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Theoretical and phase contrast microscopic eigenmode analysis of erythrocyte flicker : amplitudes
Mark Peterson, Helmut Strey, Erich Sackmann
To cite this version:
Mark Peterson, Helmut Strey, Erich Sackmann. Theoretical and phase contrast microscopic eigenmode analysis of erythrocyte flicker : amplitudes. Journal de Physique II, EDP Sciences, 1992, 2 (5), pp.1273- 1285. �10.1051/jp2:1992199�. �jpa-00247725�
J. Phys. II France 2 (1992) 1273-1285 MAY 1992, PAGE 1273
ClasswicaIion Physics Abstracts
82.65D 87.20 87.10
Theoretical and phase contrast microscopic eigenmode analysis
of erythrocyte flicker : amplitudes
Mark A. Peterson (I), Helmut Strey (2) and Erich Sackmann (2) (1) Mount Holyoke College, South Hadley, MA 01075, U-S-A-
(2) TU-Munchen, Physik-Department E22, 8046 Garching, Germany (Received 26 July I99I, accepted 23 December I99I)
Abstract. The energy eigenmodes of an erythrocyte with elastic properties described by a
curvature modulus k~ and a shear modulus p are computed, both for fixed spontaneous curvature
co, and in the bilayer coupling hypothesis (fixed (Hi ). The corresponding thickness fluctuation
profiles depend on the relative importance of k~ and p. A phase contrast microscopy determination of the thickness fluctuation profile indicates that p, at least in this process, is very small. The data favor fixed (Hi over fixed co.k~ is about 4.3 x10~~~J if co is fixed and 1.4 x 10~~~J if (H) is fixed.
1. Introduction.
The erythrocyte flicker phenomenon was first quantitatively analyzed by Brochard and
Lennon [ii in 1975, who showed convincingly that the phenomenon is Brownian shape
fluctuation. A completely quantitative understanding of this phenomenon still does not exist, however, because the equilibrium shape of the erythrocyte complicates the analysis greatly.
Brochard and Lennon modeled the erythrocyte with two infinite parallel plane membranes and a continuum of plane wave modes. It seems clear, however, because the erythrocyte is
topologically a sphere about the same size as the length scale of the flicker phenomenon itself,
that there must be only a few discrete shape modes, and that these would be strongly
influenced by the geometry of the equilibrium shape. At the end of section 2 we reanalyze
their data using the exact shape modes.
A puzzling question is the role of the membrane shear modulus ~, arising from the
cytoskeleton. If ~ were as large aS mechanical experiments [2] indicate, shape fluctuations would be much smaller than what is observed. This point is discussed further in section 2.3.
In section 2 of this paper we work out for the rust time what standard elasticity theory predicts for erythrocyte flicker amplitudes, essentially without approximation. The shape of
the thickness fluctuation profile (TFP), which can be measured by intensity-intensity
correlation in phme contrmt microscopy, i~ iound to be sensitive to the relative importance oi k~ and ~. We report in section 4 experimental data on the shape of the TFP which indicate a much lower value for ~ than is found in the mechanical experiments [2]. This may mean that
the cytoskeleton is « loose », and that small amplitude shear motion, up to some threshold shear strain, does not cost energy. We also compute the TFP under the « bilayer coupling hypothesis » (BCH) first proposed by Evans [4] and refined by Svetina and ZEks [5]. The data of section 4 seem to favor the BCH.
2. Elasticity theory.
The equilibrium shape of the erythrocyte is a stationary shape of the curvature energy [3]
k~ j ~
F~(V, A, co)
= (ci + cz co) dA (1)
2
~
Here k~ is the curvature modulus, and ci and cz are the principal curvatures. The surface
integral goes over the closed surface M, which has (fixed) volume V and area A. The spontaneous curvature co is a parameter which biases the mean curvature. It is an open question whether the erythrocyte really possesses a fixed spontaneous curvature. An
altemative, not wholly equivalent, interpretation is that co should be regarded as a Lagrange multiplier in the model to enforce fixed (cj + cz), where ( ) denotes surface average. This
suggestion of Svetina and Zlks [5], the « bilayer coupling hypothesis » (BCH), is physically plausible, since it describes a bilayer in which each monolayer has fixed area. The BCH has
observable consequences for the shape dynamics of the cell, as we show below.
The erythrocyte has a cytoskeleton which gives it a shear elasticity. The equilibrium shape
of the cytoskeleton apparently coincides with the shape found by minimizing F~. Perhaps this is because shear strain can relax by dynamic rearrangement of the protein elements of the
cytoskeleton, while curvature strain cannot wholly relax. In any case, by standard
phenomenology, the shear energy for small deformations of the erythrocyte is
F~ = ~ S,~ S'J dA (2)
M
where ~ is the shear modulus and S,j is the 2-dimensional shear strain tensor of the
deformation.
All together, then, the deformation energy of the erythrocyte is F
= F~ + F~. (3)
To describe flicker we expand this energy to second order in deformation amplitudes, find the normal modes of the resulting quadratic form, and use the equipartition theorem to compute the mean square deformation of the surface. Details of this computation are
described in the next three subsections. Results are in the last subsection.
2.I EQUILIBRIUM SHAPE. If the equilibrium surface is a surface of revolution, as we
assume, with the x-axis as rotational symmetry axis, it can be described by a curve
C
= (x(s), y(s)), with y =
0 at its endpoints, satisfying
0
= 3F~ (4)
where, specializing equation (1),
F~= [~~-~°~~-coj~yds+A ly~cosods+« jyds+ jb(y'-sine)ds.
ds y
(5)
N° 5 ERYTHROCYTE FLICKER :I. AMPLITUDES 1275
Here o is the angle between the tangent to C and the x-axis. It starts at w/2 and ends at 3 w/2. The Lagrange multipliers A and « allow one to fix the volume and area respectively.
The Lagrange multiplier b(s) maintains the parameter s proportional to arclength. The
corresponding Euler-Lagrange equations are given in [6]. The same formulation has been
used with success by Seifert, Bemdl, and Lipowsky [7] to find a surprising variety of
equilibrium vesicle shapes, and these shapes have been observed in real vesicles by Kfis and Sackmann [8]. The shape used in this paper has parameters A = -11.95045, « +
cl
= -13.51387, co
= -1.38515, and the principal curvature on the symmetry axis is 0.30501. These parameters have been scaled so that the total arclength from the symmetry axis to the equator is w/2. The shape is discocytic, with reduced volume 0.8407. It is
infinitesimally stable. This shape, shown in figure1, is our surface M. If the physical cell surface area is 135 ~Lm2, as we assume, then the unit of length is
R
=
3.02 ~Lm (6)
and the physical cell volume is 115 ~Lm3.
.6 .4 2
~ i o
~
~
O.
~ ~ ~ x in Pm
Fig. I. Model cell shape, found by minimizing curvature energy. The cell shape corresponds to a RBC at 200 mosmols with area 135 ~m2 and cell volume II 5 ~m3.
2.2 CURVATURE ENERGY. Let n :M
- R be a real valued function on the surface M
describing a deformation of M by the displacement n along the local normal to M. The
expression of F~ to second order in n at constant V and A, ignoring first order terms since M is
an equilibrium surface, is given in [9]. It is 3~F~ =
2 [(An)~ + A 'J n ; n + Bn ~] dA (7)
~
where
A'J= -2(2H-co)(C~J-Hg'J)+g~J[-6H~+4K-2coH+cj/2- (Q-T(H))/4S]
(8)
B =16H~-20H~K+4K~+2 Aj2H~-K) +cjK+4[(C~J-Hg'J)H,].~ +
+ Q ( (H) H K/2 )/S + T( (H~) H (H) K/2)/S (9)
Q
=
(4(VH(~- 8H~+4HK(2H-co)) (10) T
= (8 H~ 4 K(2 H co)) (11)
S
=
(H~) (H)~ (12)
H
= (cj + cz)/2 (13)
K
= cj cz. (14)
Here g~J is the first fundamental form (metric tensor) of M, C'J is the second fundamental
form of M, comma indicates partial derivative, semicolon indicates covariant derivative with
respect to the metric connection, ( ) indicates average over the surface M, and A is the
Laplace-Beltrami operator on M.
The normal deformation n must respect the constraints of fixed volume and fixed area to first order, I-e-, it must satisfy
0
=
j
n dA (15)
M
and
0
= lnH dA. (16)
M
The second order corrections which keep V and A fixed even in second order are built into the above expressions.
The above expressions assume co constant. The expansion under the BCH is slightly
different, but can be found by the same methods. The BCH introduces a third integral
constraint
lH dA
= constant (17)
M
3~F~ subject to this constraint has the form of equation (7), but
A~J= -4HC'J+g'J[-2H~+4K+ UP~~V~J] (18)
B =16H~-20H~K+4K~+2A(2H~-K)+4[(C'J-Hg'J)H,].~ + UP~~V (19)
where
U= ((-8H~+BHK), (4(VH(~-8H~+BH~K), (4VK.VH-BH~K+BHK~))
Ii (H) (HK) (20) P
= (HI (H~) (K) (21)
(HK) (K) (K~)
V ~~
= (0, 1/4, ~i12)~
V~~
= (0, 1/4, cj/2)~
V ~~
=
V~~
=
0 (22)
V
=
(- H, K/2, 0)~ (23)
The normal deformation n must respect the constraint of fixed average mean curvature to first
N° 5 ERYTHROCYTE FLICKER : I. AMPLITUDES 1277
order, which means, using the expansions of [9],
0
= lnK dA, (24)
M
where K is the Gauss curvature, in addition to respecting the constraints on V and A,
equations (15) and (16). Second order corrections to n are built into the above expressions.
The geometrical data in the above expressions refer to the equilibrium shape M. Although
in this paper we take M to be azimuthally symmetric, the expressions in this subsection
actually give 3~F~ at fixed V and A for any smooth closed surface M.
Now that 3~F~ is known as a quadratic form in n, with or without the BCH, one can find the
corresponding normal modes by diagonalizing it, as described in [6]. It is an extremely stringent check on the correctness of the above expressions that all the rigid Euclidean
motions emerge as zero energy modes. In order for this to happen, every term must be
correct, and, in addition, the equilibrium shape must be very accurately specified. Otherwise the first variation 3F~ is not zero, and the quadratic form does not correctly give the energy.
2.3 SHEAR ENERGY. The previous section considered only normal displacements n. The
shear energy requires one to consider also tangential displacements u. As described in [9] it is convenient to represent a small tangential displacement u by a I-form
u = da + *dp (25)
where a and p are functions on M, and * is the Hodge star operator of the surface M. The function
a is determined up to an irrelevant constant by local incompressibility
ha
= 2 nH. (26)
Thus a depends linearly on n, and the shear tensor
~ij ~~~J ~ ~ll' ~ ~~lj ~~~~
is a linear functional of n and p. Then F~ (Eq. (2)) is a quadratic form in n and p, and the normal modes of F~ are found by diagonalizing it, as described in more detail below. Once
again the rigid Euclidean motions emerge as zero energy modes, proving correctness of the
computation.
The mechanism by which a normal displacement produces 2-dimensional shear in the
membrane (cytoskeleton) requires comment. There are really two mechanisms. At a non-
umbilic point, I-e-, a point where the principal curvatures are not equal, a purely normal
displacement produces different dilations along the two different principal curvature directions. This is pure dilation plus shear. If, in addition, the membrane is incompressible, as
assumed above, then at all points where the mean curvature H is not zero there must be a
tangential displacement in the membrane to keep the local density constant. This induced tangential displacement has an associated shear. Both effects are contained in equation (27).
The second effect is associated with the function a.
Several recent papers have investigated the fluctuations of a solid membrane starting with a reference geometry which is a plane [10, 11]. This is the unique reference geometry which omits both the above effects. All points are umbilic, and the mean curvature is zero. Thus in this case, and in this case only, the shear associated with a normal displacement is higher than
first order. Renormalization by the fluctuations is then very important. One finds, for
example, that the mean square normal fluctuation amplitude at wavevector q is [10]
k~ T
~ fi 3
~ ~ 4
(~~)
The renormalization of the shear term is evident. This observation is not relevant to the red
blood cell membrane, however, which in this language would have a mean square normal
fluctuation [12]
k~ T
C (q + k~q~ ~~~~
where C depends only weakly on q, and is approximately 2 ~/R~ (R is the radius of the cell).
Because the surface is curved, the shear is already linear in normal displacement, and the
shear energy is already quadratic in normal displacement, and it is not sensitive to
renormalization effects. The problem is geometrically more complicated, but conceptually
much simpler than the corresponding problem for a plane membrane.
The puzzle mentioned in the introduction can now be restated. Lennon and Brochard
ignored ~, because their model started with a plane membrane. They deduced a value for k~ from the mean square amplitude. In fact, though, one sees from expression (29) that for a
generic shape ~ dominates the long wavelength fluctuations, and suppresses the mean square
amplitude by a factor of something like 1000, if one uses conventional values for
k~ and ~. More detailed calculations do not change this qualitative result. Of course
expression (29) above is merely schematic. In reality there are only discrete modes, eigenmodes of F (not really labeled by a wavevector q). This paper works with those exact discrete modes.
2.4 THEORETICAL FLICKER AMPLITUDES. The total elastic energy F has been expressed as
a quadratic form in n and p. Because the equilibrium shape M has rotational and inversion symmetry, the quadratic form breaks up into a direct sum of quadratic forms, one for each irreducible representation space of [- 1, II x SO (2). Practical bases in each of these spaces
are the spherical harmonics Yi~(s, ~Y) regarded as functions of the arclength s along C and azimuthal coordinate
~Y. The choice of scale mentioned at the end of subsection 2.1 facilitates
use of these functions. If n is a real linear combination of Yi~'s with I even, then p should be chosen as a linear combination, with pure imaginary coefficients, of Yi~'s with I odd.
Conversely, if n is odd, p should be even. The function n for (zero energy) translation modes has I odd, m = 0 for motion along the symmetry axis, and m
= ± for motion perpendicular
to the symmetry axis. The function n for (zero energy) rotation modes has I even,
m = ± if the rotation axis is perpendicular to the symmetry axis, and m
= 0 if the rotation is about the symmetry axis. In the last example the function n is, of course, identically zero, but p is not, and has I odd, m
= 0.
In the complex vector space spanned by the spherical harmonics F determines an Hermitian
quadratic form with respect to the L~(M) norm. Thus the normal modes (nj,~, p j,~) will be orthogonal in the L~(M) norm (although the spherical harmonics are not), and the
eigenvalues (Ej,~) will be real. Since M is infinitesimally stable, they are nonnegative.
We can normalize the modes to have norm 1. The mean square normal displacement in
thermal shape fluctuations is then
(3n)~
= £ a, n(~ (30)
N° 5 ERYTHROCYTE FLICKER I. AMPLITUDES 1279
where
a; = kT/E
j;~ (31)
and the bar denotes time average. Zero energy modes, which do not represent shape
fluctuations, but rather rigid motions, are of course omitted from the sum. If we look along
the symmetry axis at fluctuations in thickness d, we need the projection of n along the
symmetry axis. We find
(3d)~
=
4 £ A;n(~ sin~ 0 (32)
even
where
« even » denotes modes in which n is even in reflection in the equatorial plane, I-e-,
I m is
even. The m
= ± I translation modes and the m
=
0 rotation mode are omitted from the sum.
Apart from the overall scale, (3d)~ is determined essentially by only one dimensioneless parameter, which we may take to be ~R~/k~, measuring the relative importance of the two
terms F~ and F~. This parameter is thought to be about 1000. The parameter co does not
noticeably affect the profile for stable cells, and of course under the BCH there is no such parameter it is replaced by (H), which is determined by the equilibrium shape, and is not a free parameter.
In figure 2 is shown the range of possible profiles (3d)~, normalized to unity at the cell center. The profile under the BCH differs from these only in the extreme case that the shear energy is unimportant (fluid membrane). This profile for a fluid membrane under Ihe BCH is shown in figure 3, along with the profile for a fluid membrane at fixed co, this time normalized
so that k~ = x 10~~~J, showing predicted mean square displacements. It is noteworthy that
the profile is always low in the center of the cell, and high near the periphery, contrary to the way flicker is usually described. Experiments reported in the next section confirm this feature,
4
~ 3 E
~A ~
~i
0
~ ~ ~ xin#m
Fig. 2.- Thickness fluctuation profile for various values of the ratio e
= pR~/k~. The effect of
increasing this ratio is to move the peak in thickness fluctuations toward the center of the cell. For p = 0 (fluid membrane) the peak is near the rim of the cell.
60x10 ~
so
~
40
$E
bilayer coupling hypothesis spontaneous cunlature hypothesis
io
O
~ ~ xinpm
Fig. 3. Thickness fluctuation profiles with and without the bilayer coupling hypothesis (BCH). The BCH constraint essentially removes one fluctuation mode. The ordinate shows the mean square
thickness fluctuation in case k~ = I x10~~~J.
however, and there is a very simple reason for it. The integral constraints essentially eliminate
certain shape modes. To put it very roughly, the volume constraint eliminates the
Yoo mode (breathing mode), and the area constraint eliminates the Y~o mode. (We label the modes by their largest component in the basis of spherical harmonics.) The first even
m = 0 mode which is allowed is Y4O~ and this has a fairly large energy eigenvalue (the eigenvalues go roughly like i~). On the other hand, only m
= 0 modes contribute to the TFP
at the center of the cell. Thus the integral constraints suppress shape fluctuations at the
center. Under the BCH the Y~o mode is also eliminated, and the suppression in the center is
even more dramatic. The shape of this mode can be seen in figure 3 as the difference between the profile with and without the BCH.
These thickness fluctuation profiles are the main theoretical result of this paper. They show
a peak near the periphery of the cell. As shear becomes more important, this peak moves in toward the center of the cell. The amplitude of the peak relative to the amplitude in the center is much larger in the BCH than in the hypothesis of constant co. The experimental data of the
next section have the essential features of these theoretical profiles, and the location of the
peak is consistent with a fluid membrane. The size of the peak, relative to the amplitude at the center, favors the BCH.
We have also computed profiles for variants of the above (simplest) phenomenological theory. In one, the shear energy was taken to be zero as long as the shear deformation was below some threshold value, and infinite above the threshold value, so that the shear was constrained to be less than the threshold. As the threshold is made smaller and smaller the effect is very much like simply increasing the shear modulus from zero, as in figure 2. In another variant, the cytoskeleton was allowed to be compressible, and a new parameter, its
compressibility modulus, was introduced. (Of course the total area is kept constant by the
incompressible lipid bilayer.) As long as the compressibility modulus is the same order of
magnitude as the shear modulus, as one would expect from models of cross-linked proteins, it makes very little difference if the cytoskeleton is compressible. An amusing result is that if the
