(Received for publication, April 11, 1995; and in revised form, July 18, 1995)
From the
It is not well known how the kinetic constants of association
between soluble receptors and ligands may be used to predict the
behavior of these molecules when they are bound to cell surfaces.
Spherical beads were coated with varying densities of anti-rabbit
immunoglobulin monoclonal antibodies and driven along glass surfaces
derivatized with rabbit anti-dinitrophenol. Particle motion was
analyzed. The velocity, attachment frequency, and duration of binding
events were determined on individual particles. It was found that i)
beads exhibited frequent arrests lasting between a few tenths of a
second and more than one minute; ii) when antibodies were diluted, the
median arrest duration remained fairly constant (1 s) whereas
binding frequency varied as the first power of the antibody
concentration, suggesting that most particle arrests were due to the
formation of a single bond; iii) when the shear rate was increased
7-fold, the duration of transient binding events remained constant. The
disruptive force exerted on attachment points was estimated to range
between about 6 and 37 piconewtons; and iv) the distribution of arrest
durations suggested that binding was not a monophasic reaction but
involved at least one intermediate step. Therefore, transient binding
events reflected the formation of unstable associations that are not
detected with standard techniques.
An obvious requirement for a molecular understanding of cell adhesion would be to obtain a precise knowledge of the rates of bond formation and dissociation between membrane-associated receptors and ligands. Indeed, it was recently emphasized that the outcome of an intercellular contact might be more dependent on the kinetics than the affinity of interaction between ligands and receptors(1, 2, 3) . Thus, the capacity of adhesion molecules such as selectins to allow the rolling of leukocytes along endothelial cells in flowing blood was suggested to rely on a particularly high value of kinetic constants(3) . Also, when a first bond occurred between a cell and another cell or surface, a critical parameter of adhesion is the ratio between the rates of dissociation of the first bond and formation of additional interactions(4) .
However, to our knowledge, no previously reported methodology allowed a direct measurement of the lifetime of interactions between particle-bound molecules(5) . Tha et al.(6) used a travelling microtube to study the time and force dependence of rupture of antibody-mediated erythrocyte doublets. However, they did not study very transient attachments. Wattenbarger et al.(7) studied the adhesion of glycophorin-containing liposomes to a lectin-coated surface in shear flow. Although they studied the motion of individual particles, they did not present quantitative data on short-term arrests. Other experiments done with the parallel plate flow chamber yielded direct information on binding efficiency and binding strength rather than binding kinetics(8, 9) . Also, Evans et al.(10) performed micromanipulation to determine the mechanical resistance of molecular point attachments between erythrocytes. However, the contact time preceding separation was kept constant, which prevented the authors from obtaining any information on the natural lifetime of labile bonds. Recently, several authors used atomic force microscopy to study the interaction between individual surface-bound molecules(11, 12) . They reported information on binding strength rather than kinetics.
This emphasizes the importance of the theoretical framework elaborated by Bell (13) to relate the behavior of surface-bound molecules to well known kinetic and thermodynamic constants of association between soluble receptors and ligands (see also (14) for additional information). The basic idea was to represent the interaction between molecules A and B as a two-step process. The first step is a purely diffusive encounter between molecules A and B, which approach into sufficiently close proximity to allow bond formation. Kinetic parameters can be estimated with standard diffusion theory. The second step, i.e. molecular association, is assumed to be described with the same constants when molecules A and B are free or bound to surfaces. The numerical values of these parameters may thus be derived from experimental data obtained on soluble forms of receptors and ligands. The limitation of this approach is that i) the reaction is assumed to be monophasic; ii) accurate information is required on the mobility of reacting molecules; iii) drastic assumptions are required to account for the dependence of bond formation on the distance between interacting surfaces; and iv) Bell's theory could only be checked through theoretical models involving adjustable parameters(15, 16) .
It was therefore felt useful
to develop an experimental methodology allowing direct measurement of
the lifetime of individual ligand-receptor bonds involving
surface-bound molecules. The basic idea was to study the motion of
receptor-bearing cells or particles along ligand-coated surfaces under
laminar shear flow. The hydrodynamic force was less than the reported
value of the mechanical resistance of associations between biological
molecules (i.e. several tens of
piconewtons(5, 10, 11, 12) ). This
approach was applied to human neutrophils interacting with endothelial
cell monolayers (17) and murine lymphoma cells moving along
antibody-coated surfaces(4) . The wall shear rate was a few
seconds, corresponding to an hydrodynamic drag of a
few piconewtons. It was indeed possible to detect transient cell
arrests that were probably due to the formation and dissociation of a low number of molecular bonds. However, two problems were
raised by this approach. First, it was difficult to define cell arrests
with high accuracy due to spontaneous velocity fluctuations and low
velocity. Second, it was difficult to prove that observed arrests were
due to single molecular bonds. The purpose of the present work
was to overcome these difficulties with a better suited model.
Particles were small spherical beads (2.8 µm diameter). This
improved the accuracy of determination of arrest duration because the
motion of spheres was more regular than that of actual cells, and,
since the hydrodynamic drag is proportional to the square of particle
radius, whereas the velocity is proportional to the first power of this
radius(18) , it was possible to achieve higher particle
velocity without increasing the hydrodynamic force, thus improving the
accuracy of time determinations(4) . Spheres were coated with
varying amounts of anti-rabbit immunoglobulin antibodies, and they
moved along surfaces derivatized with rabbit immunoglobulin. Because
molecules were not expected to exhibit free lateral diffusion on the
sphere surface, the occurrence of multiple cell-substrate
molecular bonds became more and more unlikely when dilution was
increased. Analysis of experimental data strongly suggests that bond
formation was not monophasic and that our method allowed to detect
incomplete binding states that were not apparent with standard
approaches.
Glass coverslips were coated with rabbit immunoglobulins with a modification of a method described by Michl et al.(19) as described previously(4, 20) . Briefly, they were washed with sulfuric acid, then rinsed in distilled water, and air-dried, and they were then incubated for 30 min at room temperature with 1 mg/ml polylysine (Sigma, molecular weight > 300,000) and washed in phosphate-buffered saline. They were then incubated another 30 min in the dark with 16.8 mg/ml 2,4-dinitrobenzenesulfonic acid (Eastman Kodak, Rochester, NY) in pH 11.6 carbonate buffer. Finally, they were treated with 1.2 mg/ml rabbit anti-dinitrophenol antibodies (Sigma) and washed in phosphate buffer containing 2 mg/ml bovine albumin before use.
The chamber was set on the stage of an inverted microscope (Olympus
IM) bearing a 100 lens. The microscope was equipped with a SIT
video camera (Model 4015, Lhesa, Cergy Pontoise, France), and all
experiments were recorded with a Mitsubishi HS3398 tape recorder for
delayed analysis. All individual beads that were in apparent contact
with the chamber floor were studied. In most cases, the duration of
individual arrests was determined manually, using a computer-driven
time counter. The accuracy of time determinations was estimated to be
about 0.2 s. Further analysis was performed as described
previously(4, 21) . Briefly, the video signal was
processed with a real time digitizer (PCVision+, Imaging
Technology, Bedford, MA). Pixel size was 0.17 µm. A cursor driven
by the computer mouse was superimposed on the microscope image. Small
(32
32 pixel) images pointed with the cursor in order to
surround the analyzed bead were continuously transferred to the
computer memory for delayed determination of the cell position. In this
case, the resolution was limited by the accuracy of position
determinations, because a particle might move by less than half a
micrometer during a 0.08-s interval.
The probability P(t) for a particle bound in
state (AB) at time 0 to remain bound at time t was
derived as described in the ``Appendix.'' An analytical
formula allowed exact determination of P(t) when
parameters k
, k
, and k
were
varied.
First, the intrinsic fluorescence of a typical sample of 20-25 particles was determined. Fluorescein-labeled rabbit immunoglobulins (Jackson ImmunoResearch Labs., Inc., West Grove, PA) were then added, and another set of fluorescence determinations was performed after 30 min of incubation. Finally, the chamber was washed, and fluorescence was determined 5 min later.
Figure 1:
Typical images of flowing particles.
Spherical beads with 1.4-µm radii were driven along a glass surface
with a wall shear rate of 11 s. The velocities of
five individual beads are shown. Particles with a velocity lower than
about 25 µm/s are not markedly different from bound ones (zero
velocity). The faster particle (54.3 µm/s) is obviously out of
focus. The white bar in the lower left is 5
µm.
Figure 2:
Typical velocity distribution. The
velocity distribution of a sample of 73 beads coated with irrelevant
(anti-CD14) antibodies and subjected to a wall shear rate of 11
s is shown.
Figure 3:
Effect of wall shear rate and specific
antibody dilution on arrest duration. In 12 series of experiments,
beads coated with different proportions of specific anti-rabbit
immunoglobulin monoclonals were subjected to hydrodynamic flows of
varying shear rate. Individual particles were followed for
determination of the number and duration of transient or durable
arrests during their passage across a microscope field. The values of
arrest lengths measured in about two to three experiments were pooled
and ordered, and the fraction of cells remaining bound at time t after their initial arrest was plotted versus time in all
tested conditions. Wall shear rates were 11 s, 22
s
, 44 s
, and 72 s
as indicated. Specific antibodies were used pure (A) or
diluted at 1/10 (B), 1/100 (C), or 1/1,000 (D) with irrelevant antibodies. Note that experimental points
are not displayed as visible symbols in order to make the figure
legible.
The initial rate of bead detachment was approximated as the slope of regression lines determined with arrests lasting 1 s or less. The correlation coefficient ranged between 0.813 and 0.992 (mean 0.947). Although experimental curves sometimes displayed significant curvature over this interval, it seemed difficult to consider a shorter period of time because the number of points might be too low and the accuracy of time determinations was too low to warrant such attempts. Results are shown in Table 2. Two main conclusions were suggested: i) the rate of particle detachment was not markedly dependent on the shear rate within the studied range, and ii) the detachment rate was similar with the lowest two antibody concentrations used. An attractive interpretation of these findings would be that arrests observed with 1/100 or 1/1,000 antibody dilutions involved isolated molecular bonds and that the duration of these bonds was not affected by shear forces within the studied range, which provided a minimal value of bond strength. The consistency of this hypothesis with experimental data was thus subjected to a quantitative test.
Figure 4:
Typical fit between experimental data and
theoretical model. The distribution of arrest durations was determined
on spheres coated with 1/1,000 specific anti-rabbit immunoglobulin
antibodies antibodies and subjected to the lowest flow rate (11
s). A total number of 154 arrests were recorded, and
the fraction of arrests lasting at least time t was plotted versus t. The experimental curve is displayed as a full
line. The broken line represents the best theoretical fit
obtained with the model described in the ``Appendix.'' The
value was 13.9.
However, this agreement between theoretical and experimental data does not formally prove that we were dealing with single molecular bonds. Indeed, similar results could be obtained if a fixed minimal number of bonds were required to mediate cell arrest. Therefore, limiting dilution analysis was performed to address this point.
Figure 5:
Dependence of arrest frequency on specific
antibody concentration. Spheres were coated with a mixture of
irrelevant antibodies and anti-rabbit immunoglobulin diluted at
1/1,000, 1/2,500, 1/5,000, 1/7,500, and 1/10,000. They were then driven
along rabbit immunoglobulin-coated glass surfaces with a wall shear
rate of 11 s. The fraction of beads displaying at
least one arrest was calculated and used to determine the binding
parameter b using . Each point was determined
after studying between 69 and 321 individual beads. The uncertainty on
the determination of the fraction of beads with at least one arrest was
calculated following (23) and is shown as an error bar (± S.D.). The slope of the regression line is
1.08.
First, glass surfaces were coated with rabbit immunoglobulins as described, and the surface density of these molecules was determined with indirect immunofluorescence and confocal microscopy. No substantial release was detected during the first 3 h following preparation (not shown).
Secondly, particles were labeled on the stage of a confocal microscope. In a representative experiment, the mean fluorescence of unlabeled particles was 140 ± 9.6 (57 particles). When labeling solution was added, the fluorescence rose to 1446 ± 138 (n = 15) after a 30-min incubation on the microscope stage. Finally, when beads were washed with fresh medium, the fluorescence was not significantly changed 5 min later (1531 ± 78, n = 24). It is concluded that no significant loss of fluorescence occurred during the first 5 min following the removal of labeling molecules.
Thus, bead detachment was at least one 100-fold more rapid than that of isolated molecules. Further, the aforementioned results did not support the hypothesis that shear forces might be responsible for this rapid separation.
The main purpose of this work was to achieve a direct determination of the lifetime of ligand-receptor bonds involving particle-bound molecules.
First, when specific
antibodies were diluted 1/1000, the site density was about 3.5
sites/µm. The contact area between the bead and the
surface may be defined as the area where the distance between surfaces
is less than the sum L of the lengths of a rabbit
immunoglobulin (on the chamber floor) and a mouse immunoglobulin (on
the bead). From elementary geometrical formula, this area is
2
aL, where a is the sphere radius. Because L is about 0.02 µm (corresponding to four times the length of a
Fc or Fab fragment of an immunoglobulin molecule(26) ) and a is 1.4 µm, the contact area is about 0.17
µm
. If there is on average less than one mouse
anti-rabbit Ig molecule in this region, it is quite unlikely that there
would be two molecules simultaneously interacting with an antigen site
on the surface.
Second, if n bonds were required to mediate
a detectable arrest, the arrest probability would vary as the n power of specific antibody concentration under
conditions of limiting dilution (see Appendix 2 of (27) ). As
shown on Fig. 5, limiting dilution experiments support the
hypothesis that arrests are mediated by a single bond, because the
logarithm of arrest probability varied as the 1.08th power of the
logarithm of antibody concentration.
where a is the sphere radius and µ is the
medium viscosity. As shown on Fig. 6, the force T experienced by a single bond of length L much smaller
than a is():
Figure 6:
Distractive force experienced by a single
molecular bond holding a sphere under laminar shear flow. Four
equations were used to calculate the tension T of the bond,
reaction R of the substrate, angle between the bond and
the substrate, and angle
describing the sphere position.
Equations a and b state that the normal and parallel
components of total applied force is zero. Equation c states that the
torque at point M is zero, and Equation d is a
geometrical relationship between
and
. When a/L is much smaller than unity, angle
is close
to 90 °C and angle
is close to zero, leading to .
Using 20 nm for L (see (26) and above) and
considering spheres of 1.4-µm radius embedded in a medium of 0.001
Pas viscosity, such as water at 20 °C, we obtain:
where G is in s. Thus, under our
experimental conditions, the applied force (T) ranged between
5.6 and 36.7 piconewtons. It must be emphasized that this estimate is
only weakly dependent on the numerical value of parameter L.
The results shown on Table 2and Table 3suggest that the
lifetime of antigen-antibody bonds we studied was not
substantially reduced by this treatment. This conclusion is consistent
with previous estimates of binding
strength(6, 10, 28, 29) . Further,
our experimental system may provide additional information by allowing
simultaneous determination of applied force and bond dissociation rate.
More quantitatively, as exemplified on Fig. 4, the overall pattern of binding curves displayed on Fig. 3could be reproduced with theoretical data based on a two-step model of molecular association involving three adjustable kinetic parameters. As shown on Table 3, there was in some cases a significant discrepancy between experimental data and the best theoretical fit. We think that this did not disprove our model, because this discrepancy might be due to the infrequent formation of multiple bonds. Indeed, the agreement between experimental and theoretical curves was on average far better with the highest dilution of specific antibodies.
Further, we wish to emphasize that the existence of an
intermediate step seems required to explain the difference between the
lifetime of antigen-antibody bonds involving soluble and
particle-bound molecules. If we assume that the hydrodynamic drag
exerted on bound particles is not sufficient to substantially reduce
the bond lifetime, it is difficult to understand why the interaction
between flowing beads and substratum lasted only a few seconds. Indeed,
the lifetime of adhesions between soluble rabbit anti-dinitrophenol
antibodies and substratum may be higher than several hours, and the
lifetime of interactions between mouse anti-rabbit immunoglobulin and
these immunoglobulins is higher than several minutes (see
``Results''). This apparent discrepancy is clearly alleviated
if the arrests we detected reflected a transient binding state.
Further, other studies made on noncovalent ligand-receptor interactions
revealed such intermediate states (31, 32, 33) . Therefore, our three-parameter
model may be considered as the simplest way of interpreting
experimental data. ()
In conclusion, we visualized the formation and dissociation of individual ligand-receptor bonds between molecules linked to macroscopic bodies.
Let P and P
be the
respective probabilities for the bond to be in state (AB)
and (AB)
. At time 0, P
is equal
to 1 and P
is zero. At time t, the
probability that the bead is bound is P
+ P
. Using , we may write after simple
algebraic manipulation:
This set of equations is readily solved by looking for a linear
combination V of P and (P
+ P
) such that yields:
where a and are constants(34) . We find
two solutions: