J. Cell Sci. 41, 151-157 (1980)
Printed in Great Britain © Company of Biologists Limited ig8o
151
RED BLOOD CELL ADHESION
III. ANALYSIS OF FORCES
V.ADRIAN PARSEGIAN
Physical Sciences Laboratory, Division of Computer Research and Technology.
National Institutes of Health, Bethesda, Maryland 20014, U.S.A.
AND DAVID GINGELL
Department of Biology as Applied to Medicine, The Middlesex Hospital
Medical School, London WiP 6DB, England
SUMMARY
The results of experiments on the adhesion of glutaraldehyde-fixed red blood cells to both
polarizable metal/saline and liquid hexadecane/saline interfaces have been analysed in terms
of physical forces. The results show that the electrostatic repulsions sufficient to prevent
adhesion to these test surfaces are remarkably similar, and that a force-balance condition is
predicted at cell-substratum separations ~ 100 nm in 0 4 mM NaCl as found by interferometry.
From the repulsive force the size of the attractive force can be found. If this is viewed as an
electrodynamic attraction, the force coefficient is found to lie between 5 and 8 x io~14 erg (5
and 8 x io~ fl J), a range in reasonable correspondence with measurements in physical
systems.
INTRODUCTION
It has been shown that aldehyde-treated human red cells can adhere to a hexadecane/saline interface at concentrations exceeding 1 mM NaCl, and that further
dilution reduces adhesion (Todd & Gingell, 1980; Gingell & Todd, 1980). A concentration less than o-i mM NaCl practically abolished adhesion. In an earlier report
(Gingell & Fornes, 1975, 1976) it was found that similar red cells adhered to a metal
surface in 1 -2 mM NaF, but when the surface charge of the metal was made sufficiently
negative, exceeding 4X io4 esu/cm2, adhesion could be almost completely abolished.
Using the numerical data from these studies, we shall present estimates of the
electrostatic repulsive force sufficient to prevent adhesion. From these estimates
we deduce the size of the attractive force. A preliminary account of some of these
calculations has appeared elsewhere (Gingell, Todd & Parsegian, 1977).
METHOD OF CALCULATION
We begin by assuming that the red cells are smooth rigid discocytes which interact
en face with the metal or liquid interfaces. To facilitate the mathematics, a geometrical
simplification of the red cell shape is made. The biconcave disc is approximated
by a torus of thickness 2R, where R = o-8 /tm; the torus is then imagined to be cut
and straightened into a cylinder of circular cross-section. The length of this cylinder,
L = 18 fim, is equal to the length of the Pappus line which runs axially around the
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V. A. Parsegian and D. Gingell
torus. This simplification requires that the interaction between cell and interface be
dominated by parts of the cell nearest the wall. This condition is met provided the
minimum separation d is much less than the dimensions of the red cell and much
greater than the Debye salt screening length of the ionic medium. We shall consider
only the case where the cell is parallel to the interface, since this is its characteristic
orientation in adhesion.
In order to reduce the electrostatic interaction to a manageable form we use the
method of Brenner & Parsegian (1974) to simulate large cylinders by line charges,
enabling one to deal with cylinders bearing high surface potential. We first consider
a cylinder of radius R bearing a surface potential \jra. The potential in reduced units
is y = ei/r/kT, where e = unit ionic charge, k is Boltzmann's constant, and T
absolute temperature. For red cells, in the dilute salt solution we shall consider
surface potential sufficiently large so that 3^ > 1 (e.g. for xjra = 80 mV, y3 x 3-15).
The object of the simulation method is to mimic the electrostatic potential in the
region where y <^ 1 as though the potential emanated from a hypothetical line
charge of density vh per unit length on the cylinder axis. The potential \jr{r) in this
region is
Here r is the distance from the cylinder axis, e the dielectric constant of the
(water) medium, KQ the zero-order modified Bessel function of the second kind
and K the Debye constant in terms of the salt concentration n,
K2 = Snn^/ekT.
(2)
Equation (1) is valid only when ijr <| kT/e « 25-4 mV. It is therefore invalid to
obtain vh by equating ijf in equation 1 with i/ra at the cylinder surface r = R.
Alamov (1963) has pointed out that in cases where R ^> I/K, the operator
-\
A2 = --(
rdr\dr)
in the Poisson-Boltzmann equation can be approximated by the planar term 82/dr2
near the cylindrical surface,
Equation 3 was solved exactly for y in the region R < r ^ R + t so that y equals
ys at the cylinder surface r = R. We then solve for i^ by matching ijr(r) in value and
slope in equation (1) to the values at R+t derived from equation (3). The distance
t is taken sufficiently large such that y <^ 1 for r = R +1. One must ascertain that
all computations are insensitive to the value of t chosen.
In practice it is convenient to take the 1st integral of equation (3),
(4)
where Kx is the derivative of Ko.
Calculation of interaction forces
153
We integrate numerically from r = R + t, where y(R +1) = AK0 (n(R + t)), to
r = R in order to find that value of A for which y(R) = ys. This A automatically satisfies the condition that the slope and value of the potential be
continuous at r = R + t. From A the required vh is obtained since
The complete electrostatic potential ijrp emanating from the planar surface with
which the cell is interacting can be written exactly. At large distances x from the
surface this potential has the form
kT
*p = 4 — tanh ( V 4 ) « - r a .
(6)
where y0 is the reduced potential at the planar surface.
As explained elsewhere (Brenner & Parsegian, 1974) when there exists a continuous
region between two bodies where the sum of the reduced potentials, y = ei/r/kT <^ 1,
the force between them can be written exactly as the gradient of the potential \jrp
of one body times the hypothetical charge vh on the other. This condition is satisfied
when surface-to-surface separation d exceeds several Debye lengths. The repulsive
force on a rod of length L is then
kT
FTep = -4*L^—tanh(y o / 4 ) e -<*,
(7)
where x = R + d is the distance from planar surface to hypothetical line charge.
FORCE OF REPULSION FROM A METAL ELECTRODE
We are now equipped to consider the data of Gingell & Fornes (1975, 1976) for
percent adherent cells vs. charge density on the lead electrode. The cells adhere
progressively less as the electrode charge density increases from —25000 to
— 45 000 esu/cm2. We will convert these charges to double layer surface potentials
in 1-2 mM NaF (Debye length = 8-8 nm) and compare the electrostatic double layer
force vs. separation curves for these potentials.
The potential y0 is obtained from the surface charge density q via the GouyChapman equation
.
(8)
The surface charge densities —25000 and — 45 000 esu/cm2 translate to y0 = 7-41
and 8-5, respectively (potentials T/T0 = 185 and 215 mV, respectively). Then tanh (Jj 0 )
become 0-952 and 0-972. One therefore expects a difference of 2/0-95% = 2 - I % in
the repulsive forces at —25000 and —45000 esu/cm2 at any fixed separation.
Since the function tanh( Jy0) which determines the force (equation (7)) is insensitive
tojo at large valuesof potentialy0, the wide spread of cell adhesive behaviour seen when
percent of adherent cells is expressed as a function of electrode charge density
(Gingell & Fornes, 1976) translates into a fairly sharp cut-off when expressed as the
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41
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V• A- Parsegian and D. Gingell
force coefficient tanh (jo/4)- A repulsive force greater than (1-85 x io~2)x e~KX
dyne (0-185 x e-^/tN) (corresponding to —35000 esu/cm2 on the electrode) can
prevent half the red cell population from settling down and sticking to the metal surface.
DEDUCTION OF ATTRACTIVE FORCE
From the repulsive force we can infer the attraction that causes cells to stick to
the Pb electrode. Adhesion in the reversible region ( — 25000 to 45 000 esu/cm2)
exhibits no hysteresis, i.e. the repulsive force which prevents a given fraction of
cells from adhering as they settle onto the metal is identical to that which evicts the
same percentage of cells previously stuck under conditions of lower repulsion.
Consequently, for cells hanging under the electrode, the average attractive force is
equal to the average repulsive force plus the force of gravity on the cell. The latter
is calculated from red cell volume (85 /im3) and relative density in water (0-096 g cm-3)
to be 8 x io- 8 dyne (o-8^>N).
We make the provisional assumption that the attractive force is electrodynamic,
and calculate the attractive (Hamaker) force coefficient A(x) for cylinder-plane
geometry described above.
F att = LA(xy(iR)/8di.
(9)
For convenience in computation and ease in comparison with other estimates, we
have chosen the form of non-retarded van der Waals interaction and lumped all
retardation effects into the Hamaker function A(x). Such a procedure has been
employed and explained elsewhere (Parsegian, 1975). The actual expression used
has been taken from Langbein (1974) and written down in the limit where separation
d <^ R, the cylinder radius.
Also for convenience we use a form which assumes cylinders of uniform polarizability. (Were we to be computing cell-substratum attraction from polarizability
data, rather than inferring an attraction from experimental data, we would be obliged
to recognize the layered structure of the cell surface.) Crudely combining both
retardation and structure into A(x) should suffice for the present order-of-magnitude
estimates.
Comparing the electrostatic plus gravitational force (FTep) + mg with an attractive
force Fait at an electrode charge where half the cells fall off, we find that a secondary
minimum energy well of depth 5&T requires A(x) = 2-5 x io~14 erg (2-5 x io~21 J):
the balance between attraction and repulsion is found to occur at 140 ran. A weaker
attractive force would be unable to hold the cell while a stronger one could not be
overcome by electrostatic repulsion from the electrode since the repulsive force is
insensitive to electrode charge (equations (7) and (8)).
In principle it is possible to extract a second estimate of the attractive force from
this experiment. Since cell adhesion is irreversible at electrode charge more positive
than some critical value between o and — 5000 esu/cm2 we may assume that the
cell has been pulled into a primary minimum: in other words the repulsive barrier
to close approach has fallen sufficiently to allow molecular contact. Knowing this
critical value the attractive force just sufficient to pull the cell past the repulsive
Calculation of interaction forces
155
barrier can be calculated. For electrode charges more negative than — 1000 esu/cm2,
the force coefficient 4 tanh {\yQ) from equation 7 is insensitive to potential, but
the force varies very rapidly in the region from o to— 1000 esu/cm2 (y0 = 0 to 4).
Consequently far greater accuracy is required in the determination of the critical
charge than is available from the experiment.
FORCES OF ATTRACTION AND REPULSION AT AN O I L / \ V A T E R INTERFACE
We next treat in an analogous manner the data for the oil/water interface (Todd &
Gingell, 1980). We take first the region of reversible adhesion. In 0-3 mM NaCl half
the red cells fall off the interface. In this concentration the interface has an electrophoretic zeta potential of — 30 mV, whose origin is uncertain, while that of the
red cells is — 95 mV. The calculated electrostatic force of repulsion acting on a cell
approaching the interface is found to be 0-014 e~*x dyne (0-14 e*z/tN). Assuming
an electrodynamic force of the form given by equation 9 we find that the average
cell sits in a potential energy minimum of depth ^kT at a separation from the interface
of 230 nm. The Hamaker function for the attractive force is found to be 9 x io~14 erg
(9 x 10-21 J) for a skT well depth.
These values of A(x) inferred for both systems are comparable to attraction
coefficients computed for hydrocarbons in water (Ninham & Parsegian, 1970), polystyrene in water (Parsegian, 1975) or between phospholipid bilayers in water (Parsegian, Fuller & Rand; Lis, Rand & Parsegian, in preparation).
Using epi-illumination interferometry we have found that cells in 0-3 mM NaCl
adhere to the oil/water interface with a closest approach distance exceeding 100 nm
as judged by white light fringe colours and quantitative photometry (Gingell &
Todd, 1980). In white light the colour of the closest approach region is predominantly
pale yellow but shows a flickering effect due to colour changes as the cells oscillate
perpendicularly to the interface by Brownian motion. Progressive increase in the
salt concentration while the cells are under continuous microscopic observation
produces a colour change at the closest approach regions from pale yellow through
pale grey to dark grey and finally the image goes black as the separation falls to
around zero nm. Taken with the observation of perpendicular Brownian motion, it
shows that the cells are not rigidly held at any particular distance from the interface.
Such salt-concentration-dependent separation is in accord with the idea of a longrange force-balance involving electrostatic repulsion whose range decreases with
increasing salt concentration.
Although we cannot unambiguously exclude the presence of extended glycoprotein molecules linking cell to substratum at the very long separations detected
optically, the closeness of the experimentally deduced and theoretically predicted
Hamaker functions supports the notion of a long range force-balance.
The observation of Gingell & Todd (1975) that red cells cannot adhere to an oil/
water interface covered with behenic acid at pH > 8 in 145 mM NaCl does not
seem to fit within the framework developed here. At such salt concentrations electrostatic forces at separations greater than 15 nm are so strongly screened as to be
negligible compared to the attractive dispersion forces which are expected to be
156
V. A. Parsegian and D. Gingell
strong enough to keep the cell near the interface. This eviction of cells from a negatively charged interface is equally difficult to rationalize as a break in filamentous
attachment. The fact that in 145 mM NaCl the cells always adhere to an uncoated
interface or to one with a positively charged monolayer or to a negatively charged
behenic acid-covered interface for pH < 8 (where the behenic acid is not completely
ionized) does imply some response to electrostatic forces. Eviction from the highly
charged monolayer might reflect some mechanical idiosyncrasy of a monolayercovered interface.
A second problem is that reversibly adherent cells do not slide when the Pb/water
interface is tilted. In the case of the oil/water system, ability to slide is not a useful
sign of long-range attachment since such movement should occur also if the cell is
in molecular contact with the oil. The absence of sliding on Pb might be a consequence
of restrictions on movement near surfaces.
SIGNIFICANCE OF ATTRACTIVE FORCES
Are the attractive electrodynamic forces inferred here large enough to be significant
in adhesion? We expect Hamaker coefficients of about 5 x io~14 erg (5 x 10-21 J). This
interaction acting between planar membranes (Parsegian & Gingell, 1973) would
require 2-6 x io8 dyne/cm2 (2-6 x io 1 N/cm2) for separation from 1 nm. However
if an attractive force of this size is complemented by an electrostatic repulsion in
145 mM NaCl then the force required to separate membranes from the secondary
minimum is only ~ io 3 dyne (iomN) a force which is modest compared with
measured forces of cell adhesion (see Gingell & Vince, 1979).
We do not believe that long-range secondary minimum association is a quantifiable
concept for studying contact between cell surfaces in physiological saline, since the
predicted separation, 7-5 nm, is probably less than the distance which membrane
glycoproteins protrude from the surface. It would seem that the electrodynamic
forces which we have measured are strong enough to contribute to adhesion only at
'molecular contact' where a different theoretical formalism is required. This has
been discussed more fully by Gingell & Vince (1979).
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PARSECIAN,
(Received 8 May 1979)