Measuring the lifetime of bonds made between surface-linked molecules.

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.

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 (Ϸ1 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 membraneassociated 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. How-ever, 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 surfacebound molecules to well known kinetic and thermodynamic constants of association between soluble receptors and ligands (see also Ref. 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 Ϫ1 , 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 m 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.

Particles and Surfaces
Particles were spheres of 2.8-m diameter and 1.3 g/liter density (Dynabeads M280, Dynal, supplied by Biosys, Compiègne, France). These spheres were coated with streptavidin, a high affinity ligand for biotin. Before each experiment, 50-l aliquots of bead suspension (ϳ7 ϫ 10 8 /ml) were incubated for 20 min at room temperature with an equal volume of a mixture of biotinylated mouse monoclonal antibodies at a final concentration of 0.2 mg/ml. These antibodies were an IgG2a specific for the Fc fragment of rabbit IgG (clone IgL 173, Immunotech, Marseille, France) and an IgG2a specific for CD14 antigen (clone UCHM1, Sigma) that was used as a control with irrelevant specificity to dilute anti-rabbit immunoglobulin antibodies. Beads were washed in phosphate-buffered saline (pH 7.2) supplemented with 2 mg/ml bovine albumin in order to prevent nonspecific adhesion, and they were used at 10 7 /ml in the same solution throughout all subsequent adhesion experiments.
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 Ͼ 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
We used a previously described apparatus (4,21). Briefly, a rectangular cavity of 17 ϫ 6 ϫ 0.2 mm 3 was cut into a Plexiglas block. The bottom was a glass coverslip (22 ϫ 10 mm 2 ) that was coated with immunoglobulins as described above. The flow was obtained with a plastic syringe mounted on an electric syringe holder.
The chamber was set on the stage of an inverted microscope (Olympus IM) bearing a 100ϫ 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 (PCVisionϩ, Imaging Technology, Bedford, MA). Pixel size was 0.17 m. A cursor driven by the computer mouse was superimposed on the microscope image. Small (32 ϫ 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
The distance between flowing beads and the chamber floor was estimated with theoretical data provided by Goldman et al. (18). The numerical results displayed in Table II of this paper were used to plot cosh Ϫ1 (z/a) versus U/aG, where a is the sphere radius, G is the wall shear rate, z is the distance between the sphere center and the substrate, and U is the sphere translational velocity. This yielded a smooth plot allowing fairly accurate regression with a second order polynoma (when U/aG ranged between about 0.4 and 1.2), leading to the following formula:

Analysis of Arrest Frequency
A binding efficiency parameter b was defined as described previously (22) by writing as b⅐dx the probability that a particle rolling along the chamber floor exhibited an arrest during an elementary displacement of length dx. The probability P that a particle displayed at least one arrest during its passage across the microscope field of width w (i.e. 86 m under our conditions) was therefore:

Analysis of Arrest Duration
The numerical values of the duration of typically 100 arrests observed under given experimental conditions were ordered in order to build a numerical plot of the variations of the number of particles remaining bound after a period of time t following an initial arrest versus time. It was reasoned that a quantitative interpretation of experimental data was not possible at high binding site density, because the rate of formation of sequential bonds was probably dependent on the relative localization of binding sites in contact areas (in absence of lateral diffusion). Therefore, we only considered the low density limit. We assumed that bond formation between molecules A and B was a two-state process: The probability P(t) for a particle bound in state (AB) 1 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 d , k ϩ , and k Ϫ were varied.

Comparison between Theoretical and Experimental Distributions of Arrest Duration
Theoretical curves were fitted to experimental data with 2 test (23). Arrest durations were grouped in seven classes: 0 -0.3 s, 0.3-0.6 s, 0.6 -1.2 s, 1.2-2.4 s, 2.4 -5 s, 5-60 s, and 60 s to ϱ. Parameters k d , k ϩ , and k Ϫ were systematically varied with a step of 0.1 in order to determine the values yielding minimal 2 . This procedure was repeated with a step of 0.01 or 0.001 to refine the minimization. Note that the threshold 2 for a 0.05 significance level is 12.9 when the number of degrees of freedom is 6.

Confocal Microscopy
Kinetics of Fluorescence Release by Labeled Beads-Dynabeads were coated with biotinylated anti-rabbit immunoglobulins and deposited on rabbit immunoglobulin-derivatized glass coverslips in a custom-made flow chamber. They were examined with a confocal laser scanning microscope (Leica, Heidelberg, Germany) connected to a desk computer bearing a PCVisionϩ digitizer as described elsewhere (24).
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-First, immunoglobulin-coated surfaces were treated with an excess of fluorescent anti-rabbit immunoglobulin antibodies, and the amount of bound fluorescence/unit area was determined. Second, the fluorescence of calibrated fluorescence beads was measured under similar conditions. The surface density of fluorescent antibodies was 6.2 ϫ 10 3 molecules/m 2 .
Determination of the Density of Available Antibody Sites on Spherical Beads-The confocal microscope was used with the same settings to measure i) the fluorescence of dynabeads coated with pure anti-rabbit immunoglobulin monoclonal and then an excess of fluorescent rabbit immunoglobulins and ii) the fluorescence of a 100 g/ml solution of the same fluorescent rabbit immunoglobulins maintained as a thin layer by depositing a 12 ϫ 12-mm 2 coverslip on a 5-l droplet of this solution (25). It was found that each bead could bind about 85,450 rabbit immunoglobulin molecules (i.e. about 3,460 molecules/m 2 .

Beads Moving in Contact with the Chamber Floor Can Only
Be Recognized through Velocity Measurements-Spherical beads exposed to a shear flow exhibited a very steady motion. As shown in Fig. 1, beads that were apparently in the same plane as surface-bound particles exhibited marked velocity differences, with variations ranging within a factor of two. A typical velocity distribution of spheres seemingly located in the same focus plane is shown in Fig. 2; in the presence of a wall shear rate of 11 s Ϫ1 , the translation velocities ranged between about 7 and 14 m/s. The corresponding sphere-to-substrate distance, as evaluated with Goldman's theory (18), ranged between 4.2 and 207 nm. Because the significance of Goldman's theory is not fully demonstrated at this distance (21), we arbitrarily assumed that the particles were within binding distance from the substratum when the ratio U/aG between the velocity and the particle radius times the shear rate was lower than 0.8, corresponding to a gap width of 10% of the sphere radius. This assumption was somewhat supported by the observation that binding events could occur with all spheres falling within this velocity range.
Most Recorded Particle Arrests Are Mediated by Specific Bonds-A major problem with our experimental approach is that ill-defined "nonspecific" interactions may be responsible for a significant proportion of particle arrests. It was important to assess the importance of these interactions in the present system. As shown on Table I, when the specific antibody dilution and shear rate were varied within a wide range of numerical values, more than 90% of beads exhibited at least one arrest when specific antibody were diluted between 1/1 and 1/1,000, whereas only about 10% of these particles displayed at least one stop in the absence of specific antibody interaction. This point was made more quantitative by pooling results obtained with different values of the wall shear rate and calculating the binding efficiency parameter (i.e. mean number of arrests/m displacement). This parameter was 0.0335, 0.0404, 0.0256, 0.0231, and 0.0013 m Ϫ1 when the proportion of specific antibodies on spherical beads was 1, 1/10, 1/100, 1/1000, and 0, respectively. The binding probability was therefore between 17-and 30-fold higher in presence of specific antibody than when beads were coated with irrelevant antibodies. This finding is consistent with the hypothesis that specific bonds were responsible for most initial arrests.
The Initial Rate of Cell Detachment Is Independent of the Shear Rate-Beads were coated with different proportions of specific antibodies (between 1/1 and 1/1,000) and subjected to different flow rates for determination of the duration of transient arrests. Plots of the fraction of beads remaining bound at time t after arrest versus time are shown on Fig. 3. A total number of 1381 arrests (i.e. about 100 arrests/plot) were observed.
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 II. 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

TABLE I Dependence of arrest frequency on ligand density and wall shear rate
Individual beads were monitored during their passage across a microscope field of 86-m width in the presence of a flow of varying shear rate. The substrate was coated with rabbit immunoglobulin, and beads were derivatized with a constant amount of mouse immunoglobulins comprising a varying proportion of anti-rabbit immunoglobulin monoclonal antibodies. The percentage of beads exhibiting at least one transient or durable arrest is shown (the number of beads with arrest and the total number of monitored beads are in parentheses).

Proportion of specific antibodies
Beads with at least one arrest for a wall shear rate 11  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-In order to achieve a quantitative interpretation of experimental data, 2 analysis was performed to compare theoretical and experimental distributions of arrest durations. In a first series of calculations (not shown), it was checked that experimental data displayed on Fig. 3 could not be accounted for by a one-parameter theory involving a monophasic bonding reaction, with an adjustable off-rate. Indeed, in this case, the experimental curves should be straight lines at high antibody dilution. A three-parameter model involving an intermediate binding state was then considered ("Appendix"). It was found possible to obtain experimental curves fairly similar to experimental data. A typical fit is shown on Fig. 4. The numerical values of fitted parameters are shown on Table III as well as minimal 2 values. Interestingly, no substantial difference was found between parameters obtained with beads coated with 1/100 and 1/1000 specific antibodies, in accordance with the hypothesis that we were dealing with single molecular bonds. Further, when the hydrodynamic force was varied on a sevenfold range, no substantial change of these kinetic parameters was found, in accordance with the hypothesis that these forces were well below the threshold required to break ligandreceptor bonds.
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-The binding efficiency parameter was determined on beads coated with specific anti-rabbit immunoglobulin diluted at 1/1000, 1/2500, 1/5000, 1/7500, and 1/10000. This parameter was plotted versus  antibody concentration with a double logarithmic scale (Fig. 5). The slope of the regression line was 1.08 (correlation coefficient, 0.94), thus suggesting that arrest frequency was indeed proportional to the first power of binding site density, which supported the single bond hypothesis.

The Rate of Particle Detachment Is Much Higher Than the Spontaneous Rate of Bond Dissociation Measured on Soluble
Molecules-Particle-substrate separation might be due to a rupture of either link of the molecular chain mediating attachment. It was of obvious importance to determine the rate of bond dissociation with free ligands.
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 Ϯ 9.6 (57 particles). When labeling solution was added, the fluorescence rose to 1446 Ϯ 138 (n ϭ 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 Ϯ 78, n ϭ 24). It is concluded that no significant loss of fluores-cence 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 Receptorbearing Beads and Cells to Ligand-coated Surfaces-Recently, several authors reported quantitative data on the efficiency of adhesion between receptor-bearing cells and ligand-coated surfaces in flow chamber. When human granulocytes (17), rat basophilic leukemia cells (9) or murine lymphoid clones (4) were studied, an inverse relationship was found between cell velocity and binding probability per m displacement. However, in the present study adhesion efficiency was fairly independent of the particle velocity (Table I). We suggest that the explanation for this difference is that in cellular systems receptor-ligand interaction is mainly diffusion-driven. If the limiting parameter is the time required for cell adhesion molecules to pass through the cell-surface contact area (which may be the tip of a microvillus), the adhesion probability will be proportional to the contact time, thus making the adhesion probability per unit length of cell displacement along the surface inversely proportional to the cell velocity. However, lateral diffusion of antibody molecules is not expected on the surface of the beads used in the present study. Therefore, ligand-receptor interaction may be dependent on particle displacement, making the adhesion probability proportional to the bead displacement rather than the contact time, as found in the present study. For this reason, the bead model cannot represent actual cell behavior. However, this absence of diffusion may provide an unique opportunity to study single molecular bonds without a need for excessive receptor dilution, which would make binding events too rare to be subjected to a quantitative study.
The Arrests Observed with Particles Coated with Low Con- FIG. 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 Ϫ1 ). 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 2 value was 13.9.  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 Ϫ1 . The fraction of beads displaying at least one arrest was calculated and used to determine the binding parameter b using Equation 2. 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 Ref. 23 and is shown as an error bar (Ϯ S.D.). The slope of the regression line is 1.08. Table I, even when specific antibodies were diluted 1/1000 with irrelevant antibodies, beads displayed much more frequent arrests than when they were coated with 100% irrelevant antibodies. Therefore, even with 1/1000 antibody concentration, most arrests were mediated by specific bonds. Two arguments support the view that these arrests were mainly due to the formation of a single bond.

centrations of Specific Antibodies Are Mainly Initiated by Single Ligand-Receptor Interactions-As shown on
First, when specific antibodies were diluted 1/1000, the site density was about 3.5 sites/m 2 . 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 m (corresponding to four times the length of a Fc or Fab fragment of an immunoglobulin molecule (26)) and a is 1.4 m, the contact area is about 0.17 m 2 . 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 th power of specific antibody concentration under conditions of limiting dilution (see Appendix 2 of Ref. 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-Our approach may yield at the same time precise kinetic and mechanical data. Indeed, according to Goldman, Cox, and Brenner (18), a sphere deposited on a plane under a laminar shear flow of shear rate G is subjected to a drag force F and torque ⌫ given by: where a is the sphere radius and 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 1 : Using 20 nm for L (see Ref. 26 and above) and considering spheres of 1.4-m radius embedded in a medium of 0.001 Pa⅐s viscosity, such as water at 20°C, we obtain: where G is in s Ϫ1 . 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 Tables II and III suggest that the lifetime of antigenantibody 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-A first point supporting the validity of our model is that the slopes of experimental detachment curves (Table II) were fairly similar when the specific antibody concentration was 1/100 and 1/1000. This is in accordance with the single bond hypothesis.
More quantitatively, as exemplified on Fig. 4, the overall pattern of binding curves displayed on Fig. 3 could be reproduced with theoretical data based on a two-step model of molecular association involving three adjustable kinetic parameters. As shown on Table III, 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 1 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).
FIG. 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 Equation 6. way of interpreting experimental data. 2 In conclusion, we visualized the formation and dissociation of individual ligand-receptor bonds between molecules linked to macroscopic bodies.

Theoretical Distribution of Arrest Durations
Following Equation 3, we considered the association between two complementary binding sites (A and B) as a two-step process with an intermediate bound state (AB) 1 and a stable bound state (AB) 2 . We considered a spherical bead (coated with A-type molecules) bound to a surface coated with B-type molecules through a single bond. At time 0, the bond is in state (AB) 1 . The system evolution is dependent on three kinetic constants k d , k ϩ , and k Ϫ as recalled below: Let P 1 and P 2 be the respective probabilities for the bond to be in state (AB) 1 and (AB) 2 . At time 0, P 1 is equal to 1 and P 2 is zero. At time t, the probability that the bead is bound is P 1 ϩ P 2 . Using Equation A1, we may write after simple algebraic manipulation: This set of equations is readily solved by looking for a linear combination V of P 1 and (P 2 ϩ P 1 ) such that Equation A2 yields: V ϭ P 1 ϩ a ͑P 1 ϩ P 2 ͒ (Eq. A3) dV/dt ϭ V where a and are constants (34). We find two solutions: Ϫ͑k d ϩ k ϩ ϩ k Ϫ ͒ Ϯ ͱ͑k d ϩ k ϩ ϩ k Ϫ ͒ 2 Ϫ 4k ϩ k Ϫ 2 (Eq. A4) a Ϯ ϭ k Ϫ / Ϯ Using Equations A3 and A4, we obtain: Finally, Equations A4 and A5 yield: ͕͑1 ϩ a ϩ exp͑ ϩ t͒ Ϫ ͑1 ϩ a Ϫ ͒exp͑ Ϫ t͖͒ (Eq. A6)