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.

MATERIALS AND METHODS

Particles and Surfaces

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.

Flow Chamber and Motion Analysis

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.

Derivation of Particle-Substrate Separation

Analysis of Arrest Frequency

Analysis of Arrest Duration

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.

Comparison between Theoretical and Experimental Distributions of Arrest Duration

Confocal Microscopy

Kinetics of Fluorescence Release by Labeled Beads

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.

Determination of Ligand Density on Derivatized Surfaces

Determination of the Density of Available Antibody Sites on Spherical Beads

RESULTS

Beads Moving in Contact with the Chamber Floor Can Only Be Recognized through Velocity Measurements

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.

Most Recorded Particle Arrests Are Mediated by Specific Bonds

The Initial Rate of Cell Detachment Is Independent of the Shear Rate

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.

Experimental Values of Arrest Durations Corresponding to the Highest Antibody Dilutions May Be Fitted with a Quantitative Model of Bond Formation Involving a Single Bond and a Two-step Binding Process

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.

When Beads Are Coated with Limiting Dilutions of Specific Antibodies, the Binding Rate Is Roughly Proportional to the First Power of the Surface Density of Binding Sites

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.

The Rate of Particle Detachment Is Much Higher Than the Spontaneous Rate of Bond Dissociation Measured on Soluble Molecules

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.

DISCUSSION

The main purpose of this work was to achieve a direct determination of the lifetime of ligand-receptor bonds involving particle-bound molecules.

There Is a Basic Difference between the Adhesion of Receptor-bearing Beads and Cells to Ligand-coated Surfaces

The Arrests Observed with Particles Coated with Low Concentrations of Specific Antibodies Are Mainly Initiated by Single Ligand-Receptor Interactions

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 2aL, 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.

Our Experimental System May Yield Fairly Precise Information on the Influence of a Force on the Lifetime of Ligand-Receptor Association

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.

The Distribution of Arrest Durations Is Quantitatively Consistent with a Model Involving a Two-step Association between Individual Ligand and Receptor Molecules

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.

APPENDIX

Theoretical Distribution of Arrest Durations

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:

Using and , we obtain:

Finally, and yield:

FOOTNOTES

*

This work was supported by a grant from the Association pour la Recherche sur le Cancer. The costs of publication of this article were defrayed in part by the payment of page charges. This article must therefore by hereby marked ``advertisement'' in accordance with 18 U.S.C. Section 1734 solely to indicate this fact.

§

To whom correspondence should be addressed. Tel.: 33-91-26-03-31; Fax: 33-91-75-73-28.

()

We are grateful to Prof. Evan Evans for pointing out the importance of the difference of the forces experienced by the sphere and the bond. The problem of determining the force on a single cell exposed to a laminar shear flow on a surface was first addressed by Schmid-Schoenbein et al.(30) .

()

Note that this finding may be dependent on our model. While this paper was being submitted, an analysis of the interaction between blood neutrophils and P-selectin-coated surfaces was reported(35) . The lifetime of P-selectin-ligand bonds was estimated at about 1 s in accordance with our estimate of about 2 s for neutrophil-E-selectin interaction(17) , but no transient state was detected.

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