As previously reviewed (Bongrand, 1999; Zhu et al., 2002; Robert et al., 2007), the interaction between two molecules A and B in solution is well accounted for by two numbers, the association rate k_on and dissociation rate k_off: (1)A + B⇄konkoffAB

d[AB]/dt = k_on [A][B] − k_off [AB], the ratio k_on/k_off is the affinity constant K_a.

However, this conventional framework could not account for interactions between membrane-bound receptors and ligands: Firstly, bonds formed between surface-bound molecules are often subjected to external forces, and until recently no information was available on the effect of forces on bond lifetime. Secondly, as emphasized earlier, even the dimension of association rate between surface-bound molecules (corresponding to so-called 2D conditions) is different from the dimension of conventional (3D) association rates as defined in Eq. 1 (Pierres et al., 2001). Thirdly, 2D conditions impose special constraints on multivalent associations. We shall address these points sequentially.

During the last two decades, experiments based on laminar flow chambers (Kaplanski et al., 1993; Pierres et al., 1995), atomic force microscopes (Florin et al., 1994), the biomembrane force probe (Merkel et al., 1999), or optical tweezers (Nishizaka et al., 1995) allowed us to study single-bond formation and rupture between surface-attached molecules subjected to controlled forces. Bond rupture under force often followed a simple formula (Chen and Springer, 2001; Evans et al., 2010) previously suggested by Bell (1978): (2)koffF=koff0expF/F0 where k_off(F) is the rupture frequency (in s⁻¹) of a single-bond subjected to a distractive force F. A simple interpretation of this formula can be obtained by viewing bond rupture as the passage of a molecular complex AB from a bound state at zero separation to a free state that is reached by crossing an energy barrier of height W at separation distance d (Figure 2). According to Boltzmann’s law, the probability of barrier-crossing should be proportional to exp(−W/k_BT), where k_B is Boltzmann’s constant and T is the absolute temperature. Applying a force F will decrease W by the product Fd (Figure 2) thus multiplying the rupture frequency k_off by exp(Fd/k_BT). Bell estimated at 0.5 nm the order of magnitude of parameter d for an antigen-antibody interaction corresponding to the depth of an antibody binding site, leading to an estimate of ∼8 pN for parameter F°= k_BT/d. More detailed discussion may be found in a number of papers following Eyring’s (1935) and Kramer’s (1940) seminal papers (Hänggi et al., 1990; Evans and Ritchie, 1997; Dudko et al., 2008). The rupture frequency and force coefficient F° for a number of receptors including selectins, integrins, cadherins, or antibodies were often on the order of 1–100 pN and 0.01–10 s⁻¹. Depending on molecule conformation, the force-free rupture frequency of LFA-1/ICAM-1 bond varied between 0.008 and 2 s⁻¹, with a force coefficient of 7–10 pN (Evans et al., 2010). However, the above results are only an approximation and single molecule studies confirmed that bond rupture is a complex process requiring multiple barrier-crossing events (Pierres et al., 1995; Merkel et al., 1999; Derenyi et al., 2004).

The catch-bond phenomenon, which is highly relevant to leukocyte-endothelium interaction, was predicted on the basis of thermodynamical reasoning by noticing that a disruptive force might decrease bond rupture frequency k_off, although it had to decrease binding affinity k_on/k_off. Bonds displaying such a strange behavior were dubbed “catch bond,” in contrast with “ordinary” bonds that were called “slip bonds,” responding to disruptive forces with increased rupture frequency (Dembo et al., 1988). A few years later, it was reported that L-selectin-mediated rolling required a minimal shear level, suggesting that L-selectin might form catch bonds (Finger et al., 1996). More recently, it was demonstrated with flow chambers that a lectin-like bacterial adhesin formed catch bonds (Thomas et al., 2002), and a similar property was demonstrated on P-selectin/PSGL-1 interaction with both flow chamber and atomic force microscopy (Marshall et al., 2003): Bond lifetime displayed a fairly sharp maximum in presence of a pulling force close to 30 pN. P-selectin/PSGL-1 thus displayed catch-bond behavior in presence of a force ranging between 0 and 30 pN. Theoretical studies led to the conclusion that actual biomolecules interactions are much more complex that sketched on Figure 2. Thus, a catch-bond behavior might be accounted for by the existence of two dissociation pathways (Pereverzev et al., 2005).

The rate of bond formation between two surfaces bearing known receptors and ligands cannot be derived from a “2-dimensional on-rate constant” since it is dependent on a number of parameters that are extrinsic to the receptor and ligand, including distance between surfaces, lateral mobility of receptors and ligands, length and flexibility of the links between binding sites and membranes, and behavior of surrounding molecules. First, it was suggested that the 3D k_on (a number expressed in μm² molecule⁻¹ s⁻¹) had to be replaced with a function k_on(d) representing the frequency (in s⁻¹) of bond formation between a ligand and a receptor molecules maintained at distance d (Pierres et al., 1996). The function k_on(d) could in principle be derived experimentally by simultaneous determination of the binding frequency of receptor-bearing microspheres and ligand-coated surfaces and microsphere-to-surface distance (Pierres et al., 1998). However, other experiments show that this seemingly straightforward method may be difficult to use. Indeed, robust receptor-ligand association may not be immediate, and require a non-negligible amount of time for progressive crossing of barriers from less stable to more stable binding states (Pierres et al., 1995; Marshall et al., 2005; Pincet and Husson, 2005). This point was addressed experimentally in a model system (Robert et al., 2009): The formation of an ICAM-anti-ICAM-1 bond required a minimal contact time of about 10 ms to resist a disruptive force of order of 100 pN during at least 200 ms. This challenges the current framework used to describe bond formation (Eq. 1): the probability of bond formation between a ligand and a receptor is not proportional to the contact time. It is 0 if contact is shorter than some threshold, and 1 above this threshold. The threshold is dependent on the sensitivity of bond detection. More experiments are needed to check the relevance of these results to integrin-ligand associations. This is made more difficult to study experimentally by the dependence of integrin conformation on interactions with underlying membranes. However, since antigen-antibody association is very rapid, it is likely that kinetic effects may be found on most biological systems.

Theoretical studies (Seifert, 2000) have long revealed the difficulty of relating the properties of multivalent attachments to single bonds. This difficulty is a consequence of two important processes: force-sharing and rebinding. This point can be made easier to grasp by comparing the lifetime of attachments mediated by one or two identical bonds.

First, let us consider the effect of an external force F: if the force is equally shared between both bonds, the lifetime of each bond will be divided by exp(F/2F°), where F° is the aforementioned force constant. After the rupture of a first bond, the whole force will be applied on the remaining one, thus inducing rapid failure. The dissociation rate of the divalent attachment will thus vary as exp(F/2F°). In absence of force-sharing, the force is expected to divide attachment lifetime by exp(F/F°). As a numerical example, a disrupting force of 40 pN is expected to reduce the lifetime of a single-bond attachment mediated by an integrin of force constant F°∼ 7 pN by 300, while the lifetime of a force-sharing divalent attachment will be reduced by only 17.

The importance of rebinding may be still more impressive. Let us consider an attachment involving high affinity receptors such that bond formation is much more frequent than bond rupture. If a bond has a high probability to reform after spontaneous rupture, provided a second bond maintains surfaces in close contact, the lifetime of a divalent attachment may be nearly infinite, and in any case much higher than that of a monovalent attachment.

Recently, this point was addressed experimentally by comparing the lifetime of monovalent and divalent attachments formed between ICAM-1-coated surfaces and anti-ICAM-1-coated microspheres (LoSchiavo et al., 2012): the proportion of divalent attachments resisting a force of 30 pN for at least 5 s was 3.7-fold higher than that of monovalent attachments. This was due to a combination of force-sharing, bond maturation and rebinding. Importantly, rebinding requires a tight proximity between receptors.

Remarkably, clustering has been recognized by many authors as a key feature of integrin function (Cambi et al., 2006; Selhuber-Unkel et al., 2008; van Zanten et al., 2009), although other experiments were compatible with the assumption that conformational activation of individual molecules might suffice to initiate adhesion in absence of any significant modulation of clustering (Kim et al., 2004).

The physical background we summarized will help us understand the mechanisms of integrin-mediated leukocyte arrest.

Integrins likely need lateral mobility to align along ligands on opposing surfaces, and the mobility requirement may be higher as lower ligand density (Chan et al., 1991). A positive correlation between lymphocyte adhesiveness to ICAM-1-coated surfaces and LFA-1 membrane mobility was reported (Kucik et al., 1996). More recently, Bakker et al. (2012) concluded that monocytes required a mobile population of surface integrins to adhere to ICAM-1-coated surfaces under static or flow conditions.

The difficulty of relating integrin-cytoskeletal association to cell adhesiveness (Lub et al., 1997) may be due to (i) heterogeneity of mobilities of LFA-1 molecules on a given cell, (ii) contradictory need for mobility (to allow ligand-receptor contact) and integrin-cytoskeleton attachment (to ensure mechanical strength), (iii) dependence of integrin-cytoskeleton interaction on cell differentiation and activation status (Cairo et al., 2006).

Since the lifetime of a newly formed LFA-1/ICAM-1 bond may be quite short if full activation has not been triggered, a pre-clustering of LFA-1 molecules might strongly enhance the duration of initial attachment and allow for the formation of additional bonds. This may be less necessary if integrins are in a fully activated state. Note that the precise chronology of clustering remains controversial. Studies made on phagocytes strongly suggest that integrin clustering preceded ligand binding (Detmers et al., 1987; Cambi et al., 2006; van Zanten et al., 2009). Other authors concluded that the binding of multivalent ligands was required to induce a clustering of lymphocyte integrins (Kim et al., 2004).

In conclusion, integrin-mediated cell arrest likely results from a combination of conformational changes (within seconds), nanotopographical membrane rearrangement to allow contact with ligand-bearing surfaces (within tens of seconds), and lateral diffusion of integrin molecules to align along ligands, form clusters, or both. However, other experiments suggest that initial arrest is followed within minutes by an attachment strengthening phase including several actions, as described below.

Monitoring cell-surface detachment under flow revealed significant cell deformation during detachment (Mège et al., 1986; Cao et al., 1998). Flow induced detachment is a peeling process, with a sequential detachment of membrane stripes involving a few bonds. The rupture force therefore increases in parallel with membrane stiffness (Evans, 1985). This reasoning is supported by experimental evidence (Rees et al., 1977; Badley et al., 1981). This supports the functional importance of microfilament concentration in cell-surface contact areas (André et al., 1991).

Strong integrin-mediated cell attachment requires that integrins be strongly attached to the cell-surface. Microfilaments indeed enhanced integrin-mediated attachment in some experiments (Lub et al., 1997). Thus, integrin-cytoskeleton interaction may be a multiphasic, time-dependent process: initial integrin release should favor alignment with ligand and clustering, binding of ligand-attached integrins to cytoskeletal elements would then strengthen overall attachment. This is consistent with the multiplicity of integrin states (free versus immobile, isolated versus clustered) on the cell membrane (Cairo et al., 2006).

Spreading may follow and markedly stabilize cell adhesion when this is mediated by suitable receptors (Pierres et al., 2002). A frequent consequence of integrin-ligand association is the generation of signaling cascades (this is outside-in signaling) inducing rapid spreading (Abram and Lowell, 2009). This was demonstrated not only with LFA-1 (Feng et al., 2012) but also β1 (Zeller et al., 2010) or β3 (Kasirer-Friede et al., 2007) integrins.

In conclusion, LFA-1-mediated leukocyte arrest on endothelial cells is a key step of inflammation. This strongly depends on a combination of integrin-mediated processes that are likely to involve a network of tens of proteins and hundreds of interactions (Ley et al., 2007; Zaidel-Bar et al., 2007). A possible way of understanding the functions of these networks is to analyze the perturbation (Ku et al., 2012) generated by the deficiency of a specific component. A recently characterized defect of FERMT3 gene resulting in abnormal or absent kindlin-3 protein provides a good example.