and the barbed end of the actin ﬁlament.The number of actin ﬁlaments in section
m
is related to thetotal probability of ﬁnding a pointed end in the fascicle
p
e
(
n
)
¼
p
c
(
n
)
1
p
u
(
n
) by
N
ð
m
Þ¼
+
N
n
p
e
ð
n
Þ
:
Imposing thetotal number of barbed ends at
z
¼
0, we ﬁnd
N
ð
m
Þ¼
c
s
p
r
20
k
d
k
p
k
d
k
u
k
p
k
p
k
p
1
k
u
m
k
u
k
d
k
u
k
p
k
p
k
d
m
:
(10)The incompressibility of the actin fascicle then directlygives the shape of the fascicle. The radius
r
at a distance
z
from the plane of the polymerizing barbed ends is deducedfrom the relation
N
(
m
)
¼
c
s
p
r
2
(
z
):
r
ð
z
Þ¼
r
0
k
d
k
p
k
d
k
u
k
p
k
p
k
p
1
k
u
z
=
a
k
u
k
d
k
u
k
p
k
p
k
d
z
=
a
" #
1
=
2
:
(11)We show, in Fig. 3, a typical fascicle stem shape calculatedfrom this equation.The weight average length of the actin ﬁlaments can becalculated from the monomer distribution
Æ
m
æ
¼
+
mN
ð
m
Þ
+
N
ð
m
Þ ¼
k
p
k
u
ð
k
d
k
p
Þ
3
ð
k
d
k
p
Þ
2
ð
k
p
1
k
u
Þ
1
k
u
ð
k
d
k
p
Þð
k
p
1
k
u
Þ
1
k
p
k
2u
k
d
ð
k
p
1
k
u
Þ
k
2p
:
(12)We consider, in the following, the limit where depoly-merization is faster than polymerization and where uncap-ping is the slowest event
k
p
ð
k
d
k
p
Þ
$
k
u
:
In this case, at large distances the radius of the cilium decreases exponen-tially as
r
¼
r
0
ð
k
d
k
p
=
k
d
k
p
k
u
Þ
1
=
2
exp
z
=
2
l
;
where
l
¼
Æ
m
æ
a
¼
v
T
k
u
1
. The length of the fascicle can be deﬁnedby imposing that the fascicle radius must be larger than a monomer size and is given by
L
¼
2
v
T
k
1u
log
r
0
ak
d
k
p
k
d
k
p
k
u
1
=
2
:
(13)We here recover the law found experimentally byRzadzinska al. (10), stating that the treadmilling velocity ina stereocilium is proportional to the fascicle length. The ratiobetween the length and the velocity deﬁnes a timescale gov-erning the length of the ﬁlaments. This time is the detach-ment time
k
u
1
of the capping protein from the actinﬁlaments pointed ends required for depolymerization.In the vicinity of the tip, the proﬁle is almost perfectlycylindrical and the curvature is oriented toward the ciliuminside. If
k
d
k
p
and
k
u
are not very close (as in Fig. 3), theshape is similar to the experimental shapes observed in
FIGURE 3 Two-dimensional shape and three-dimensional reconstructionof a cilium calculated from Eq. 11. The tip has been added. The parametersare
k
u
/
k
p
¼
0.05 and (
k
d
k
p
)/
k
p
¼
0.055. These values have been chosen toreproduce the experimental shapes of Fig. 6 of Rzadzinska et al. (10).
1128 Prost et al.Biophysical Journal 93(4) 1124–1133

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