PHYSICAL REVIEW E 85, 051904 (2012)
Reversibility of red blood cell deformation
Maria Zeitz* and P. Sens†
Laboratoire Gulliver (CNRS UMR 7083), ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France
(Received 22 November 2011; published 7 May 2012)
The ability of cells to undergo reversible shape changes is often crucial to their survival. For red blood cells
(RBCs), irreversible alteration of the cell shape and flexibility often causes anemia. Here we show theoretically
that RBCs may react irreversibly to mechanical perturbations because of tensile stress in their cytoskeleton. The
transient polymerization of protein fibers inside the cell seen in sickle cell anemia or a transient external force
can trigger the formation of a cytoskeleton-free membrane protrusion of µm dimensions. The complex relaxation
kinetics of the cell shape is shown to be responsible for selecting the final state once the perturbation is removed,
thereby controlling the reversibility of the deformation. In some case, tubular protrusion are expected to relax via
a peculiar “pearling instability.”
DOI: 10.1103/PhysRevE.85.051904
PACS number(s): 87.17.Rt, 87.17.Aa
I. INTRODUCTION
Red blood cells (RBCs) have been extensively studied by
physicists as a relatively simple example of biological cells.
Their composite mechanical properties reflect the structure of
the cell interface, including the (fluid) plasma membrane (PM)
and the two-dimensional cytoskeleton (CSK) network attached
to it [1,2]. RBCs exhibit a complex dynamical response to
mechanical stress, characterized by viscoelasticity, and plastic
deformation at high strain [3,4]. Some RBCs undergo major
deformation as they cycle through the cardiovascular system,
and it is important to understand the extent to which such
large deformation are reversible. In sickle cell anemia, a
mutation in the hemoglobin (Hg) gene leads to the formation
of fibers of deoxygenated Hg and results in cells presenting
extended tubular protrusions (spicules) or sickled shapes [5].
The cells loose their flexibility and may obstruct small blood
capillaries, with serious medical consequences [6]. Although
cell sickling is reversible (fibers depolymerize in the lung
because of oxygen intake, but polymerize again as oxygen
is released), the decrease of flexibility is not, possibly due
to the irreversible loss of membrane area due to the release
of spectrin-free microvesicules during repeated cycles of
cell sickling and unsickling [7,8]. These microvesicles are
thought to result from the failure of the long (CSK-free)
spicules present in deoxygenated cells to reincorporate the cell
upon reoxygenation. In vitro, (very) long membrane tethers
extracted from a RBC by an external force (e.g., applied by
optical tweezers) may also sometimes fail to retract into the
cell when the force is switched off [9].
Our goal is to determine the conditions under which a
transient mechanical perturbation can give rise to irreversible
cellular modifications. We study the model sketched in Fig. 1
of a highly deformed RBC presenting a tubular protrusion
generated by the polymerization of a long fiber [10] or
the application of a localized external force on the cell
membrane [11] and investigate the cell’s relaxation after the
force is removed. We first show that the tight mechanical
coupling between the CSK and the cell membrane can
*
†
[email protected]
[email protected]; http://www.pct.espci.fr/∼pierre
1539-3755/2012/85(5)/051904(9)
generate mechanical frustration and the appearance of several
metastable cell shapes. We then show that the kinetics of shape
relaxation strongly influences the relaxed shape. Finally, we
briefly discuss more complex relaxation routes, including the
peculiar pearling of a long and thin protrusion.
II. METASTABLE SHAPES OF A RBC
The phase diagram of equilibrium RBC shapes is very rich
[2]. An important control parameter is the area-to-volume ratio
s̄ ≡ S/[(36π )1/3 V 2/3 ] (where S and V are the total area and
volume of the cell). Details of the cell shape (biconcavity, spiculation) and transition between different equilibrium shapes
(e.g., the stomatocyte-discocyte-echinocyte sequence) result
from the interplay between CSK elasticity (both tensile and
shear stresses) and membrane elasticity (including membrane
tension and bending stresses) [2]. Here, we wish to study
the existence of alternative metastable shapes, where some
membrane separates from the cell body or fails to reincorporate
it when extracted. This could be due to the mechanical
frustration caused by CSK-PM coupling. Indeed, the CSK is
known to contract when the entire RBC membrane is removed
[12], which suggests that it is stretched by its attachment to the
cell membrane. The formation of a membrane bulge detached
from the CSK would thus reduce the CSK stretching energy, at
the expense of the membrane deformation energy [13,14]. The
simplest way to address this question at the phenomenological
level is to assume that tensile stress dominates the CSK energetics and that all membrane contributions can be lumped into
a constant effective membrane tension σ [15]. CSK elasticity
should also includes resistance to shear, but substantial shear
stress is only found in highly spiculated cell shapes [2]. As
discussed further in Appendix A, it is omitted here since
the CSK-covered cell body is assumed to always retain a
rather smooth shape. The true membrane energetics include
bending torques which define a local membrane preferred
curvature and a global preferred area difference between the
two membrane leaflets. Both effects are included in a more
complete description given in Appendix A and are shown to
produce qualitatively similar conclusions than the constant
membrane tension considered here. Within our approximation,
the energy of a cell body of area Sc connected to a protrusion
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©2012 American Physical Society
PHYSICAL REVIEW E 85, 051904 (2012)
MARIA ZEITZ AND P. SENS
shows a typical nucleation profile if the CSK is sufficiently
stiff or sufficiently prestressed in the reference state. Prestress
is characterized by #0 ≡ 1 − (S0 /S)3/2 (the excess cell volume
compared to the volume for vanishing prestress). The location
of the critical line is approximately #0 = 25/4 (σ/Kc )3/4 −
3σ/2Kc . In practice, σ/Kc % 0.1 (σ % 10−6 J/m2 and Kc %
10−5 J/m2 [2]) and extracellular membrane protrusion can be
stable for fairly weak prestress (#0 % 25%). The nucleation
barrier is typically quite large (∼ 500kB T ) so the protrusion
should not form spontaneously but can be triggered by an
external force.
osmolytes
cytoskeleton
plasma membrane
f
f →0
A
B
C
III. RELAXATION KINETICS
FIG. 1. (Color online) A strongly deformed red blood cell
(top) contains a long, cytoskeleton-free membrane tube filled with
hemoglobin, created by an internal force (the polymerization of an
hemoglobin fiber) or an external force (the action of optical tweezers).
Upon force removal (f → 0) the cell may relax to its initial shape
if the tube completely retracts (a), or to a different state where some
membrane area remains outside the cell body and forms a spherical
bulge (b). In the latter case, the tube might exhibit pearling (c) during
relaxation.
We assume that a localized force has created a long tubular
protrusion of length L and initial radius ri (with L & ri ).
The force is switched off at time t = 0, and the perturbation
relaxes. We assume for now that the protrusion shape relaxes
smoothly toward a spherical bulge (without pearling) and
can be described by two parameters (its volume V and
area S) continuously evolving from the initial values of a
thin tube (Vi ∼ ri2 L and Si ∼ ri L) to those of a spherical
protrusion (V ∼ S 3/2 ). Cytosol volume and membrane area
are transferred between the cell body and the protrusion
during relaxation, leading to energy dissipation. If dissipation
is dominated by cytosol hydrodynamics, volume exchange is
slow and the protrusion area decreases under almost constant
volume to form a small sphere. If dissipation is dominated by
membrane flow, the protrusion volume increases under almost
constant area to form a large sphere. When the protrusion
becomes a sphere, the system evolves in the energy landscape
shown Fig. 2, which may exhibit an energy barrier for a
protrusion volume v ∗ (v = V /V is the normalized protrusion
volume). The initial relaxation dynamics thus determines on
which side of the barrier the system falls, and whether the final
state is an intact cell (v < v ∗ ) or a cell with a “permanent”
spherical bleb (v > v ∗ ).
In order to study this situation more quantitatively, we write
the balance of generalized elastic and dissipative forces using
of area S reads
Eel =
Kc (Sc − S0 )2
+ σ (S + Sc ),
2
S0
(1)
where Kc is the CSK stiffness and S0 its area of reference.
We further simplify the model and consider that the
CSK-covered cell body adopts a simple spherical shape (see
Fig. 1; it is shown in Appendix A that allowing for more
complex cell shapes does not alter our conclusions). The
CSK area is then directly related to the protrusion volume
by Sc = (36π )1/3 (V − V )2/3 , where the total cell volume V is
assumed constant (relaxing this constraint does not alter our
results). The cell energy Eq. (1) is a function of two variables
only, the protrusion area S and its volume V . The case of a
spherical protrusion Esph = Eel |S=(36πV 2 )1/3 , presented Fig. 2,
0.3
1.15
or
1.10
0.2
1.05
0.1
0.0
0
(a)
1.20
Energy (unit of σS ∼ 104 kB T )
Cytoskeleton prestress 0
0.4
0.0
10
20
30
40
Relative cytoskeleton rigidity K c /σ
50
(b)
v∗
0
0.1
0.2
Protrusion
(dimensionless) volume
0.3
= 1/4
0.4
v
FIG. 2. (Color online) (a) Static phase diagram in the parameter space Kc /σ (elastic ratio) and #0 (the CSK prestress), showing the region
where a (meta)stable protrusion can exist (shaded gray). The dashed line is the approximation given in the text. (b) Energy of a RBC with a
spherical protrusion as a function of the relative protrusion volume v, for different #0 (Kc /σ = 10). An energy barrier exists at v = v ∗ beyond
a critical prestress.
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REVERSIBILITY OF RED BLOOD CELL DEFORMATION
a Lagrangian description [16,17],
∂P
∂E
+
= 0,
∂{S,V } ∂{Ṡ,V̇ }
(2)
including elastic forces derived from the system’s energy E
and dissipative forces derived from the energy dissipation (per
unit time) P. The energy includes the CSK and membrane
tension energy Eel , Eq. (1), and a contribution from the
membrane bending rigidity κ, approximated by the bending
energy of a tubular protrusion (the only limit where it is
relevant): Eκ % κ/8S 3 /V 2 . More details can be found in
Appendix B.
The energy dissipation for membrane and cytosol flows
are derived in detail in Appendix C. The former is dominated
by the membrane flow over the (immobile) cytoskeleton in
the cell body. It is characterized by a friction parameter αl ≈
107 −1011 Pa · s/m [18]. The latter is dominated by the cytosol
flow through the neck connecting the body and the protrusion,
assumed to be of small dimension of order the CSK mesh
size rn ∼ 100 nm [19]. It is characterized by a viscosity η ≈
10−2 Pa/m [20]:
η
αl 2
Ṡ .
PV =
(2V̇ − rn Ṡ)2 , PS =
(3)
3
π rn
4π
Inserting E = Eel + Eκ , Eq. (1), and P = PV + PS ,
Eq. (3), into Eq. (2) yields two coupled dynamical equations
for S and V . Using normalized variables and omitting various
numerical prefactors and less important terms for readability
(x̄ is the normalized form of x; all details and the complete
equations can be found in Appendix D), these equations read
!
" #2 $
s
1
ṡ = −
1 + κ̄
,
τs
v
!
"
#
" #3 $ (4)
1
(1 − v)2/3 − s0
s
1
K̄c
,
+ 1 + κ̄
v̇ =
τv (1 − v)1/3
s0
v
where s and v are the area and volume of the protrusion
(normalized by those of the intact cell) and s0 = (1 − #0 )2/3
is the optimal CSK area. The important elastic parameter in
Eq. (4) is the ratio of CSK to membrane elasticity K̄c = Kc /σ
(% 10). The bending rigidity κ̄ = κ/(σ S) only intervenes
at the very early stage of tubule relaxation. The important
dynamical parameters are the typical exchange times for
area (τs ∼ αl S/σ ) and volume [τv ∼ ηS 2 /(σ rn3 )]. Interpreting
Eq. (4) is rather straightforward: Once the force that caused
the protrusion disappears, area is drawn toward the cell body
(ṡ < 0) by membrane tension and to some extent by membrane
bending stress, and volume is pushed out of the cell (v̇ > 0)
by the stretched CSK (if v < #0 ), by the membrane tension,
and to some extent by the bending stress in the protrusion.
The dynamical equations (4) are highly nonlinear and must
be solved numerically. A spherical protrusion is formed once
s(t) = v(t)2/3 (we call the corresponding volume vx ), at which
point v and s are no longer independent, and the protrusion
evolves as a sphere described by a single dynamical equation
(given in Appendix D). The shape of a cell with a spherical
protrusion evolves in the energy landscape of Fig. 2, which
possesses two minima and a barrier at v = v ∗ for large enough
prestress. If vx > v ∗ , the deformation is irreversible and a
permanent protrusion remains after relaxation. Figure 3(a)
illustrates the impact of the cell dynamics on the critical
tether volume beyond which cell shape change is irreversible.
As discussed earlier, the protrusion is most likely to relax
toward a large bulge with vx > v ∗ if cytoplasm exchange
with the cell body is fast compared to membrane exchange.
For very fast cytosol exchange (τv /τs → 0), the protrusion
evolves with a constant area, and the irreversible shape change
can be triggered by extracting a protrusion of volume vi >
v ∗ % 1% of the cell volume. This can be done by extracting a
100-nm-radius membrane tether of length L ∼ 100 µm (with
V = 100 µm3 ), a large value but within experimental range. A
larger value of τv /τs (for RBCs, one expects τv ∼ τs ) renders
cell deformation more reversible. The relaxation time to a
spherical protrusion taking up 1% of the total cell area is of
order 10 seconds (see Appendix D).
IV. THE CASE OF A DEPOLYMERIZING FIBER
In the physiologically relevant case of a protrusion formed
by the polymerization of a protein fiber (an Hg fiber in sickle
cell anemia), which is then rapidly depolymerized at t = 0, the
Hg concentration and the osmotic pressure are initially vastly
larger in the protrusion than in the cell body. This strongly
drives the increase the protrusion volume and irreversible
shape changes are much more likely. Let us consider a
protrusion containing N of the total N monomers in the
cell. Assuming that the density difference between the two
compartments varies over the size rn of the neck connecting
the protrusion to the cell body (rn ) L), Fick’s law predicts
−N
an equilibration kinetics of the form Ṅ % Drn ( N
− NV−V
), or
V
with n ≡ N/N :
"
#
1
1 n 1−n
Drn
,
ṅ = −
∼
−
.
(5)
τD v
1−v
τD
V
The system’s free energy appearing in Eq. (2) must also include
the entropic contribution of both compartments: E = Eel +
Eκ + EN with (using the ideal gas law)
#
"
1−n
n
+ const. .
EN = kB T N n ln + (1 − n) ln
v
1−v
(6)
Immediately after fiber depolymerization, the osmotic pressure
in the protrusion is very large, leading to a fast inflation of
the tether (by volume transfer from the cell body with our
assumption of constant total volume V). This additional force
thus increases the likelihood of ending with a permanent
protrusion, and the more so if (i) the protrusion volume
is large, (ii) the protein density in the protrusion is large,
and (iii) protein diffusion through the connecting neck is
slow: τD /τs & 1. In sickled RBCs, protrusions are formed
by bundles of Hg fibers. Figure 3(b) shows the minimal
volume a protrusion must have in order not to retract after fiber
depolymerization as a function of the number of fibers in the
bundle, for different values of the protein diffusion coefficient.
Note that water can permeate through lipid membranes [21],
and the more so through the membrane of cells, which contain
water channels. Water permeation can be readily included in
PV in Eq. (3) (with Pperm ∼ αp V̇ 2 /S; αp is a permeation
coefficient) and does not qualitatively modify our conclusions.
It is however likely to have a strong quantitative effect if the
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PHYSICAL REVIEW E 85, 051904 (2012)
D=∞
Protrusion volume µm3
Protrusion volume µm3
MARIA ZEITZ AND P. SENS
8
4
6
D = 100µm2 /s
3
4
2
(a)
10
20
30
Volume to Surface dissipation
40
2
D = 10µm2 /s
1
D = 1µm2 /s
20
40
60
τv /τs (b) Number of fibers in the initial bundle
FIG. 3. (Color online) Critical initial protrusion volume beyond which a transient protrusion fails to retract inside the cell body, (a) for a
protrusion created by an external force, as a function of the ratio of the volume to area exchange times (with Kc /σ = 10, #0 = 1/4), and (b) for
a protrusion created by a bundle of protein fibers (proteins of size 4 nm, with τv /τs = 15), as a function of the number of fibers in the bundle,
for different values of the protein diffusion coefficient.
osmotic pressure difference between the protrusion and the
extracellular medium is large. Increase of the protrusion’s
volume by osmotic water influx should strongly promote
irreversible shapes change for fiber-induced protrusions or
under hypotonic conditions.
V. PEARLING OF A RBC TETHER
The previous analysis showed that if a sufficiently long
membrane tether is extracted from the cell by a transient
force, elastic stress in the CSK can prevent the membrane
from reintegrating the cell when the force is switched off,
eventually leading to the formation of a “permanent” spherical
protrusion. The protrusion may however not relax directly to
a spherical shape. Experiments using optical tweezers have
shown evidence of a pearling instability, during which the
tube grows as a necklace of slowly inflating bubbles [9,11].
Furthermore, some of the vesicles shed during sickle cell
reoxygenation are initially rod shaped and typically relax into
a string of spherical vesicles [7].
The most favorable case for irreversibility is when relaxation occurs with little area exchange (S % const.). If
the areas of both leaflets of the protrusion membrane are
also constant, any variation of the initial curvature Ci = 1/ri
of the protrusion creates differential stretching in the two
leaflets, with an energy EBL = Km S/2(dC − dCi )2 , where
C is the local membrane curvature, d (% 5 nm) the membrane thickness, and Km (% 0.1 J/m2 ) the bilayer stretching
modulus. The differential stretching effect (included in the
extended version of the global cell energy [Eq. (1)] found
in Appendix A) is equivalent to the membrane having a
spontaneous curvature Ci which hinders the relaxation toward
a spherical protrusion. Assuming that the protrusion (initially
a cylinder of radius ri , length L) relaxes with time as a string
of close-packed bubbles of radius r(t) = 2/C(t), the driving
force for protrusion growth is fV = −∂EBL /∂V + Peff . Here,
Peff includes contribution from the CSK tensile stress, and the
osmotic pressure difference with the extracellular medium if
water permeation is present. With V ∼ S/C, the driving force
for protrusion growth at constant area thus reads
fv = Peff − Ks d 2 C 2 (Ci − C).
(7)
If Peff > Ks d 2 Ci3 , differential stretching is unimportant, and
the protrusion grows steadily toward a sphere. If not, the force
vanishes for a string of bubbles of size of order ri . With
ri % 100 nm, pearling is expected below a pressure of order
1 kPa, or an osmotic imbalance of order 1 mMol. Protrusions
generated by protein fibers or in hypotonic environment should
relax directly toward a spherical shape, but a tether drawn from
a cell close to iso-osmolarity should relax through pearling
[Fig. 1(c)], provided it is long enough for the assumption
of constant area to be reasonable. This phenomenon is
indeed observed experimentally [9,11]. In the absence of any
membrane exchange with the cell body, relaxation during
pearling is controlled by interleaflet lipid exchange (flip-flop)
and can be slow. It should be noted that RBCs, like in most
other cells, maintain a preferred lipid asymmetry lipid between
the two membrane leaflets by ATP-dependent flippases [2], so
the kinetics of relaxation by pearling could be ATP dependent.
VI. CONCLUSION AND OUTLOOK
Irreversible loss of membrane area is one of the hallmarks
of sickle cell anemia, and is thought to be one of the factors
that considerably shorten the RBCs’ lifetime in this pathology.
Such irreversible loss can also be observed in healthy RBCs
subjected to extreme mechanical perturbations. In this paper,
we have shown that the origin of membrane loss might be
traced to the mechanical properties of the cell, and in particular
to the mechanical frustration between the cytoskeleton and the
plasma membrane. We have also shown that the dynamics of
cell shape relaxation plays an important role in selecting the
final cell state following a transient mechanical perturbation.
These results have been obtained using fairly crude mechanical
models for the RBC. It is shown in the appendices that
our conclusion should not be invalidated when using more
advanced models using more realistic cell shapes and including
additional energetic contributions. It is our hope that this
paper will foster experimental and numerical work to test our
predictions.
Beyond their conceptual and practical interest for sickle
cell anemia, our results may be extended to more complex and
larger cells. In this context, they illustrate the crucial role of
cytoskeleton remodeling and surface area regulation (e.g., via
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REVERSIBILITY OF RED BLOOD CELL DEFORMATION
endo- and exocytosis) by which more complex cells may deal
with mechanical frustrations. Interestingly such regulation
necessarily occurs over a wide range of time scales (from
30 sec. to 30 min.) [22], and one may thus expect that fast
mechanical perturbation on more complex cells could lead to
irreversible shape changes not dissimilar to those exposed in
the present paper.
S = V 2/3 , V
Sc , Vc
V 1/3
H
ACKNOWLEDGMENTS
R. W. Briehl, P. Nassoy, and M. S. Turner are gratefully
acknowledged for stimulating discussion.
APPENDIX A: MORE COMPLEX CELL SHAPES
Here, we do not address the dynamics of cell shape
relaxation, but aim at showing that the relevant features of
the equilibrium energy landscape for cell shape obtained
assuming a spherical cell (Fig. 2) are universal and exist
for more complex cell shapes. In the main text, the cell’s
energy is assumed to be dominated by two contributions: a
(constant) membrane tension σ , and the tensile elasticity of
the cytoskeleton, characterized by a stiffness Ks and a rest area
S0 [see Eq. (1)]. RBC shapes are usually not spherical, except
if the osmotic pressure is larger inside the cell than outside
(hypo-osmotic conditions). The shape of a RBC results from a
minimization of the cell’s elastic energy, subjected to particular
constrains. It has been shown in [2] that one can reproduce all
known shapes (including the normal biconcave shape) by an
energy minimization at constant volume and area. We write
the total volume V and total area S.
The main contributions to the cell’s energy is the elasticity
of the cytoskeleton and the membrane bending energy. The
latter includes a spontaneous term C0 and the energy from the
area-difference elasticity (ADE), which accounts for the fact
that the two membrane leaflets might have a different optimal
area (δS0 is the optimal area difference) [2]. The total energy
reads (at the scaling level)
#
"%
Kc
κ
δS
(Sc − S0 )2
Etot =
C 2 dS − 2 C0 +
2
d
2S
Km
+
(A1)
(δS − δS0 )2 + constant,
2S
where C is the
& local curvature, d is the membrane thickness,
and δS = d CdS is the area difference between the two
membrane leaflets. Kc is the CSK stretching modulus, κ the
bending rigidity, and Km the membrane stretching modulus.
The CSK elasticity also includes resistance to shear,
which can be relevant for certain cell shape changes. It has
however been shown numerically in [2] that sizable shear
stress only develops in regions of high curvature such as
the spicules present in the echinocyte shape. The CSK shear
elasticity basically prevents membrane regions attached to
the CSK from undergoing budding (the formation of a small
spherical outgrowth), as a simple close membrane would in
the case of high (positive) spontaneous curvature or area
difference (in the main text and below, the bud is CSK
free). Under the same conditions, shear elasticity favors the
formation of spicules. In more regular shapes, shear stress
can be neglected [2]. In the case studied below, the detached
protrusion
cell body
L
FIG. 4. (Color online) More complex cell shape considered in the
main text.
membrane is connected to the cell body via a (possibly small)
membrane neck. Shear elasticity could be relevant to the neck’s
shape and energy, but the neck is only a small contribution
to the global energy budget, and its shape may not change
during the relaxation process (except at the very end in case of
complete retraction). In different but related circumstances, it
was shown numerically that a CSK cortex with shear elasticity
does not prevent the formation of a bud in regions where the
membrane is detached from the cortex [23].
The cell’s aspect ratio is mostly fixed by the surface-tovolume ratio s̄ ∝ S/V 2/3 (equal to unity for a sphere and larger
otherwise). Details of the shape (biconcavity, spiculation)
result from an elastic balance between the stresses appearing
in Etot . The phase diagram of RBC shapes involves a large
number of parameters and is very rich. The origin of the
biconcave shape of normal RBCs is thought to be due either
or both to the existence of a negative spontaneous curvature
C0 < 0 and a negative preferred area difference (δS0 < 0) [2].
We restrict ourselves to these conditions here, and to cases
where the CSK is stretched by its attachment to the cell
membrane in an intact cell: S0 < S.
The lowest level of complexity that can inform us on complex cell shapes is to consider a nonspherical cell characterized
by two parameters: H along the axes of symmetry and L
normal to it (Fig. 4). We are interested in oblate shapes such as
the typical biconcave shape, with L > H , and we will explore
the limit on highly nonspherical cells at the lowest order in
H /L, since the other limit is the spherical case described in
the main text. In this limit, the conservation relations for the
total volume V and total area S impose (at the scaling level)
Vc + V % L2 H + V = V = const.,
Sc + S % L2 + V 2/3 = S = const.,
(A2)
where the cell body area and volume are Sc and Vc and the
protrusion volume is V . Using the rescaled variable v ≡ V /V
for the protrusion volume (and with s̄ ≡ S/V 2/3 ), we have
Vc = V(1 − v), Sc = V 2/3 (s̄ − v 2/3 ),
'
1−v
L % V 1/3 s̄ − v 2/3 , H % V 1/3
.
s̄ − v 2/3
(A3)
At the lowest order in H /L, the quantities
relevant to the
&
energy Eq. (A1) are (at the scaling level) dSC 2 = L/H + 1,
δS = d(L + V 1/3 ), and Sc = L2 . Using Eq. (A3), the total
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MARIA ZEITZ AND P. SENS
energy reads
!
$
'
(s̄ − v 2/3 )3/2
κ
1+
− 2(v 1/3 + s̄ − v 2/3 )c0
Etot =
2
1−v
kc
km 1/3 '
+ (s̄ − v 2/3 − s0 )2 +
(v + s̄ − v 2/3 − δs0 )2
2
2
+ const.,
(A4)
with
S0
δS0
, δs0 =
,
2/3
V
dV 1/3
km = Km d 2 , kc = Kc V 2/3 .
c0 = C0 V 1/3 , s0 =
(A5)
Etot may be studied numerically to determine the parameter
range where one expects a nucleation-type behavior such as
the one described in Fig. 2(b). In the main text, the critical line
was found to be located at (1 − s0 ) ∼ (σ/Kc )3/4 in the limit
Kc & σ . Expansion of the energy Eq. (A4) for v ) 1 shows
that the critical line is here approximately given by
√
"
#
3 km ( s̄ − δs0 ) + κ(−c0 ) 2/3
s̄ − s0 ∼ 4/3
.
(A6)
2
kc
This expression is conceptually similar to the one obtained
in the main text for a spherical cell. The CSK prestress must
be larger than the ratio of a quantity that characterizes the
membrane elasticity (σ in the main text, and a combination
of bending rigidity and ADE here), over the CSK stiffness.
So the surface tension used in the main text should be seen
as an effective parameter that includes bending stresses. The
optimal area difference δs0 and the spontaneous curvature
c0 are thought to be negative for RBCs. If they are not, a
spherical protrusion may be metastable even in the absence of
CSK prestress, in which case Eq. (A6) is not valid. Expected
values of the parameters for RBCs are kc /κ % 100 and
km /κ % 1 [2].
The precise location of the critical line, shown Fig. 5,
is found as in the main text by numerically solving for the
appearance of an inflection point: ∂v Etot = 0 and ∂v2 Etot = 0.
The more nonspherical a cell (large value of s̄), the smaller
the critical CSK prestress for the appearance of a metastable
spherical protrusion, so the effect described in the main text
is certainly not restricted to spherical cells. In real RBCs, one
expects kc /κ > 100, so the effect we predict could be observed
for area mismatch (s̄ − s0 )/s̄ as low as 10%.
It is clear that the model outlined in this section is still
very crude, but it serves to show that (i) contributions to the
elastic energy not included in the main text (bending rigidity +
spontaneous curvature, or ADE) do not fundamentally alter the
picture, even at the order of magnitude level, and (ii) the precise
shape of the cell does not either. A more thorough analysis
based on the realistic model of [2] is certainly desirable, but is
numerically much more complicated.
APPENDIX B: COMPLETE ENERGY FUNCTION FOR
A SPHERICAL CELL
Energies for the elastic deformation of the cell cytoskeleton
(CSK) and plasma membrane (PM) are readily obtained by
assuming an elastic CSK (Hookean, with a stretching modulus
Kc and a rest area S0 ) and a PM under constant membrane
tension σ . They are given in the paper [Eq. (1)].
1. Bending energy
The additional contribution from the membrane bending
energy requires the knowledge of the full protrusion shape
during relaxation. The protrusion shape evolves from an
initially thin tube to a sphere and its bending energy is in
principle a complex function of the protrusion shape. Since
it is clear that the contribution from the membrane’s bending
rigidity is only relevant for highly curved membranes, namely
in the tubular state, we choose to write the bending energy as a
simple function of the variables S (= 2π Lr) and V (= π Lr 2 )
which is valid in this limit:
1 1
1 S3
(B1)
κ 2S = κ 2.
2 r
8 V
In the opposite limit of a close-to-spherical protrusion, the
numerical factor 1/8 in Ebend is replaced by 2/9, but the energy
is completely negligible compared to the contribution from
membrane tension.
Ebend =
0.6
2. Complete energy function
0.5
s̄ − s0
s̄
s̄ = 1.5
0.4
s̄ = 2
s̄ = 3
0.3
0.2
0.1
0.0
0
20
40
60
80
100
Adding the contributions from the PM tension, bending
energies, CSK stretching energy, and an unbalanced osmotic
pressure between the cell body and the protrusion arising from
the different concentration of hemoglobin (Hg) proteins in
both compartments [assumed to behave as an ideal gas for
simplicity, Eq. (6)], the total energy reads
kc
κ
E=
FIG. 5. (Color online) Equilibrium phase diagram for a nonspherical cell with a protrusion. The critical CSK prestress is plotted as
a function of the CSK stiffness kc /κ, for different values of the cell
area-to-volume ratio s̄. The asymptotic expression given in the text
[Eq. (A6)] is plotted for s̄ = 3 (dashed red line). The protrusion
corresponds to a local energy minimum in the shaded region of the
diagram. The values of the parameters are kd /κ = 1, δs0 = 0, and
c0 = −5.
1 S3
+ σ (S + (36π )1/3 (V − V )2/3 )
κ
8 V2
1 [(36π )1/3 (V − V )2/3 − S0 ]2
+ Kc
2
S0
N
(N − N)
,
+ kB T N ln + kB T (N − N ) ln
V
(V − V )
(B2)
where the variables are S,V , and N (surface and volume of the
protrusion and number of Hg molecules in the protrusion), and
051904-6
PHYSICAL REVIEW E 85, 051904 (2012)
REVERSIBILITY OF RED BLOOD CELL DEFORMATION
S, V, and N are their (constant) counterparts for the whole
cell (protrusion and cell body).
Two relaxation phases must be distinguished in the relaxation of the protrusion. Initially, S and V are independent
from one another as the shape evolves from a tube to a
sphere. When the protrusion becomes spherical, they are
not independent anymore, and one evolves with the other
according to S = (36π )1/3 V 2/3 . It follows for the energy
∂P/∂ V̇ . The volume dissipative force reads
fVd =
(C2)
The area dissipative force reads
Esphere = σ (36π )1/3 [V 2/3 + (V − V )2/3 ]
1 [(36π )1/3 (V − V )2/3 − S0 ]2
+ Kc
2
S0
N
(N − N)
+ kB T N ln + kB T (N − N ) ln
.
V
(V − V )
(B3)
"
#
rn Ṡ
8ηln
V̇
−
.
π rn4
2
fSd =
#$
!
"
2V̇
4ηln
αl Ṡ
.
1
−
1+
2π
αl rn2
rn Ṡ
(C3)
Under relevant physiological conditions [with ηln /(αl rn2 ) %
10−6 to10−2 ], the latter force is dominated by PM-CSK
friction, so that fSd % α2πl Ṡ . This approximation was used in
the text.
APPENDIX C: ENERGY DISSIPATION
APPENDIX D: COMPLETE DYNAMICAL EQUATIONS
The dissipative forces in Eq. (2) are obtained from the
so-called Rayleigh dissipation function P [16], which can be
calculated from microscopic models of viscous dissipation
within the system. In our case, the system is made of a PM, a
2D spectrin CSK, and a cytosol containing Hg proteins. The
CSK is essentially immobile during the relaxation process,
since the radius of the cell body varies only little. Membrane
and cytosol on the other are exchanged between the protrusion
and the cell body during shape relaxation. The membrane
dissipation PS is dominated by the PM friction over the
immobile CSK imposed by the variation of the protrusion area
(Ṡ). This dissipation was calculated in [18] and is proportional
to a friction coefficient α [see Eq. (C1) below] which was
measured for various cell types, including RBC, in the same
publication (α % 107 −1011 Pa · s/m).
The viscous dissipation in the cytosol PV occurs all over
the cell, but is dominated by the neck region connecting the
protrusion of the cell body (a small neck of radius rn and length
ln , with rn % ln % 100 nm; in the paper, we have assumed
ln = rn for simplicity). In this region of space, the membrane
flows from the protrusion into the cell body (mostly driven
by surface tension, Ṡ < 0) and the cytosol flows from the
cell body toward the protrusion (driven by bending rigidity
and a higher concentration of Hg protein in the protrusion,
V̇ > 0). The membrane velocity vm is given by Ṡ = 2π rn vm ,
and the cytosol velocity profile is a Poiseuille flow (low
Reynolds number) v(r) = vc − (vc − vm )(r/rn )2 , where r is
the radial distance from the neck’s axis of revolution, and
the cytosol velocity
& vc on the axis is related to the net
volume flux by d 2 rv(r) = V̇ , or V̇ = 1/2π rn2 (vc + vm ).
Integrating the cytosol &viscous dissipation over the neck
volume [24], PV = η/2 dV (∂r v)2 , where η (%10−2 Pa/m
[20]) is the cytosol viscosity, we obtained P = PS + PV
with
"
#
rn Ṡ 2
αl 2
4ηln
V̇
−
Ṡ , PV =
PS =
,
(C1)
4π
π rn4
2
1. Nonspherical protrusion
For nonspherical protrusions, S and V evolve independently. Using the Lagrangian formalism Eq. (2) and the energy
Eq. (B2), one can obtain the dynamical equations describing
the evolution of the protrusion surface and volume:
!
" # $
3κ S 2
2π σ
1+
,
(D1)
Ṡ = −
αl
8σ V
!
#
" #
"
rn
π rn4
N −N
κ S 3
N
V̇ = Ṡ +
kB T
−
+
2
8ηln
V
V −V
4 V
#$
1/3 "
1/3
2/3
2(36π ) σ Kc (36π ) (V − V ) − S0
+
+1 .
3(V − V )1/3 σ
S0
(D2)
The dynamical equation for the protrusion area, Eq. (D1),
is rather straightforward and is explained in the main text.
For the evolution of the protrusion volume, the first term
in Eq. (D2) comes from the fact that the volume variation
is hydrodynamically coupled to the area variation and that
solvent is entrained by the membrane movement. This term
has a limited influence on the relaxation process and has
been omitted for simplicity in the main text. The second
term describes the osmotic pressure difference between the
protrusion and cell body, which drives protrusion growth, the
third term comes from the bending energy, and the last term is
due to the membrane and CSK tension in the cell body.
2. Spherical protrusion
Once the protrusion has reached a spherical shape, the
system’s energy is given by Eq. (B3). The evolution of
the protrusion’s shape proceeds subjected to the constraint
S = (36π )1/3 V 2/3 , or
1/3
with αl ≡ α ln Vrn [see Eq. (3)]. The generalized dissipative
forces can then be calculated as fSd = ∂P/∂ Ṡ and fVd =
051904-7
Ṡ =
∂S
2
V̇ = (36π )1/3 V −1/3 V̇ .
∂V
3
(D3)
PHYSICAL REVIEW E 85, 051904 (2012)
MARIA ZEITZ AND P. SENS
Including this constraint in the dissipation function Eq. (C1)
and using the generalized force balance ∂V E + ∂V P = 0 gives
a single dynamical equation:
!
"
#
$
8ηl
(36π )1/3 rn
2(36π )2/3 αl
1
−
+
V̇sphere
rn4
V 1/3
9π V 2/3
#
!
"
N −N
2
σ
N
−
+ (36π )1/3
= kB T
V
V −V
3
(V − V )1/3
"
#$
Kc (36π )1/3 (V − V )2/3 − S0
×
+1 .
(D4)
σ
S0
3. Relaxation of monomer concentration differences
For the case in which the protrusion is created by a
bundle of polymerized fibers, and its relaxation follows sudden
fiber depolymerization, the concentration of monomers is
initially very different in the protrusion and in the cell body.
This difference relaxes by diffusion. While concentration
inhomogeneities might also exist within the protrusion and
the cell body, the bottleneck to concentration equilibration
is likely to be the thin tubular connection between the
protrusion and the cell body. The diffusion then becomes
quasi-unidimensional, and the flux JN of monomers from
the cell body to the protrusion is obtained from Fick’s
law [24]
"
#
D N
N −N
∂C
=−
−
,
(D5)
JN = −D
∂x
ln V
V −V
where D (∼ 0.5−4 µm2 /s) is the monomer diffusion constant.
The variation of the number of monomers in the protrusion then reads Ṅ = π rn2 JN , and the dynamical equation
for N is
"
#
N −N
πr2 N
−
,
(D6)
Ṅ = −D n
ln V
V −V
τv , respectively, with
τs =
Sαl
,
2π σ
τv =
ηln S 2
2 ηln S
=
τs .
2
4
3π σ rn
3π αl rn4
(D7)
We rewrite the three dynamical equations for the dimensionless
variable v ≡ V /V, s ≡ S/S, and n ≡ N/N :
!
" #2 $
27π
1
s
1+
,
(D8)
ṡ = −
κ̄
τs
2
v
!
"
#
1
1
(1 − v)2/3 − s0
v̇ = r̄n ṡ +
K̄
+1
c
τv (1 − v)1/3
s0
" #3
#$
"
s
27π
n 1−n
κ̄
−
,
(D9)
+
+ T̄
2
v
v
1−v
"
#
1 n 1−n
−
,
(D10)
ṅ = −
τD v
1−v
with the renormalized parameters
κ
3 kB T N
Kc
K̄c =
, κ̄ =
, T̄ =
,
σ
σS
2 σS
#
"
9π rn3 1/3
ln V
r̄n =
.
, τD =
2V
π rn2 D
(D11)
(D12)
In the case of a sphere,
!
# $
" #1/3 "
4 τs
2
4π
1 − r̄n τv v̇
+4
9 v 2/3
3
3
!"
#
1
(1 − v)2/3 − s0
1
=
−
+
K̄
+
1
c
(1 − v)1/3
v 1/3
s0 (1 − v)1/3
#$
"
n 1−n
−
.
(D13)
+ T̄
v
1−v
We express the protrusion volume relative to the total cell
volume V and its area relative to the area of a sphere of same
volume S = (36π )1/3 V 2/3 . The evolution of the protrusion
area and volume then exhibit characteristic time scales τs and
The typical parameter values given above and in the main
text lead to τv % 104 s, τs % 102 −106 s, and τD ∼ 106 s.
These values are connected to the evolution of the entire cell
area and volume. The actual relaxation time of a protrusion is
weighted by the protrusion relative dimensions as compared
to that of the whole cell. A protrusion taking up 1% of the cell
area relaxes to the spherical shape in about 10 seconds.
The set of equations Eqs. (D8), (D9), (D10) or Eqs. (D10),
(D13) can be solved numerically with various initial conditions
to determine the parameter space in which cell shape change
is irreversible, as shown Fig. 3.
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4. Normalization
051904-8
PHYSICAL REVIEW E 85, 051904 (2012)
REVERSIBILITY OF RED BLOOD CELL DEFORMATION
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