Size, Kinetics, and Free Energy of Clusters Formed by Ultraweak Carbohydrate-Carbohydrate Bonds

Abstract

Weak noncovalent intermolecular interactions play a pivotal role in many biological processes such as cell adhesion or immunology, where the overall binding strength is controlled through bond association and dissociation dynamics as well as the cooperative action of many parallel bonds. Among the various molecules participating in weak bonds, carbohydrate-carbohydrate interactions are probably the most ancient ones allowing individual cells to reversibly enter the multicellular state and to tell apart self and nonself cells. Here, we scrutinized the kinetics and thermodynamics of small homomeric Lewis X-Lewis X ensembles formed in the contact zone of a membrane-coated colloidal probe and a solid supported membrane ensuring minimal nonspecific background interactions. We used an atomic force microscope to measure force distance curves at Piconewton resolution, which allowed us to measure the force due to unbinding of the colloidal probe and the planar membrane as a function of contact time. Applying a contact model, we could estimate the free binding energy of the formed adhesion cluster as a function of dwell time and thereby determine the precise size of the contact zone, the number of participating bonds, and the intrinsic rates of association and dissociation in the presence of calcium ions. The unbinding energy per bond was found to be on the order of 1 k_BT. Approximately 30 bonds were opened simultaneously at an off-rate of k_off = 7 ± 0.2 s⁻¹.

Introduction

In biological adhesion, ultraweak interactions play a pivotal role to allow for transient assembly and disassembly processes fostering fast and reversible attachment to diverse substrates and adjacent cells. Prominent examples are cell adhesion and immunological recognition processes relying on the fast formation and rupture of many multiple bonds organized in a parallel fashion (1, 2, 3, 4, 5, 6). It is clear that cell tissue formation in embryogenesis and cell migration of motile cells rely on fast and cooperative assembly and disassembly of a large number of parallel bonds formed by specialized adhesion molecules. These bonds need to be sufficiently weak with large off-rates to permit bond failure already at low mechanical forces or on short timescales. Large numbers of bonds with small binding constants but fast kinetics are necessary to efficiently but reversibly glue together rather large objects such as cells where many thousands of bonds participate and at the same time allow these bonds to open quickly if necessary. Prime candidates for ultraweak bonds are protein-carbohydrate or early in evolution employed carbohydrate-carbohydrate interactions (CCIs) (1, 2, 7, 8, 9, 10, 11). Although known for many years, CCIs are among the less investigated and less understood interactions in the context of cell-cell recognition. Early multicellular organisms such as marine sponges employ CCIs allowing the organisms to aggregate through calcium-mediated interactions of sulfonated carbohydrates displayed by cell-surface proteoglycans (1, 2, 12). In eukaryotic cells, glycans are attached to both proteins and lipids, with the cell surface being the basis of the glycocalyx. Among the most prominent candidates for CCIs are the glycan-specific and Ca²⁺-mediated homomeric interactions with Lewis X (Le^X). Le^X is a terminal trisaccharide on cell surface glycans and its carbohydrate determinant 1,3-fucosyl-n-acetyl-lactosamine was initially described on blastomeres of mouse embryos and in embryonal carcinoma cells (13). Eggens et al. (14) were the first researchers to our knowledge, to report on self-associating properties relying on homophilic Le^X-Le^X interactions mediated by Ca²⁺. The molecular basis of this carbohydrate-carbohydrate interaction has been shown by its crystal structure and nuclear magnetic resonance studies (15, 16), while adhesion has also been assessed by atomic force microscopy (AFM) and isothermal titration calorimetry (17, 18).

Although the ultraweak self-association in carbohydrate systems is of great physiological importance, the number of successful approaches to quantify the number of bonds and thermodynamics, as well as kinetics of CCI formation in the context of membranes and therefore confined geometries, is only sparse. While a few single molecule studies exist, which address the dynamic binding strength of potentially only two carbohydrates in contact using atomic force microscopy (19, 20), only a limited number of studies managed to access the free energy of many parallel bonds in the context of lipid bilayers (12, 21). Patel et al. (22) were among the first to address weak self-interactions of heteroxylans by measuring the sedimentation velocity in an analytical ultracentrifuge. More recently, Kunze et al. (21) analyzed the equilibrium fluctuations of single liposomes to reveal how cholesterol and Ca²⁺ influence trans-interactions of Le^X-glycosphingolipids.

The measurement of association kinetics in the context of real membranes poses a great challenge for today’s analytical arsenal comprising SPR, QCM, FT-IR, ellipsometry, etc., all techniques that are either not sensitive enough or unable to mimic the native situation closely enough. Recently, Bihr et al. (23) introduced a new method based on the growth curves of an adhesion domain to assess the association rates of various ligand-receptor pairs embedded in a giant liposome and on the surface. Albeit the dynamic binding strength could not be measured, their approach closely mimics the native situation of cell adhesion.

In the context of model systems, knowledge of the intrinsic rate constants and the participating number of bonds is of prime interest to understand early binding cluster formation in cell adhesion and performance in cell migration.

Here, we present a method based on membrane-coated colloidal beads attached to an AFM cantilever (Fig. 1 a) that allows assessing the number of bonds, the association rate, and binding constant of weak bonds such as those between carbohydrates in the context of supported lipid bilayers. Measuring the adhesion energy as a function of contact time between the membrane-coated probe and the membrane-coated surface, we could model the formation and dissociation of bonds using a kinetic scheme involving lateral diffusion of the lipids. The method allows measuring bond cluster dynamics of extremely weak bonds of only a few k_BT such as that found for homophilic Le^X-Le^X interactions in the presence of calcium. We found that ∼30 bonds are formed in the contact zone, in which each bond contributes only 1 k_BT. Our data also explains the lifetime of tethers pulled from the surface due to adhesion mediated by carbohydrates. The model can be easily extended to assess the binding strength of cell adhesion molecules using even living cells as probes. The data also suggests that cooperativity other than stochastic bond breakage under an external force transmitted through a soft transducer is not necessary to explain the dynamic binding strength. Lateral cis-interactions between adjacent carbohydrate moieties are not necessary to explain a cluster formation that is simply due to a diffusion-reaction scheme.

Figure 1.

(a) Scheme of the membrane probe force spectroscopy setup. (b) Sample force distance curves; marked are the adhesion force $f_A$ and the tether length $l_t$. (c) In situ synthesis of the glycolipids using thiol-maleimide click chemistry. To see this figure in color, go online.

Materials and Methods

Membrane probe force spectroscopy

All lipids were purchased from Avanti Polar Lipids (Alabaster, AL). Lipid mixtures of the desired composition were prepared from stock solutions in HCCl₃ (0.25 mg of total lipids). The solvent was removed in a stream of nitrogen and the resulting lipid film was dried in vacuo at 55°C for at least 3 h. Addition of spreading buffer (250 μL, 20 mM HEPES, 150 mM KCl, 2 mM EDTA, pH = 5.9) and rigorous mixing produced a cloudy suspension of multilamellar vesicles that were transformed into small unilamellar vesicles by 30 min of sonification (Sonopuls HD 2070; Bandelin, Berlin, Germany). As the main membrane lipid, POPC (1-palmitoyl-2-oleoyl-sn-glycero-3-phosphocholine) was used to obtain fluid membranes to ensure receptor mobility. A quantity of 10 mol % of MalChex DOPE (1,2-dioleoyl-sn-glycero-3-phosphoethanolamine-N-[4-(p-maleimidomethyl)-cyclohexane-carboxamide] was included for in situ synthesis of glycolipids, while 1 mol % of BODIPY-labeled lipids (2-(4-4-difluoro-5-methyl-3-bora-3a,4a-diaza-s-indacene-3-dodecanoyl)-1-hexadecynoyl-sn-glycero-3-phosphocholine) was used as a fluorescent dye.

Colloidal probe cantilevers were prepared by gluing a borosilicate microsphere (15 μm diameter; Duke Scientific, Palo Alto, CA) with epoxy resin (Epikote 1004; Brenntag, Mülheim/Ruhr, Germany) to a tipless silicon nitride cantilever (MLCT-O-C; Bruker, Billerica, MA, with a nominal spring constant k_c = 10 pN/nm) using a micromanipulator (Kleindiek Nanotechnik, Reutlingen, Germany) to handle the cantilever (12, 24).

Silicon substrates for SLBs were cut from silicon wafers (Crystec, Berlin, Germany), oxidized in a solution of hydrogen peroxide (30%, 10 μL) and ammonia (25%, 10 μL) in ultrapure water (50 μL) for 20 min at 70° C and cleaned in oxygen plasma for 1 min immediately before membrane spreading to obtain a flat, hydrophilic silicon oxide surface. A quantity of 50 μL of the vesicle suspension (1 mg/mL) were diluted to 500 μL with spreading buffer and incubated on the wafer for 2 h at room temperature followed by extensive rinsing with EDTA-measuring buffer (same composition as spreading buffer, pH = 7.4). The membrane integrity was controlled by fluorescence and AFM imaging and only SLBs with a surface coverage over 95% were used for subsequent experiments.

The membrane covering the colloidal probe was prepared immediately before carrying out force measurements by incubating the colloidal probe in a hanging droplet of small-unilamellar vesicle suspension (80–100 μL, 1 mg/mL) for 15 min followed by rinsing with at least 1 mL of EDTA-measuring buffer.

Synthesis of Le^X-thiol was carried out as described in Tromas et al. (17) and Geyer et al. (25), while Le^X-glycolipids were synthesized in situ by coupling of Le^X-thiol to the MalChex headgroups (Fig. 1 c). The substrate and the probe were incubated simultaneously by rinsing with coupling buffer (same composition as spreading buffer, pH = 6.8, 7 × 1 mL), followed by addition of a solution of the Le^X-thiol in the same buffer (200 nmol in 0.5 mL) to a final concentration of ∼0.08 mM. After 2 h of incubation at room temperature, excess Le^X was removed by rinsing with Ca²⁺-measuring buffer (7 × 1 mL, 20 mM HEPES, 150 mM KCl, 10 mM CaCl₂, pH = 7.4).

Coupling efficiency of Le^X-thiol to the lipid membrane was measured by means of ellipsometry (model No. EP3; Accurion, Göttingen, Germany). Because the refractive index of carbohydrates is roughly equal to the refractive index of water, the binding can hardly be observed directly by ellipsometry. Therefore, competition experiments were performed utilizing binding of a small peptide, H₆WGC, that can easily be followed by ellipsometry. The peptide was added to a bilayer that first reacted with the Le^X thiol. The amount of H₆ WGC covalently binding to the remaining unreacted maleimide groups allows us to determine the amount of reacted Le^X (see the detailed experimental procedure in the Supporting Material).

Force measurements

Force measurements were carried out with a commercial atomic force microscope (model No. MFP3D; Asylum Research, Santa Barbara, CA). The spring constants of the cantilevers were calibrated before the measurements using the thermal noise method, and were found to be in the range of 7–14 pN/nm. The approach velocity was kept constant at 0.5 μm/s with a load force of 200 pN. The retraction velocity was kept constant at 1 μm/s, while the contact time between probe and substrate was varied between 0 s and 5 s. For later analysis, the contact times were corrected for the time span between the initial probe substrate contact to reach the loading force (∼40 ms). For every set of parameters, at least 200 force-distance curves were collected using two different substrates and probes. For each freshly prepared substrate and probe, respectively, a set of measurements was performed before coupling of Le^X-thiol to determine the nonspecific interactions. The influence of calcium ions was estimated by performing control measurements under calcium-free conditions, i.e., in EDTA-measuring buffer, as well as in Ca²⁺-measuring buffer (250 μL). After recording of control curves, Le^X-thiols were covalently coupled to the maleimide bearing lipid bilayers as described above. Thereafter another set of measurements was performed in EDTA- and Ca²⁺-measuring buffer, respectively.

Model

The following subsection describes how the number of bonds, the kinetic parameters, and free energy per bond are estimated from solving reaction-diffusion equations in the contact zone between the colloidal probe and the solid supported membrane.

Unbound membrane-anchored receptors and ligands (identical in our case) are freely diffusing within the opposing lipid bilayers. When two membranes are brought into close contact transient trans-oriented pairs of receptors/ligands might form within a defined contact area. Because the binding is reversible, i.e., spontaneous, thermally driven bond rupture can occur on experimental timescales, an equilibrium between bound and unbound molecules is reached eventually. Notably, the overall free energy of the system comprises the free energy of binding N_bΔF° and an entropic term TΔS° that captures the configuration change of receptors/ligands after partially forming pairs (26, 27). The key step is now to pull apart the bonds quickly enough that the number of bonds that have been formed within the given dwell time rupture adiabatically but slowly enough that the rupture force does not depend on the loading rate, i.e., each single bond (except for the last one) ruptures under equilibrium conditions. Hence, we measure the rupture force of N_b bonds in local binding equilibrium allowing us to compute the free binding energy ΔF° of a single bond (see below).

When the nonspecific work of adhesion in the absence of binding molecules is small compared to the free energy of binding, it can be assumed that the binding energy of two carbohydrate moieties is the major contribution to the work of adhesion. Because the externally enforced unbinding in a colloidal force experiment is fast compared to lipid diffusion, the bond rupture can be described as an adiabatic process with respect to lipid reorganization allowing us to neglect entropic contributions to the free energy during bond rupture and therefore link the work of adhesion W_A directly to the number of bonds $N_b$:

$$W_A\approx N_b\Delta F^{\circ }.$$ (1)

The free binding energy $\Delta F^{\circ }$ can be calculated directly from the reaction rates k_on and k_off at zero external force using $\Delta F^{\circ }=k_{\text{B}}T\text{ln}\left(k_{\text{on}}/k_{\text{off}}\right)$. The number of bonds can be calculated by integrating over the concentration $c_b\left(t,r\right)$ of bound receptors/ligands, which is the product of the molar fraction ${\chi }_b\left(t,r\right)$ of bound molecules and the density of lipids in a lipid membrane ρ. Because we include diffusion and binding kinetics, the molar fractions are functions of both position and time, leading to a dwell time-dependent work of adhesion $W_A\left(t\right)$.

When the AFM-cantilever is retracted from the surface, the bonds connecting the colloidal probe and substrate are cleaved if the restoring force of the cantilever exceeds the critical adhesion force $f_A$ of the bonds (see Appendix). If the contact stiffness, $N_b{k}_s$, with $k_s$ the stiffness of a single bond, is larger than the transducer stiffness $k_c$, a mechanical instability occurs leading to a jump out of contact, preventing the direct determination of the work of adhesion from integration of the force distance curve. In fact, the area under the force curve depends on the transducer stiffness $k_c$ as $W=f_A^2/k_c$ and therefore is not a thermodynamic quantity (28). Therefore, we use a different approach to assess the free binding energy using a self-consistent contact model. We employ the Johnson, Kendall, and Roberts (JKR) (29) model for the adhesive elastic contact, which is suitable for soft materials and adhesion occurring only in a small gap between the opposing surfaces. As outlined in the Appendix for flat cylindrical punches, this continuum approach corresponds to a molecular description of equilibrium rupture of binding clusters in the limit of large bond numbers and low pulling speed. It can be readily shown for flat cylindrical punches that in either case the critical adhesion force scales with the square route of the free binding energy times the bond stiffness $f_A\approx N_b\sqrt{k_s\Delta F^{\circ }}$. For a spherical probe with radius R in contact with a planar surface, the situation is slightly more complicated because the adhesion radius depends also on the adhesion energy. The JKR theory connects the adhesive energy per unit area ${\tilde{w}}_A$ to the critical unbinding force $f_A$ of the cluster measured experimentally (28):

$$f_A=-\frac{3}{2}\pi {\tilde{w}}_{A}R.$$ (2)

The energy surface density ${\tilde{w}}_A$ can be calculated from the work of adhesion as obtained from Eq. 1 and the contact area. In the case of circular symmetry of the contact area, we can use polar coordinates to obtain:

$${\tilde{w}}_A=\frac{W_A\left(t\right)}{\pi r_{\text{JKR}}{\left(t\right)}^2}\approx 2\Delta F^{\circ }\rho r_{\text{JKR}}{\left(t\right)}^{-2}\stackrel{r_{\text{JKR}}\left(t\right)}{\underset{0}{\int }}\text{d}rr{\chi }_b\left(t,r\right).$$ (3)

Staying in the framework of JKR theory, the contact radius $r_{\text{JKR}}\left(t\right)$ between the membrane-coated colloidal probe i and the planar membrane j can be calculated, if the reduced Young’s modulus, $E^{\ast }={((1-{\nu }_i^2)/E_i+(1-{\nu }_j^2)/E_j)}^{-1}$ (with $E_{i,j}$ and ${\nu }_{i,j}$ as the corresponding Young’s moduli and Poisson’s ratios) and the load force $f_L$, are known (28):

$$r_{\text{JKR}}^3=\frac{3R}{4E^{\ast }}\left(f_L+3\pi {\tilde{w}}_{A}R+\sqrt{6\pi {\tilde{w}}_{A}R{f}_L+{\left(3\pi {\tilde{w}}_{A}R\right)}^2}\right).$$ (4)

Because the upper limit of the integral in Eq. 3, $r_{\text{JKR}}$, is therefore a function of W_A, an iterative, self-consistent scheme was used to solve Eq. 3 with Eq. 4. The essential code illustrating the computational approach can be found in the Supporting Material.

The time evolution of the molar fraction ${\chi }_b$ of bound molecules can be described by a rate equation considering both binding and unbinding:

$$\frac{\partial {\chi }_b}{\partial t}=k_{\text{on}}\left(r\right){\chi }_{f,i}{\chi }_{f,j}-k_{\text{off}}{\chi }_b.$$ (5)

The explicit dependency of both ${\chi }_b$ and the molar fraction ${\chi }_f$ of unbound receptors on time t and the radius r was omitted for the sake of readability. Because the unbound receptor is freely diffusing in contrast to the bound receptors/ligands, a diffusion term is added to the rate equation for the molar fraction ${\chi }_f$. The expression is valid for both bilayers covering the colloidal probe i and the planar substrate j:

$$\frac{\partial {\chi }_{f,i}}{\partial t}=\nabla \left(D\nabla {\chi }_{f,i}\right)-k_{\text{on}}\left(r\right){\chi }_{f,i}{\chi }_{f,j}+k_{\text{off}}{\chi }_b.$$ (6)

Here we assume that the diffusion coefficient D of the unbound (free) receptors/ligands is the same inside and outside of the contact area. Binding can only occur in the contact area. The rate of binding $k_{\text{on}}\left(r\right)$ is a function of the radius because it is $k_{\text{on}}=0$ for $r>r_{\text{JKR}}$.

The resulting initial value problem was solved numerically using the Crank-Nicolson method, assuming vanishing flux boundary conditions for r = 0 and a constant value of the molar fractions for $r\gg r_{\text{JKR}}$ to model an infinite reservoir of receptors within the bilayers. A nonuniform grid with quadratically increasing grid spacings was used to increase the accuracy of the numerical integral for both molar fractions in the contact area. The size of the grid was chosen such that the concentration at large radii remains unchanged to prevent artifacts. To capture the fast dynamics of initial binding and rebinding of receptor and ligand compared to the timescale of their diffusion, logarithmic time steps were used. An initial assumption of a constant $r_{\text{JKR}}$ reached reasonable convergence within five iterations.

Results and Discussion

Coupling efficiency and Lewis X self-recognition

The binding efficiency of Lewis X-thiol to the lipid bilayer equipped with maleimide-bearing phospholipids was measured using ellipsometry. After covalent coupling of the Le^X-thiol, 79 ± 3% of the maleimide headgroups were still accessible for the coupling of cysteine-terminated peptides (averaged over three measurements; see Fig. S1 in the Supporting Material). Because the membrane contained 10 mol % maleimide functionalized lipids, a final glycolipid concentration of 2.1 ± 0.3 mol% was therefore reached.

Force distance curves between nonfunctionalized and Le^X-equipped membranes in the presence and absence of calcium were recorded to exclude contributions from nonspecific interactions. Typical force distance curves showing retraction of the membrane probe from the membrane-coated surface are shown in Fig. 1 b. The probability density function of the adhesion forces obtained at contact times of 1 s was estimated by kernel density estimation shown in Fig. 2. The nonfunctionalized membranes show very little interaction between the membranes even in the presence of calcium with a maximum in the unbinding force distribution at 40 pN. In contrast, after coupling of Le^X, strong adhesion forces up to several hundred pN with the most likely unbinding force f ∼120 pN were observed in calcium-containing buffer. This interaction can therefore be unequivocally attributed to specific Le^X-Le^X self-recognition. The interaction was not observed in the absence of calcium, where unbinding j are dropped to 27 pN, further supporting the assignment to specific Le^X interactions—although the forces are surprisingly strong in light of the supposedly weak nature of carbohydrate interactions, suggesting that the bonds act cooperatively to the applied load (1, 2).

Figure 2.

(a) Kernel density estimate of the rupture force distribution. After coupling of Lewis X, rupture forces increase to ∼120 pN. Rupture forces vanish in the absence of calcium. (b) Kernel density estimate of the rupture force distribution at different contact times showing a strong increase of adhesion for longer contact times. To see this figure in color, go online.

Adhesion forces show a strong dependency on the contact time between the probe and the substrate (Fig. 3 a). Adhesion forces increase from a most likely unbinding force of 30–120 pN, showing a saturation for contact times longer than 0.5 s, while mean adhesion forces increase from 90 to 230 pN.

Figure 3.

(a) Contact time dependency of the rupture forces fitted with the numerical solution of the model adhesion forces. The fit yields reaction rates of $k_{\text{on}}=18\pm 3{\text{s}}^{-1}$ and $k_{\text{off}}=7\pm 0.2{\text{s}}^{-1}$ and a free binding energy of $\Delta F^{\circ }=1.0\pm 0.1k_{\text{B}}T$. (b) Work of adhesion and contact radius as a function of the contact time as obtained from the fit. (Dashed lines) Range of uncertainty. To see this figure in color, go online.

Modeling reaction kinetics

The measured contact time dependency of the adhesion force can be described by this model if the binding strength $f_A$ is proportional to the number of bonds in the contact zone. It is possible to fit the parameters of the kinetic scheme so that the model closely matches the experimental rupture force as a function of time. Three conditions need to be satisfied to model the critical adhesion force measured in the experiment. First, the complete work of adhesion needs to be transferred to cantilever deformation, meaning that the spring constant of the parallel bonds including linker is large compared to that of the cantilever and therefore extension of bonds is small compared to the cantilever’s deflection. This is a reasonable assumption, because a very short tether was used to anchor Le^X to the membrane acting as a transducer, while a very soft cantilever was used. To verify this assumption experimentally, the slope of every force distance curve immediately before bond rupture was linearly fitted to determine the effective spring constant $k_{\text{eff}}={\left(k_c^{-1}+{\left(N_b{k}_s\right)}^{-1}\right)}^{-1}$, which was found to be close to the cantilever spring constant (see Fig. S3). Second, the binding energy needs to be the main contribution to the work of adhesion. Because very little nonspecific interaction was measured between opposing nonfunctionalized membranes, this condition is reasonable. Third, the measured rupture force needs to be largely independent of loading rate (pulling velocity) to ensure quasi-equilibrium conditions of the enforced separation. Detachment experiments performed as a function of increasing pulling velocity did not show a significant change in the measured adhesion energy of CCIs (see Fig. S2), which we attribute to a soft cantilever connected to stiff parallel aligned bonds that hold the probe in place after rupture of individual bonds (30). If the lifetime $\tau =f_A/k_{c}v$ of the cluster is reached, the colloidal probe detaches from the surface and rebinding is no longer possible (31, 32).

The only unknown parameters used in the model are the reaction rates of Lewis X binding $k_{\text{on}}$ and unbinding $k_{\text{off}}$. A least-squares fit of the binding energies obtained from the model (Fig. 3 a) to the experimental data (adhesion force) gives reaction rates of $k_{\text{on}}=18\pm 3$ s⁻¹ and $k_{\text{off}}=7.0\pm 0.2$ s⁻¹ assuming a diffusion coefficient of $D=1.4\pm 0.1$ μm²/s, a Young’s modulus of $E=20$ MPa for the supported lipid bilayers, a Poisson’s ratio of $\nu =0.5$, and a lipid density of $\rho =1.46{\text{nm}}^{-2}$ (24, 33, 34). From this, the binding energy for a single Le^X-Le^X bond is obtained as $\Delta F^{\circ }=1.0\pm 0.1k_{\text{B}}T$. The standard error was estimated by error propagation of the uncertainties of the diffusion coefficient and the glycolipid concentration, respectively.

The binding energies previously reported for the Le^X self-recognition vary over more than one order of magnitude from 0.17 to 5.1 k_BT (21, 25, 35, 36). A common problem in most of these studies is the difficulty to assess the number of Le^X-molecules involved in the multivalent binding process, which is naturally crucial to extrapolate from the measured data to the free binding energy of a single bond. Our approach overcomes this problem because it actually accounts for the accumulation of binding sites as a function of dwell time. This is accomplished by directly measuring the buildup of adhesion clusters in the contact zone as a function of time, modeling it by solving the reaction diffusion equation. Because over- and underestimation of the number of bonds has a severe effect on the observed binding energy, this could explain the deviation of the reported data in literature. Geyer et al. (25) circumvented this problem by titration of a Le^X solution of known concentration while monitoring binding by NOESY-NMR. They reported a binding affinity of 2–3 M⁻¹ relating to a binding energy of 1 k_BT, which is in excellent agreement with our results.

The energy release of forming a single Le^X-Le^X pair is only ∼1 k_BT, which is more than one order-of-magnitude smaller than that of a streptavidin-biotin bond (35 k_BT). The dissociation rate on the order of seconds is extraordinarily large compared with a typical lifetime of biologically relevant bonds, which ensures the necessary dynamics of the system to allow fast changes in the aggregation state. CCIs such as homomeric Le^X-Le^X pairs therefore allow for dynamic, highly fluctuating adhesion sites that put the cells in a position to respond to changes in bond number very quickly (see below). By changing the transducer stiffness, cells might indeed switch from a shared load condition where the bonds act cooperatively (i.e., a soft transducer) to a situation where the force per bond is not shared (i.e., a stiff transducer). This allows the system to open many bonds with very low force, while soft transducers ensure high lifetime of the cluster through shared loading.

The average number of closed bonds is also an outcome of the fitting procedure, thereby solving a major problem of membrane-based adhesion models, i.e., the intrinsically unknown number of bonds formed in the contact zone (Fig. 3 b) (12, 24). The first 50 ms show a steep increase that is essentially controlled by the reaction kinetics (Eq. 5), before diffusive transport of additional receptors into the contact area dominates the kinetics. After roughly 1 s saturation is reached, i.e., the flux of Le^X into the contact area equals the flux out of the contact area and the average number of bonds stays constant. At this point, on average $33\pm 2$ closed bonds have been formed between the two membranes. Neglecting diffusive transport of receptor molecules turned out to be not feasible because the dependency of $f_A$ on the contact time is rather weak. Fitting the adhesion work as a function of dwell time with a simple rate equation $\left(f_A=f_A^{\circ }\text{exp}\left(-t/\tau \right)\right)$ provides a timescale τ orders-of-magnitude larger than using Eq. 6.

Interestingly, the contact radius reaches its final value earlier than the number of bonds reaches its equilibrium value. The parallel action of a considerable amount of closed bonds in the contact zone is therefore responsible for the adhesion forces measured by force spectroscopy. Interestingly, a single streptavidin-biotin bond generates the same force and binding energy. However, rupture of a single Le^X bond has virtually no effect on the lifetime of the cluster, while rupture of a single streptavidin-biotin bond would inevitably lead to loss of the contact.

Using the linear scaling between the unbinding force and the number of bonds for equilibrium unbinding (see Appendix) to extrapolate to the rupture force of a single bond, we find that it requires only 7 pN to rupture a single Le^X bond. This is in contrast to previous force spectroscopy studies of Le^X reported by Tromas et al. (17), who observed forces of 20 pN. However, loading conditions used in these experiments were not reported and the authors used surfaces of probe and substrate covered completely with carbohydrates, so that multiple bonds cannot be excluded, especially because forces of individual bonds are very small and the tip radius is in the regime of several nanometers—sufficient space to host several molecules. The very low unbinding forces measured here are much more in line with the expected weak character of the Le^X self-recognition.

In principle, this approach can easily be extended to single-cell force spectroscopy experiments in which cells are used as probes (37). However, the molecular intricacy displayed by the plasma membrane of living cells paired with a less defined lateral mobility of receptors renders quantification of adhesion cluster formation more erroneous than in the case of membrane-coated colloidal probes.

Formation of lipid tethers

A frequent feature observed in the retraction curves between Le^X-equipped bilayers are force plateaus indicating the formation of membrane tethers, i.e., nanotubes connecting the probe and the substrate (Fig. 4 a). Because tether lifetimes are inherently linked to the number of closed bonds in the contact zone, we expected them to share properties governed by ligand/receptor assembly as discussed above, but on a much smaller length scale. The observed tether force $f_t$ of $81\pm 6$ pN remains constant for different contact times. This is reasonable because tether forces $f_t=2\pi \sqrt{2\kappa \sigma }+C\eta v$ should only depend on intrinsic features of the membrane such as bending modulus κ, tension σ, and viscosity η (C is a constant on the order of unity) (38). The radius of the tether $r_t=2\pi \kappa /f_t$ can be calculated from the tether force and the bending modulus of the membrane κ. Employing thin-shell theory, the bending modulus κ of a fluid lipid membrane with a thickness of h = 4–5 nm and a Young’s modulus E = 20 MPa gives $\kappa =\left(8-14\right)\times {10}^{-20}$ J. This corresponds to a tether radius of r_t = 6–11 nm, in agreement with previous findings (24).

Figure 4.

(a) Scheme of the formation of a lipid tether pulled from the substrate. (b) The average number of tethers observed per force distance curve and the mean lifetime of the tethers as a function of the contact time. (c) The average number of tethers observed per force distance curve as a function of the number of closed bonds showing a linear trend. (d) The mean tether lifetime as a function of the average separation between two bonds. When the separation falls below 25 nm, a steep increase in the tether lifetime is observed. As a guide for the eye, the lines in (b–d) represent exponential or linear fits, respectively. To see this figure in color, go online.

The lifetime $t_t$ of the tethers can be calculated by dividing the tether length $l_t$ through the retraction velocity and should depend on the lifetime of bonds attaching the tether to the substrate. The distribution of tether lifetimes can be described with a monoexponential decay $\text{exp}\left(-t_t/{\tau }_t\right)$ using the mean lifetime ${\tau }_t$. However, very short-lived tethers cannot be observed experimentally, because the characteristic force plateau is concealed by the adhesion peak leaving the experimentally determined mean tether lifetime an upper bound for the lifetime of the bonds connecting the tether to the opposing bilayer.

Although both the probability for tether formation and the mean lifetime are growing with increasing contact times, they show a very different progression (Fig. 4 b). The tether formation probability shows a steep increase followed by saturation for contact times longer than 0.5 s, resembling the contact time-dependency of the work of adhesion. In contrast, the mean tether lifetime shows a more shallow increase with contact time. The differences in their behavior can be rationalized by setting the mean lifetime in relation to the average distance $d\left(t\right)$ between two closed bonds, which can easily be calculated from the contact area $2\pi r_{\text{JKR}}{\left(t\right)}^2$ and the number of closed bonds $N_b\left(t\right)$ as $d\left(t\right)=r_{\text{JKR}}\left(t\right)\sqrt{2\pi /N_b\left(t\right)}$. Tether lifetime remains constant until the bond-bond-distance falls below 25 nm, which corresponds roughly to the diameter of a single tether calculated above (Fig. 4 c). This follows a simple reasoning: only when two bonds are closer to each other than the tether diameter can they share the load acting on the tether and thereby increase its lifetime. The mean lifetime of a tether at large bond-bond distances, which is held by only a single bond, is ∼70 ms. Although this is an upper bound for the actual bond lifetime, the load $f_t$ leads to a decreased lifetime compared to the mean bond lifetime $k_{\text{off}}^{-1}\approx 140$ ms of a single Le^X-bond in the absence of force, as predicted by Bell’s theory.

Besides, the tether formation probability increases linearly with the number of closed bonds (Fig. 4 d), meaning that every bond, regardless of the position or the distance to neighboring bonds, contributes equally to tether formation. These results are in line with the expected cooperative behavior of parallel bonds and thereby further support the validity of the proposed model above. The approach allows us, in principle, to perform measurements with only a very few weak bonds participating in tether formation, thereby ensuring a defined probe geometry and contact area giving access to lifetimes; this in turn offers a more precise way to determine the off-rate at zero force as opposed to recording rupture force histograms for different pulling velocities (38).

Conclusions

Colloidal probes, coated with fluid lipid bilayers that are equipped with a defined number of glycolipids, offer a unique way to study the dynamics of adhesion cluster formation in the contact zone with a planar-supported membrane. It also allows us to assess kinetic parameters such as rates of association and dissociation along with the number of participating bonds.

The interest in weak and ultraweak bonds such as those formed between carbohydrates or carbohydrates and lectins originates from their role in transient cell adhesion as it occurs in early metazoans, or in tumorigenesis or embryogenesis (1, 2, 36). With interaction energies on the order of k_BT, many bonds are needed to cooperatively join into supramolecular clusters that eventually ensure overall free adhesion energies to be large enough to hold cells together over a long time. Low individual binding energies of merely 1 k_BT of many parallel bonds go hand in hand with short lifetimes of the bonds, allowing cells to quickly change their environment on short timescales. Here, we found that the lifetime of a single homotypic carbohydrate bond is on the order of seconds—tremendously short-lived compared with other noncovalent bonds, such as several hours for streptavidin-biotin at the other end of the scale (39). Comparably fast dissociation rates are also found in highly dynamic systems such as the actin cortex (40). Similar to cell-cell contact, the actin cortex is highly dynamic, with actin-binding proteins attaching and detaching from the actomyosin network and actin filaments being polymerized and depolymerized. The system is stabilized in the steady state, where actin assembly and disassembly are balanced. We assume that highly dynamic cell-cell contacts with low bond lifetime also have the advantage that short contact times are not sufficient for cellular aggregation, thereby preventing migrating cells from sticking to each other. However, once a comfortable situation is reached to form a multicellular state and eventually an organism, cells might limit their mobility and cluster with time, thereby reaching a dynamic strength that is sufficiently large to easily withstand external forces. The timescale of this reaction is set not just by rates of association and dissociation but also by the lateral mobility of the carbohydrates. Final cluster size is ultimately limited by diffusion, and thereby less energy per bond is needed if the system is sufficiently fluid.

Our study explores the full parameter space of ultraweak CCIs in the context of membrane models, giving access simultaneously to the energy of adhesion, adhesion forces, the contact size, reaction rates, the free energy per bond, and the number of bonds as a function of time.

Author Contributions

H.W. and M.O. performed the AFM measurements; H.W., F.S., B.G., and A.J. developed the kinetic model; F.S. implemented the fitting procedure; S.I.A. performed the synthesis of Lewis X-thiol; D.B.W. and A.J. designed the research; and H.W. and A.J. wrote the article.

Editor: Klaus Gawrisch.

Appendix

The following treatment shows to what extent we can assume equivalence between classical contact mechanics of cluster dissolution and stochastic bond rupture of parallel bonds close to equilibrium. Fig. 5 a illustrates the envisioned scenario, where $N_b$ bonds out of a total of $N_t$ possible bonds form the adhesion cluster subjected to a pulling force.

Figure 5.

(a) Adhesion cluster modeled by a coupled spring model subject to force transmission via a soft transducer $k_c$ representing a typical AFM cantilever. Bond stiffness is $k_s$; cantilever deflection is $z_c$; and piezo movement is $z_p=vt$, in which v is the pulling speed. (b) Model potential employed to illustrate how to calculate the mean rupture force (blue lines). Distortion of the potential due to application of a force $f=k_{c}vt$ (Magenta lines). (c) Solving Eq. 16 gives the mean equilibrium force $\langle f\left(vt\right)\rangle /N_t$ as a function of piezo movement $z_p=vt$. To see this figure in color, go online.

Contact mechanics

According to the JKR model, which considers the effect of contact pressure and adhesion only inside the area of contact, all springs detach simultaneously as soon as the critical length $\Delta x_c=\sqrt{2\pi a^2{\tilde{w}}_A/c}$ is reached. a is the contact radius and $c=2a{E}^{\ast }$ is the contact stiffness. Therefore, the total force required to detach the indenter from the substrate is (41)

$$f_A=c\Delta x_c=2a{E}^{\ast }\sqrt{\frac{W_A}{a{E}^{\ast }}}.$$ (7)

Here, a is the radius of the cylindrical punch; and ${\tilde{w}}_A$ is the separation energy of the two planar substrates per unit area corresponding to $W_A=\pi a^2{\tilde{w}}_A$, the free energy of adhesion under equilibrium conditions. $E^{\ast }$ is the reduced E-modulus that in the case of two membranes opposing each other is $E^{\ast }={\left({(E_1/(1-{\nu }_1^2))}^{-1}+{\left(E_2/(1-{\nu }_2^2)\right)}^{-1}\right)}^{-1}=0.5E_1{\left(1-{\nu }_1^2\right)}^{-1}$ with $E_1=E_2$, the Young’s modulus of the respective surface (membrane) and ${\nu }_1={\nu }_2$, the corresponding Poisson’s ratio. Note that according to Fig. 5 a, the contact stiffness is also given by $c=2a{E}^{\ast }=\text{d}{f}_A/\text{d}x=N_b{k}_s$, leading to the compact formula:

$$f_A\approx \sqrt{2N_b{k}_s{W}_A}.$$ (8)

Note that $W_A=N_b\Delta F^{\circ }$, where $\Delta F^{\circ }$ is the free energy of a single bond and $N_b$ is the number of closed bonds in equilibrium. The approach is identical to assuming that all bonds in the contact zone rupture simultaneously and behave as one single adhesive bond with $N_b$ times the strength of the molecular bond.

Molecular approach

The following approach assumes a reasonable bond potential (Fig. 5 b) coupled to a soft transducer with regard to the contact stiffness $\left(N_b{k}_s\gg k_c\right)$. It is convenient to use the distance $x$ as the control variable corresponding to an (N,x,T-ensemble) rather than force (equivalent to the N,f,T-ensemble if $N\gg 1$). The N,x,T ensemble ensures decoupling of bonds, in contrast to using the force as the control variable where we have $f=\left(N_b{k}_s{k}_c/(k_c+N_b{k}_s)\right)vt$ (42). The molecular potential is assumed to obey (Fig. 5 b):

$$V_0\left(x\right)=\{\begin{array}{cc}\frac{1}{2}{k}_s{\left(x_0-x\right)}^2& \text{for}x\le x_u\\ \Delta F^{\circ }& \text{for}x\ge x_u.\end{array}$$ (9)

In Fig. 5 b, we assumed $x_0=0$. The full potential requires us to add the cantilever potential $V_c\left(x,vt\right)=\left(1/2\right)k_c{\left(vt-x+x_0\right)}^2$, leading to

$$V_t=V_0\left(x\right)+V_c\left(x,vt\right).$$ (10)

Generally, the average rupture force $\langle f_{\text{rup}}\rangle$ as a function of z-piezo movement is obtained from the probability distribution (43, 44):

$$\langle f_{\text{rup}}\rangle ={\int }_0^{\infty }f{p}_{\text{rup}}\left(f\right)\text{d}f={\int }_0^{\infty }{n}_b\left(f\right)\text{d}f,$$ (11)

with $p_{\text{rup}}\left(f\right)$ as the probability density of rupture:

$$p_{\text{rup}}\left(f\right)=-\frac{\text{d}{n}_b\left(f\right)}{\text{d}f}=k_{\text{off}}{n}_b\left(f\right)-\left(1-n_b\left(f\right)\right)k_{\text{on}}$$ (12)

and $n_b\left(f\right)=N_b\left(f\right)/N_t$, the population of the bound state (probability to be in the bound state) at a given force. We can now form the average of $\langle f_{\text{rup}}\left(vt\right)\rangle =k_c\langle vt-x\rangle$ (32, 43):

$$\langle f\left(vt\right)\rangle =k_c\left({\langle vt-x\rangle }_b+{\langle vt-x\rangle }_u\right).$$ (13)

The first term refers to the bound state and the second to the unbound state. For potentials with a long attractive tail, even the unbound state may generate a noticeable force (32). Here, in virtue of our model potential, we ignore the second term. We obtain for the average force:

$$\langle f_{\text{rup}}\left(vt\right)\rangle =N_b\left(f\right)\frac{{\int }_0^{x_u}{k}_c\left(vt-x\right)\text{exp}\left(-\frac{V_t\left(x,vt\right)}{k_{\text{B}}T}\right)\text{d}x}{{\int }_0^{x_u}\text{exp}\left(-\frac{V_t\left(x,vt\right)}{k_{\text{B}}T}\right)\text{d}x},$$ (14)

with

$$N_b\left(f\right)=N_t\frac{{\int }_0^{x_u}\text{exp}\left(-\frac{V_t\left(x,vt\right)}{k_{\text{B}}T}\right)\text{d}x}{{\int }_0^{\infty }\text{exp}\left(-\frac{V_t\left(x,vt\right)}{k_{\text{B}}T}\right)\text{d}x},$$ (15)

and $N_t$, the total number of possible bonds. Altogether, this gives for the average force per bond:

$$\frac{\langle f_{\text{rup}}\left(vt\right)\rangle }{N_t}=\frac{{\int }_0^{x_u}{k}_{\text{c}}\left(vt-x\right)\text{exp}\left(-\frac{V_t\left(x,vt\right)}{k_{\text{B}}T}\right)\text{d}x}{{\int }_0^{\infty }\text{exp}\left(-\frac{V_t\left(x,vt\right)}{k_{\text{B}}T}\right)\text{d}x}.$$ (16)

Fig. 5 c shows a typical force curve as obtained from the solution of Eq. 16. The maximal rupture force is called the “critical” force.

For an equilibrium separation process, we can write $\text{d}{n}_{\text{b}}\left(f\right)/\text{d}f=0$ to get:

$$n_{\text{eq}}\left(f\right)=\frac{N_b\left(f\right)}{N_t}=\frac{1}{1+\frac{k_{\text{off}}\left(f\right)}{k_{\text{on}}\left(f\right)}}.$$ (17)

For the sake of simplicity, we assume that the rates can be approximated according to Bell (31), where $x_u$ is the distance from the bound state to the energy barrier (see Fig. 5 b):

$$k_{\text{off}}\left(f\right)=k_{\text{off}}^0\text{exp}\left(\frac{f{x}_u}{k_{\text{B}}T}\right)$$ (18)

and force-independent rebinding rates:

$$k_{\text{on}}\left(f\right)=k_{\text{on}}^0.$$ (19)

The average force in equilibrium is therefore

$$f_{\text{eq}}={\int }_0^{\infty }\frac{\text{d}f}{1+\frac{k_{\text{off}}\left(f\right)}{k_{\text{on}}^0}}=N_t\frac{k_{\text{B}}T}{x_u}\text{ln}\left(1+\frac{k_{\text{on}}^0}{k_{\text{off}}^0}\right)$$ (20)

and the critical force $f_c^{\ast }$ for which $p\left(f\right)=-\text{d}{n}_{\text{eq}}\left(f\right)/\text{d}f$ is maximal is

$$f_c^{\ast }=N_t\frac{k_{\text{B}}T}{x_u}\text{ln}\left(\frac{k_{\text{on}}^0}{k_{\text{off}}^0}\right)=N_t\frac{k_{\text{B}}T}{x_u}\frac{\Delta F^{\circ }}{k_{\text{B}}T}=N_t\frac{\Delta F^{\circ }}{x_u}.$$ (21)

Because we can assume that for a single bond $x_u=\sqrt{2\left(\Delta F^{\circ }/k_s\right)}$ holds, we introduce the bond stiffness:

$$f_{\text{c}}^{\ast }=N_t\frac{\Delta F^{\circ }}{x_u}=N_t\sqrt{\Delta F^{\circ }{k}_s/2}.$$ (22)

Substitution of $N_b=N_t/(1+K_{\text{eq}}^{-1})=N_t/(1+\text{exp}\left(-\Delta F^{\circ }/k_{\text{B}}T\right))$ gives:

$$f_c^{\ast }=N_b\left(1+e^{-\Delta F^{\circ }/k_{\text{B}}T}\right)\sqrt{\Delta F^{\circ }{k}_s/2}.$$ (23)

For $\Delta F^{\circ }\ge k_{\text{B}}T$, we get the same result as derived for the contact model in Eq. 8:

$$f_A=f_c^{\ast }\approx N_b\sqrt{2\Delta F^{\circ }{k}_s}=\sqrt{2W_A{N}_b{k}_s}.$$ (24)

A more thorough derivation is given by Lin et al. (42), but with the same outcome for binding energies $\Delta F^{\circ }\ge k_{\text{B}}T$. This result shows the correspondence between continuum mechanics and the stochastic treatment of cooperative bond failure under external force for a large number of bonds and low pulling velocities.

Supporting Citations

References (45, 46, 47, 48) appear in the Supporting Material.