Biomechanics of leukocyte rolling

Abstract

Leukocyte rolling on endothelial cells and other P-selectin substrates is mediated by P-selectin binding to P-selectin glycoprotein ligand-1 expressed on the tips of leukocyte microvilli. Leukocyte rolling is a result of rapid, yet balanced formation and dissociation of selectin-ligand bonds in the presence of hydrodynamic shear forces. The hydrodynamic forces acting on the bonds may either increase (catch bonds) or decrease (slip-bonds) their lifetimes. The force-dependent ‘catch-slip’ bond kinetics are explained using the ‘two pathway model’ for bond dissociation. Both the ‘sliding-rebinding’ and the ‘allosteric’ mechanisms attribute ‘catch-slip’ bond behavior to the force-induced conformational changes in the lectin-EGF domain hinge of selectins. Below a threshold shear stress, selectins cannot mediate rolling. This ‘shear-threshold’ phenomenon is a consequence of shear-enhanced tethering and catch-bond enhanced rolling. Quantitative dynamic footprinting microscopy has revealed that leukocytes rolling at venular shear stresses (> 0.6 Pa) undergo cellular deformation (large footprint) and form long tethers. The hydrodynamic shear force and torque acting on the rolling cell are thought to be synergistically balanced by the forces acting on tethers and stressed microvilli, however, their relative contribution remains to be determined. Thus, improvement beyond the current understanding requires in silico models that can predict both cellular and microvillus deformation and experiments that allow measurement of forces acting on individual microvilli and tethers.

1. Introduction

Inflammation is an acute response coordinated by the innate arm of the immune system and involves leukocyte recruitment from the blood stream into the inflamed tissue [100,112,134,147,149,176,187]. Although the first set of observations of leukocyte recruitment across the blood-tissue barrier date back to the early nineteenth century [1,38,203-204,207], the studies done over the last two decades [7,10,11,53,76,77,95,96,105,111,118,191,201,223] have inspired the creation of a paradigm for the leukocyte recruitment to the inflammatory foci [112,147]. Neutrophil, or polymorphonuclear (PMN) leukocytes, are the most abundant leukocytes in the peripheral blood of humans and are recruited to the site of acute inflammation [3]. The neutrophil recruitment takes place primarily in the postcapillary venules of the inflamed tissue and involves neutrophil adhesion to the vessel endothelium against the hemodynamic shear forces exerted by the flowing blood [10,11,95]. Neutrophil adhesion to the vessel endothelium requires coordination of several steps, each dependent on non-covalent interactions of specific adhesion receptors (AR) on the neutrophil or endothelium with their respective ligands on the apposing cell [21,112,187]. Rolling initiates with the first step of capture which can result in rolling along the vessel endothelium followed by slow rolling, activation, and firm arrest [112]. Collision with red blood cells (RBC) and RBC aggregates cause neutrophils to marginate towards the blood vessel wall [55,137,164]. The free flowing marginated neutrophils tether and roll along the vessel endothelium through reversible and rapid interactions between the selectin family of adhesion molecules expressed on (P, E- selectin) endothelial cells (EC) or neutrophils (L-selectin) with their carbohydrate ligands expressed on the apposing cell to mediate the primary steps of capturing and rolling [33,36,49,126,196]. Neutrophil rolling in the postcapillary venules is predominantly mediated by P-selectin expressed on the EC binding to its sialo-mucin ligand P-selectin glycoprotein ligand (PSGL)-1 constitutively expressed on the neutrophils [20,76,87,96,111,133,138-139]. Neutrophil rolling along the vascular endothelium of the cremaster or mesenteric postcapillary venules has been studied extensively using in vivo intravital microscopy [10,11,32,33,49,75-77,96,111,149,182].

Flow chamber assays have proved to be an excellent model to replicate the selectin-mediated rolling of neutrophils in an in vitro setting [7,53,102,103,105,142,155,162,166,180,191]. Neutrophils in a physiological buffer or whole blood are allowed to flow in laminar flow regime (Reynolds number, R_e <<1) over a glass or polystyrene substrate either coated with EC, recombinant or purified selectins or selectin ligands. The hydrodynamic forces experienced by the rolling cells are estimated from the shear stress exerted by the streaming fluid on the flow chamber wall [54,92] and the volumetric flow rates are adjusted to achieve venular wall shear stresses (0.12 to 3.5 Pa) [33]. These in vivo and in vitro studies have profoundly contributed to the current understanding of the biomechanics of neutrophil rolling on selectin substrates.

The measurements of cellular and molecular biophysical parameters generated in these studies have inspired the creation of several in silico theoretical models [22,23,26,36,37,60,72,84,85,88, 93,145,150,152,196,226] which are capable of simulating neutrophil rolling at temporal resolutions 10-10⁴ times higher than those possible experimentally. These models predict that the phenomenon of rolling is a result of balanced but rapid formation and dissociation of selectin-carbohydrate bonds at the front and rear of the rolling cell, respectively, under the influence of shear forces exerted by the streaming media. This molecular mechanism responsible for neutrophil rolling, which was introduced two decades ago, [35,60,196] has also been observed experimentally [191]. Quantitative dynamic footprinting (qDF) microscopy revealed that the selectin-carbohydrate bonds in the footprints of rolling neutrophils are relaxed in the front where they form, compressed in the center, and stressed at the rear of the footprint where they finally break away from the substrate [191]. The same study also revealed that the footprint of a neutrophil rolling at high wall shear stress (> 0.6 Pa) is four times larger than that reported by the in vivo studies as well as theoretical models and that rolling neutrophils also form three to four long tethers at the rear which can extend up to 16 μm behind the cell footprint [191]. In addition to verifying the molecular mechanism of rolling predicted by theoretical models, these findings also provide an insight into the discrepancy between the rolling behavior predicted theoretically and that observed experimentally. Stable neutrophil rolling is observed in venules in vivo [33] and in whole blood flow chambers in vitro [31,98,191] at shear stresses higher than 0.6 Pa. However, theoretical models predict rolling only at small shear stresses (< 0.2 Pa) and fail to reproduce stable rolling at high shear stresses (> 0.6 Pa) commonly observed in vivo [88,150]. Neutrophils are deformable cells and their surface is covered with interconnected ridges of microvilli which are membrane structures ~200 nm long [48,163]. With increasing wall shear stress, rolling neutrophils deform thus creating a larger footprint [33,49,191], the microvilli undergo extension, and may form tethers [142,155,166,191]. In tether formation, the cytoskeleton is separated from the plasma membrane [209,210]. None of the currently available theoretical models [72,85,88,150] take all three biophysical phenomena into consideration. This may limit the ability to model rolling of neutrophils at high shear stress. This review provides an overview of the current understanding of the biomechanics of neutrophil rolling on selectin substrates.

2. Adhesion molecules

2.1. Selectins

Neutrophil rolling in postcapillary venules in vivo [76,87,111] and flow chambers in vitro [53,102,104-105,129, 205] is mediated by the selectin family of adhesion molecules, P-, E-, and L-selectin. Models of gene-deficient mice have revealed that deletion of a single selectin leads to moderate [8,99,125,190,194] while deletion of more than one leads to severe inflammatory disorders [18,19,50]. Selectins are type I transmembrane glycoproteins that share a common structural organization across all mammalian species [80,185,193,200]. The N-terminus of each selectin has a Ca²⁺-dependent carbohydrate recognition domain which is characteristic of C-type lectins, followed by an epidermal growth factor (EGF) like domain, a series of short consensus repeats which vary in number among the three selectins within species as well as for the same selectin within different species [127,128,200]. The number of consensus repeats determines the length of the extracellular domain of each selectin. Selectins also have a transmembrane domain followed by a short cytoplasmic domain. The length of extracellular domains, expression levels, molecular density, role in leukocyte rolling, and physiological ligands for different selectins are shown in Table 1.

Table 1.

Adhesion receptors mediating neutrophil rolling and tethering in vivo and in vitro.

Receptor	Extracellular length (nm)	Expression	Molecular Density	Role in leukocyte recruitment	Physiological ligand
Hu P-selectin	38	Inducible	20-50 molecules/μm2 on cultured HUVEC	Tethering and rolling	PSGL-1 on neutrophils, monocytes, and Th1 cells*
Mu P-selectin	34	Inducible	No data	Tethering and rolling	PSGL-1 on neutrophils, monocytes and Th1 cells*
Hu E-selectin	27	Inducible	750 molecules/μm2 on cultured HMVEC	Slow rolling	PSGL-1, sialylated glycosphingolipids, and sLex bearing glycoproteins on neutrophils
Mu E-selectin	27	Inducible	No data	Slow rolling	PSGL-1, ESL-1, CD44, and core O-derived glycans bearing glycoproteins on neutrophils
Hu L-selectin	12	Constitutive	Neutrophils (30,000-40,000 molecules), monocytes, lymphocytes	Tethering, secondary capture, and rolling (only in HEVs)	PSGL-1 on neutrophils and monocytes, PNAds (GlyCAM-1, CD34, and podocalyxin) on EC in HEVs of lymph nodes
Mu L-selectin	12	Constitutive	Neutrophils, monocytes, lymphocytes	Tethering, secondary capture, and rolling (only in HEVs)	PSGL-1 on neutrophils and monocytes, PNAds (GlyCAM-1, CD34, and podocalyxin) on EC in HEVs of lymph nodes
Hu PSGL-1	50	Constitutive	Neutrophils (18,000-25,000 molecules), monocytes, Th1 cells*	Tethering, rolling, and secondary capture	P- and E-selectin on EC and L-selectin on neutrophils, and monocytes
Mu PSGL-1	50	Constitutive	Neutrophils (75,000 molecules), monocytes, Th1 cells*	Tethering, rolling, and secondary capture	P- and E-selectin on EC and L-selectin on neutrophils, and monocytes

Hu, human; Mu, murine; HUVEC, human umblical-cord vascular endothelial cells; HMVEC, human micro-vascular endothelial cells; HEVs, high endothelial venules; PSGL-1, P-selectin glycoprotein ligand-1; ESL-1, E-selectin ligand-1; sLe^x, sialyl Lewis x; EC, endothelial cells.

^*PSGL-1 is expressed on other leukocytes, but is correctly glycosylated only on myeloid cells and Th1 cells.

2.1.1. P-selectin

P-selectin mediates neutrophil rolling in the postcapillary venules of mice in vivo [87,96,111] and flow chambers in vitro [20,105,205]. It also mediates interactions of activated platelets with neutrophils [225] and monocytes [119]. Although P-selectin is undetectable on the surface of EC of most tissues of mice in vivo and cultured EC in vitro under baseline conditions [64,75], it is stored preformed in the Weibel-Palade bodies of EC and is rapidly translocated to the plasma membrane following stimulation with histamine or thrombin in vitro [52,64] or surgery-induced trauma in vivo [41,75,87]. Cultured human umbilical vein EC (HUVEC) are known to express P-selectin at a site density of 20-50 molecules/μm² within 5-10 min following treatment with histamine [64]. In addition, P-selectin transcription and translation are upregulated in EC within 2 h following cytokine treatment [41,57,59,75,160,211]. P-selectin is an extended molecule with an extracellular domain that varies in length among different species based on the number of consensus repeats present [200]. The extracellular domain of human P-selectin has nine short consensus repeats and is 38 nm long (refer Table 1) [199]. However, the mouse P-selectin has eight short consensus repeats and is 34 nm long [199-200]. P-selectin expressed on EC forms homodimers through noncovalent interactions of its transmembrane domains [14,199]. P-selectin clustering on EC through interaction of its cytoplasmic tails with clathrin-coated pits has been shown to enhance neutrophil rolling [168,170]. In vivo studies using gene-deficient mice and function blocking antibodies have revealed that neutrophil rolling along the vascular endothelium of postcapillary venules during the first one hour following surgery-induced trauma or cytokine induced inflammation is predominantly mediated by endothelial expressed P-selectin with an average rolling velocity of 40 μm/s [33,74,76,96,111]. Rolling of human neutrophils on different site densities of P-selectin has been studied in vitro as a function of wall shear stress [105] and the data from this study is shown in Fig. 1.

Fig. 1.

In vitro rolling velocity of human neutrophils plotted as a function of P-selectin site density and wall shear stress. Data reproduced from Ref. [105]. Closed triangle (400 sites/μm²), closed square (200 sites/μm²), and closed circle (50 sites/ μm²).

2.1.2. E-selectin

E-selectin expression is restricted to EC and is undetectable in most tissues under baseline conditions but gets upregulated within 2 h following cytokine treatment [41,75]. Cultured human microvascular EC (HMVEC) have been shown to express E-selectin at a site density of 750 molecules/μm² following 5 h treatment with interleukin (IL)-1α [110]. E-selectin mediates slow rolling of neutrophils on cytokine activated EC in vivo [76,182] and immobilized E-selectin in vitro [101,104,179]. The slow rolling of neutrophils (rolling velocity ≤ 5 μm/s) following tumor necrosis factor (TNF)-α induced inflammation is absent in mice deficient in E-selectin, and this is reproduced in wild type mice treated with anti-E-selectin mAb [97]. E-selectin dependent slow leukocyte rolling requires the β₂ integrin LFA-1 [223]. The extracellular domain of human and mouse E-selectin contains one lectin domain, one EGF domain, and six short consensus repeats and extends 27 nm above the plasma membrane (refer Table 1) [199,200]. E-selectin expressed on EC colocalizes as clusters in membrane rafts and clathrin-coated pits and this clustering has been shown to stabilize neutrophil rolling [169].

2.1.3. L-selectin

L-selectin is constitutively expressed on neutrophils, monocytes, and lymphocytes [80,185] and rapidly shed from the cell surface following cellular activation by chemoattractants and activating factors, which involves proteolytic cleavage at a membrane proximal site [58,78,89]. L-selectin monomers are clustered on the tips of neutrophil microvilli (30,000-40,000 molecules per human neutrophil [177]) which facilitates capturing of free flowing neutrophils by presentation of L-selectin to EC-bound ligands [44,48,101,202]. L-selectin is the smallest of the three selectins with an extracellular domain that contains only two short consensus repeats in addition to the lectin and EGF domains and extends ~12 nm from the plasma membrane in both mice and humans (refer Table 1) [199,200]. Studies using gene deficient mice and function-blocking antibodies have shown that L-selectin mediates fast rolling (rolling velocity > 100 μm/s [74]) of neutrophils in inflamed as well as noninflamed peripheral post capillary venules [8,76,97,111,113-114]. However, studies done with mice in vivo [43,94,182] and flow chambers in vitro [13,205] have also shown that most of the L-selectin mediated fast rolling of neutrophils is a result of secondary capture or tethering of free flowing neutrophils by adhered or rolling neutrophils.

2.2. PSGL-1 and other selectin ligands

P-selectin glycoprotein ligand-1 (PSGL-1) constitutively expressed on neutrophils is a ligand for all three selectins (P-, E-, L-selectin) and mediates neutrophil rolling in the peripheral microvasculature in vivo [42,67,138,182-183,214,222] and flow chambers in vitro [31,98,124,205,214]. PSGL-1 is the dominant ligand on neutrophils to mediate P-selectin dependent rolling both in vivo [138-139,183] and in vitro [105,133,144,156,191]. PSGL-1 also serves as a ligand for L-selectin during secondary capture of neutrophils by already adhered neutrophils both in vivo and in vitro [6,42,43,79,94,117,143,182,184,188,197,205]. PSGL-1 is a transmembrane disulfide-linked homodimeric sialo-mucin constitutively expressed on the tips of microvilli at a site density of 75,000 molecules per mouse neutrophil, 18,000-25,000 molecules per human neutrophil, and 36,000 molecules per HL-60 cell [24,127,129,133,138,185,199,200, 224]. The localization of PSGL-1 on the microvilli tips has been shown to enhance P-selectin-PSGL-1 mediated tethering by increasing the probability of interactions between the two molecules [115,133,218]. PSGL-1 is an extended molecule with an extracellular domain which extends out ~50 nm above the plasma membrane and consists of decameric repeats rich in threonine, serine, and proline [2,12,115,159,192]. Threonine and serine residues throughout the extracellular domain of PSGL-1 are glycosylated to bear sialylated, fucosylated O-glycans capped with the sialyl Lewis-x or sLe^x [NeuAcα2–3Galβ1–4(Fucα1–3)GlcNAcβ1-R] tetrasaccharide determinant [127-129,132,187,200]. P- and L-selectin binding to PSGL-1 is the result of interactions of lectin domain residues and the Ca²⁺ ion in the lectin domain with the sLe^x determinant on short core 2 O-glycan expressed on the N-terminal threonine (Thr17 in mice and Thr57 in humans), adjacent to sulfated tyrosine residues (Tyr13 in mice and Tyr46, Tyr48, and Tyr51 in humans) [24,40,107-109,120,153,181,213]. E-selectin binding to PSGL-1 does not require sulfated tyrosine residues and is mediated by the lectin domain and Ca²⁺ ion interacting with the sLe^x determinant expressed on core 2-O glycans throughout the PSGL-1 ectodomain [116,181]. Uncharacterized L-selectin ligands expressed on EC of postcapillary venules of mouse cremaster also mediate neutrophil rolling in vivo [174].

L-selectin also binds to a family of sialylated, fucosylated, and sulfated mucins known as peripheral node addressins (PNAds) expressed in high endothelial venules [129,157]. PNAd is defined by its reactivity with MECA-79, a mAb that recognizes 6-sulfo-N-acetylglucosamine (GlcNAc-6-sulfate) [66,157]. The protein composition of PNAd includes glycosylated cell adhesion molecule-1 (GlyCAM-1), CD34, and podocalyxin. PNAd mediates L-selectin dependent rolling of lymphocytes in high endothelial venules of lymph nodes by presenting a sulfated form of sLe^x on extended core 1 O-, branched core 2 O-, and N-glycans expressed throughout the ectodomain [83,129,131,157,198]. In addition to PSGL-1, E-selectin mediated rolling of neutrophils in the peripheral microvasculature of mice is also mediated by N-glycans on E-selectin ligand-1 (ESL-1) and CD44 expressed on murine neutrophils [67,82,106]. ESL-1 is localized on microvilli but away from the tips [189] and has been shown to synergize with PSGL-1 to mediate initial tethering on E-selectin and with CD44 localized on the cell body [202] to mediate slow-steady rolling [67]. Recently, core 2 O- and extended core 1 O-glycans on uncharacterized glycoproteins other than PSGL-1, ESL-1, and CD44 expressed on mouse neutrophils have also been proposed to mediate tethering and rolling on E-selectin both in vivo and in vitro [217]. Human neutrophils have been shown to express sialylated glycosphingolipids which mediate E-selectin dependent tethering and rolling in vitro [136].

3. Forces acting on a rolling neutrophil

Rolling is a characteristic ‘jerky-tumbling’ motion of neutrophils along the blood vessel wall in vivo and selectin substrate in vitro in the presence of hydrodynamic shear flow [11,28,105]. Shear flow exerts a hydrodynamic shear force F_s (in the x direction), and torque T_s (in the y direction about the cell center), on a neutrophil adhered to a P-selectin coated substrate [60,196] (Fig. 2). Neutrophils interact with a P-selectin substrate through noncovalent bonds between PSGL-1 expressed on the microvilli tips and P-selectin on the substrate [15,133]. As the adhered neutrophil under shear flow moves forward, the hydrodynamic shear force and torque is translated into force on the bonds at the rear of the neutrophil [35,60,129,196]. The force acting on the bonds increases with the forward motion of the cell until it balances the hydrodynamic shear force and torque, causing the cell to stop. Eventually, the stressed bonds at the rear fail, allowing the cell to undergo a jerky rolling motion in the direction of flow. As the cell rolls forward, it forms new bonds in the front, while the unloaded bonds, which were earlier in the center of the cell footprint, are now exposed at the rear and start to bear stress, causing the cell to retard. Since PSGL-1 is clustered on the tips of microvilli, the force acting on all bonds on a microvillus can be collectively summed up as the force acting on the microvillus, F_mv, causing it to stretch [150,152]. Neutrophils rolling at high shear stress are known to form long membrane tethers (discussed in section 4.2.3) at the rear of the cell and these tethers have been suggested to contribute synergistically with the stressed microvilli to balance the hydrodynamic shear force and torque [142,155,166,191]. Recently, qDF microscopy revealed that neutrophils rolling at high shear stress (τ_w ≥ 0.6 Pa) form 3-4 long tethers behind the cell footprint and these tethers synergize with less than 10 stretched microvilli in the rear portion of the cell footprint to stabilize rolling [191].

Fig. 2.

Forces acting on a rolling neutrophil. F_s and T_s are the shear force (along x-axis) and torque (along y-axis about the cell center), respectively, exerted by the streaming fluid on the rolling cell. A single long tether ‘te1’ (representative of n tethers) and a stressed microvilli ‘mv1’ (representative of m stressed tetherless microvilli) are shown. The distance of the tether and microvillus bonds anchorage points from the center of the cell is denoted as L_i (i = 1 to n) and l_j (j = 1 to m), respectively. The tethers and the stressed microvilli are shown to form an angle φ_i (i = 1 to n) and θ_j (j = 1 to m) with the P-selectin substrate, respectively. The hydrodynamic force acting on the tethers and stressed tetherless microvilli is denoted as F_te,i (i = 1 to n) and F_mv,j (j = 1 to m), respectively. The cell radius is denoted by r_c. The normal reaction force exerted by the substrate on the cell is denoted as F_N. P-selectin on the substrate and PSGL-1 on the microvilli shown in blue and green color, respectively. The Cartesian coordinate system is shown in grey. v_x(z) is the fluid velocity along x-axis. Not drawn to scale. Symbols of biophysical parameters are also explained in Table 2.

A schematic of a neutrophil rolling on P-selectin coated substrate under a shear flow (τ_w ≥ 0.6 Pa) is shown in Fig. 2. As the Reynolds number is very much less than unity, the inertial forces are negligible [54,63]. The shear force F_s and torque T_s acting on a stationary sphere close to the wall in a semi-infinite linear shear flow are calculated from Goldman et al.'s expressions [54] shown in Eqs. 1 and 2, assuming a cell body to substrate separation distance of 25 nm [191]:

$$F_s=31.97{\tau }_{\mathrm{w}}{r}_c^2$$ (1)

$$T_s=11.87{\tau }_{\mathrm{w}}{r}_c^2$$ (2)

where τ_w is the wall shear stress and r_c is cell radius. A force balance on the rolling neutrophil along the x and y-axis results in Eqs. 3 and 4:

$$F_s=\sum _{i=1}^n{F}_{te,i}\text{cos}{\phi }_i+\sum _{j=1}^m{F}_{mv,j}\text{cos}{\theta }_j$$ (3)

$$F_N=\sum _{i=1}^n{F}_{te,i}\text{sin}{\phi }_{\mathrm{i}}+\sum _{j=1}^m{F}_{mv,j}\text{sin}{\theta }_j$$ (4)

where F_N is the normal reaction force, F_te,i is the force on the i^th tether, F_mv,j is the force on the j^th stressed microvillus (the symbol is used to denote the force when there is no tether), and φ_i and θ_i are the corresponding angles as shown in Fig. 2. The torque in the y direction about the cell center due to F_te,i is denoted by T_tei, and due to F_mv,j by T_mv,j. A moment balance about the center of the rolling neutrophil results in Eq. 5:

$$T_s=\sum _{i=1}^n{T}_{te,i}+\sum _{j=1}^m{T}_{mv,j}$$ (5)

The angles φ_i and θ_j can be calculated from L_i and l_j, respectively. The solution for the forces acting on the individual tethers and stressed microvilli and the torques generated by these forces requires the knowledge of their relative contribution in counterbalancing the hydrodynamic drag.

4. Biomechanics of selectin-mediated rolling

4.1. Molecular biomechanics

As discussed in the previous section, hydrodynamic shear forces acting on the rolling neutrophils are translated into molecular forces of disruption that stretch the P-selectin-PSGL-1 bonds at the rear of the rolling neutrophil [35,60,196]. Stretched P-selectin-PSGL-1 bonds behave like Hookean springs and the force F_b acting on a stretched bond is a linear function of the strain in the direction of the force as shown in Eq. 6 [22,34,35,152]:

$$F_{\mathrm{b}}=\sigma (L_{\text{sep}}-\lambda )$$ (6)

where L_sep and λ are the stressed and unstressed length of the P-selectin-PSGL-1 molecular complex, respectively and σ is the bound state spring constant of the complex. F_b = 0, if L_sep ≤ λ. Thus, neutrophil rolling on a P-selectin substrate in the presence of hydrodynamic shear flow is a consequence of the temporal balance between the hydrodynamic and Hookean forces and the torques generated by them. The strain in P-selectin-PSGL-1 bonds in response to tensile forces in the range that are generated during rolling of neutrophils under physiological conditions has been measured using atomic force microscopy [51]. The P-selectin-PSGL-1 bonds were shown to withstand tensile forces up to 165 pN and extend up to 125 nm [51]. Molecular strain measurements in response to piconewton forces in live rolling neutrophils were made by qDF microscopy [191]. qDF micrographs were used to generate 3D footprints of rolling neutrophils (representative example shown in Fig. 3) which show the x- and z-coordinates of individual microvilli tips in the footprint. These 3D footprints (Fig. 3) revealed that the P-selectin-PSGL-1 bonds are unstressed (L_sep = λ = 70 nm [144,150,152]) in the front where they form, compressed in the center (L_sep < 70 nm), and stressed beyond the equilibrium length (L_sep > 70 nm) at the rear of the footprint where they finally fail. An average maximum strain, L_sep(max), of 133 nm was measured for P-selectin-PSGL-1 bonds at the rear of the footprint, which is 190% of the unstressed length, λ, of the P-selectin-PSGL-1 molecular complex. Based on Eq. 6, these results experimentally validate that the molecular bonds only at the rear and not in the front or center of the footprint of a rolling neutrophil are under tensile stress.

Fig. 3.

Stressed, compressed, and unstressed bonds in the footprint of a rolling neutrophil [3D-reconstruction adapted from Sundd, P. et al. Nat Methods 7, 821-824 (2010) with permission]. 3D reconstruction of the footprint of an EGFP-expressing neutrophil rolling on P-selectin coated substrate generated using qDF microscopy. A microvillus is defined as a conical protrusion longer than 25 nm in z-direction. Only three microvilli (mv1, mv2, and mv3) are labeled. Schematics of P-selectin-PSGL-1 molecular complex (not to scale) drawn on the tips of microvilli. As the cell rolls from left to right, the P-selectin-PSGL-1 bonds are formed in the front at the unstressed length (L_sep ~ 70 nm; mv3), compressed in the center (L_sep < 70 nm; mv2) and stressed beyond the unstressed length (L_sep > 70 nm; mv3) at the rear of the footprint where they finally break. P-selectin and PSGL-1 shown as blue and green, respectively. Wall shear stress 0.6 Pa. P-selectin 20 molecules/μm².

Neutrophil rolling requires rapid formation of new bonds in the front and rapid dissociation of stressed bonds at the rear under the influence of shear forces [35,60,196]. The forces acting on the bonds can regulate the dynamics of neutrophil rolling by facilitating or retarding the dissociation of the stressed bonds at the rear. The selectin-carbohydrate molecular interaction is reversible and regulated by the chemical kinetics of the bond formation and dissociation as shown in Eq. 7 [15,35]:

$$\mathrm{R}+\mathrm{L}\underset{k_r}{\overset{k_f}{\rightleftarrows }}\mathrm{RL}$$ (7)

where R is the receptor (P-selectin), L is the ligand (PSGL-1), RL is the receptor-ligand (P-selectin-PSGL-1) molecular complex, and k_f and k_r are the forward and reverse reaction rates, respectively. The reverse reaction follows first order chemical kinetics, but the lifetime of the selectin-carbohydrate bond (1/k_r) is also modulated by the force acting on the bond. Depending on the bond force, selectin-carbohydrate bonds may behave as ‘catch bonds’ for which the life time increases, or ‘slip bonds’, for which the lifetime decreases with increasing bond force [34,35,123]. These bonds have been shown to regulate neutrophil rolling under physiological wall shear stresses [27,123,129,143,219].

4.1.1. Slip bond kinetics

Neutrophil rolling on P-selectin substrates at venular wall shear stresses (τ_w > 0.12 Pa) [33] in vitro is predominantly mediated by slip bonds [123]. Slip bonds exhibit the intuitive behavior of decreasing their lifetime in response to increasing bond forces [34-35,123]. The bond force is expected to reduce the free energy barrier between the bound and free states, thereby promoting receptor-ligand bond dissociation [15,35,47]. However, the optimal force above which different selectin-carbohydrate bonds behave like slip bonds differ between P- and L-selectin [129]. The P-selectin-PSGL-1 bonds that mediate neutrophil rolling in in vitro flow chambers behave as slip bonds at wall shear stresses in excess of 0.02 Pa, which corresponds to a force of at least 20 pN per bond [123]. In contrast, L-selectin-PSGL-1 bonds that mediate secondary capture of neutrophils in vitro [13,143,205] behave as slip bonds at wall shear stresses in excess of 0.1 Pa, which corresponds to a bond force of at least 100 pN [129,219,228]. The dissimilarity in the slip bond behavior exhibited by different selectins binding to the same ligand is a result of the difference in intrinsic kinetics of bond formation and dissociation which is a consequence of differences in their molecular structure and tensile force-induced conformational change in lectin-EGF domain interface (discussed in section 4.1.2) [121,122,129,186]. Over the last two decades, several in vitro studies have experimentally validated the kinetics of slip-bond assisted neutrophil rolling on P-selectin substrates [7,27,156,179]. These experiments involved tethering and rolling of isolated neutrophils on very low densities of immobilized P-selectin at venular wall shear stresses, conditions which favor unimolecular interactions between neutrophil and P-selectin substrate. In these experiments, the lifetime of transient tethers were measured at different wall shear stresses and the rate of detachment of the tethered neutrophils from the substrate was used as an estimate for the rate of P-selectin-PSGL-1 bond dissociation. The experimental data was matched with a theoretical model of receptor-ligand interaction in the presence of tensile force, and the reverse reaction rate was extracted as a function of bond force. Two theoretical models that have been widely used over the last two decades to describe the force dependence of slip bond dissociation kinetics are the model for specific adhesion of cells to cells proposed by Bell (1978) [15] and the Hookean spring model for adhesion of a membrane to surface proposed by Dembo (1988) [35]. Within the ranges of forces and distances found during rolling, the two models yield nearly identical results (Pospieszalska and Ley, unpublished data).

Bell model

The Bell model [15] states that the lifetime of a receptor-ligand bond decreases exponentially with increasing tensile force; therefore, a receptor-ligand bond subjected to tensile force behaves as a ‘slip’ bond. The forward and reverse reactions are shown in Eq. 7 and the tensile force acting on a bond is shown in Eq. 6. The dependence of the reverse reaction rate, k_r, on the tensile force F_b acting on the bond is defined by:

$$k_r=k_r^o\text{exp}\left[\frac{\gamma F_b}{{\kappa }_B\mathrm{T}}\right]$$ (8)

where $k_r^o$ is the unstressed reverse reaction rate at zero force (F_b = 0) and γ is the reactive compliance with units of length. The reactive compliance describes the susceptibility of the bond to fail under tensile force. Large γ implies a more compliant bond. κ_B is Boltzmann's constant and T is the absolute temperature. Based on the Boltzmann distribution at equilibrium, the forward reaction rate, k_f, was proposed to be dependent on the separation distance, L_new, between the bases of the receptor and the ligand [22] as shown in Eq. 9:

$$k_f=k_f^o\text{exp}\left[\frac{\sigma \mid L_{\text{new}}-\lambda \mid \left(\gamma -\frac{1}{2}\mid L_{\text{new}}-\lambda \mid \right)}{{\kappa }_{\mathrm{B}}\mathrm{T}}\right]$$ (9)

Where L_new is the length of the new bond, λ is the equilibrium unstressed length of the receptor-ligand bond, $k_f^o$ is the unstressed forward reaction rate which is defined as the forward rate when the new bond is of the same length as the equilibrium bond length, i.e. L_new = λ. Several in vitro flow chamber studies have validated that the force dependence of the selectin-ligand reverse reaction rate follows Bell model in the force regime that favors slip bonds [5,7,28,156,179].

Dembo model

The model proposed by Dembo et. al. (1988) [35] is based on the assumption that the unstressed transition state and the bound state of the receptor-ligand complex only differ in their spring constants, and the equilibrium constant for the formation of the stretched bond can be defined as:

$$\frac{k_f}{k_r}=\text{exp}\left[\frac{\text{-}{\sigma }_{\mathrm{ts}}{\left(L_{\text{sep}}-\lambda \right)}^2}{2{\kappa }_{\mathrm{B}}\mathrm{T}}\right]$$ (10)

where k_f and k_r are forward and reverse reaction rate, respectively; $k_f^o$ and $k_r^o$ are the unstressed forward and reverse reaction rate, respectively, σ is the bound state spring constant, L_sep is the stretched length of the bond, λ is the equilibrium length of the bond, κ_B is the Boltzmann's constant, and T is the absolute temperature. The forward reaction rate is defined as:

$$k_f=k_f^o\text{exp}\left[\frac{\text{-}{\sigma }_{\mathrm{ts}}{\left(L_{\text{sep}}-\lambda \right)}^2}{2{\kappa }_{\mathrm{B}}\mathrm{T}}\right]$$ (11)

where σ_ts is the transition state spring constant. Substitution of Eq. 11 in Eq. 10 results in the reverse reaction rate, k_r, shown in Eq. 12:

$$k_r=k_f^o\text{exp}\left[\frac{\left(\sigma \text{-}{\sigma }_{\mathrm{ts}}\right){\left(L_{\text{sep}}-\lambda \right)}^2}{2{\kappa }_{\mathrm{B}}\mathrm{T}}\right]$$ (12)

Bonds behave as ‘slip’ bonds for σ > σ_ts and ‘catch’ bonds (discussed in section 4.1.2) if σ < σ_ts. The problem with the Bell and Dembo models is that γ and σ_ts cannot be measured experimentally. Therefore, estimates based on curve fitting are widely used [4-5,7,39,123,156,179].

The symbols used for biophysical parameters are defined in Table 2. The values of $k_f^o$ and $k_r^o$ for different selectin-ligand pairs have been measured experimentally using in vitro rolling assays [5,7,27,156,179], surface plasmon resonance [90,130,135], atomic force microscopy [61,62,123], and micropipette manipulation [195]. Values of these parameters are shown in Table 3. Although the Bell and Dembo models predict the force dependence of slip bond dissociation kinetics, they are not suitable for validating the catch-slip bond transitional behavior exhibited by selectin-ligand bonds from small to large forces. The two pathway model (discussed in section 4.1.2) proposed by Evans and coworkers (2004) is currently the most widely used model to study the catch-slip bond transition [46].

Table 2.

Nomenclature of biophysical parameters.

Symbol	Parameter
F s	Hydrodynamic shear force acting on rolling neutrophil along x-axis (pN)
T s	Hydrodynamic torque acting on rolling neutrophil (about the cell center) along y-axis (pN.μm)
F N	Normal reaction force acting on rolling neutrophil (pN)
r c	Cell radius (μm)
τ w	Wall shear stress (Pa)
νx(z)	x-component of fluid velocity as a function of z distance from the substrate
F te,i	Force on ith tether (pN)
F mv,j	Force on jth stressed tetherless microvillus (pN)
φ i	Tether angle with the P-selectin substrate (degrees)
θ j	Angle of stressed tetherless microvillus with P-selectin substrate (degrees)
T te,i	Torque (about the cell center) acting on rolling neutrophil along y-axis due to the tether force Fte,i (pN μm)
T mv,j	Torque (about the cell center) acting on rolling neutrophil along y-axis due to the microvillus force Fmv,j (pN μm)
L i	Distance (along x-axis) of the anchorage point of tether bonds from the center of the cell (μm)
l j	Distance (along x-axis) of the anchorage point of stressed (tetherless) microvillus bonds from the center of the cell (μm)
F b	Bond force (pN)
σ	Bound state spring constant of the bond (pN/μm)
L sep	Stressed length of the bond (nm)
λ	Unstressed length of the bond (nm)
R	Receptor
L	Ligand
RL	Receptor-ligand complex
$k_f^o$	Unstressed forward reaction rate at Fb = 0 (M-1s-1)
$k_r^o$	Unstressed reverse reaction rate at Fb = 0 (s-1)
kf	Forward reaction rate at Fb > 0 (M-1s-1)
kr	Reverse reaction rate at Fb > 0 (s-1)
γ	Reactive compliance (Å)
κ B	Boltzmann's constant (J/°K)
T	Absolute temperature (°K)
L new	Length of the possible new bond (nm)
σ ts	Transition state spring constant (pN/μm)
k1rup	Reverse reaction rate for pathway 1 (s-1)
k2rup	Reverse reaction rate for pathway 2 (s-1)
$k_2^o$	Unstressed reverse reaction rate for the pathway 2 at Fb = 0 (s-1)
S1/S2	Occupancy ratio of the two bound states
Φ o	Occupancy ratio of the two bound states at Fb = 0
f 12	Bond force at which occupancy ratio changes 2.7 fold (pN)
ΔE 21	Energy barrier between two bound states (J)
Fmv(t)	Pulling force on the microvillus (pN)
Ḟmv(t)	Loading rate (pN/s) for microvillus
F th	Threshold force for dissociation of membrane from cytoskeleton (pN)
Lmv(t)	Microvillus extension (μm)
L̇mv(t)	Rate of microvillus extension (μm/s)
σ mv	Microvillus spring constant (pN/ μm)
κ mv	Spring constant of the second elastic unit in Fig. 8(c) (pN/ μm)
η eff_mv	Effective viscosity of microvilli (pN s/ μm)
Fte(t)	Pulling force on tether (pN)
Ḟte(t)	Loading rate (pN/s) for tether
Lte(t)	Tether extension (μm)
L̇te(t)	Rate of tether extension (μm/s)
η eff_te	Effective viscosity of tether (pN s/ μm)
k c	Cell membrane curvature modulus (pN μm)
d int	Interfacial drag coefficient (pN s/μm3)

Symbols for different biophysical parameters are defined. Units are shown in parenthesis.

Table 3.

Kinetic parameters for different receptor-ligand pairs.

Receptor-ligand	$k_f^o\left({\mathrm{M}}^{\text{-}1}{\mathrm{s}}^{\text{-}1}\right)$	$k_r^o\left({\mathrm{s}}^{\text{-}1}\right)$	Kd (μM)	γ (Å)	Ref.
P-selectin-PSGL-1	4.4 × 106	1.4	0.3		[130]
P-selectin-PSGL-1	-	1	-	0.5	[7]
P-selectin-PSGL-1	-	0.95	-	0.39	[5]
P-selectin-PSGL-1	9.3 × 105	1.4	1.5	-	[90]
P-selectin-PSGL-1	-	2.68	-	-	[29]
P-selectin-PSGL-1	-	0.55	-	-	[227]
P-selectin-PSGL-1	-	0.9	-	-	[70]
P-selectin-PSGL-1	-	0.84	-	0.68	[27]
P-selectin-PSGL-1	-	0.19	-	1.2	[61]
P-selectin-PSGL-1	-	0.18	-	1.35	[62]
sP-selectin-PSGL-1	-	1.1	-	0.7	[156]
mdP-selectin-PSGL-1	-	1	-	0.42	[156]
L-selectin-PSGL-1	-	10.2	47	-	[29, 90]
L-selectin-PSGL-1	-	7	-	0.24	[4]
L-selectin-PSGL-1	-	0.83	-	1.51	[62]
L-selectin-PNAd	-	6.6	-	0.19	[5]
L-selectin-GlyCAM-1	≥ 105	≥ 10	108	-	[135]
E-selectin-PSGL-1	-	0.24	-	1.11	[62]
E-selectin-ESL-1	7.4 × 104	4.6	62	-	[212]
E-selectin-ESL	-	0.7	-	0.3	[5]
E-selectin-sLex	-	0.82	-	0.34	[195]

For the different ligand pairs, $k_f^o$ – unstressed forward reaction rate, $k_r^o$ – unstressed reverse reaction rate, K_d – equilibrium dissociation constant, and γ – reactive compliance were estimated by different studies mentioned under Ref. sP-selectin, soluble monomeric P-selectin, mdP-selectin, membrane dimeric P-selectin and ESL, an E-selectin ligand on human neutrophil.

4.1.2. Catch bond kinetics and shear threshold phenomenon

As a theoretical possibility, Dembo and coworkers (1988) suggested that the lifetime of a receptor-ligand bond may decrease, increase or remain unchanged upon application of an external force; these bonds were referred to as ‘slip’, ‘catch’ and ‘ideal’ bonds, respectively [35]. The first experimental demonstration of the counter-intuitive phenomenon in which P-selectinligand bonds resist breakage and become stronger under the influence of a “low” external force was made in 2003 [123]. Application of higher forces dissociates the P-selectin-PSGL-1 bond via the slip dissociation pathway (discussed in section 4.1.1). To date, the catch-to-slip bond transition under progressively increasing forces has been demonstrated experimentally for P-selectin-PSGL-1 [46,123] and L-selectin binding to PSGL-1 and endoglycan [161]. It is noteworthy that the L-selectin-PSGL-1 bond strengthening due to the catch bond behavior, as quantified by the decrease in k_r relative to the unstressed reverse reaction rate constant, $k_r^o$, is more pronounced than that of P-selectin-PSGL-1 bond (5-fold versus 2-fold decrease) [123,161]. Moreover, this catch bond behavior is detected over a wider range of external forces for L-selectin (up to 50 pN per bond) relative to P-selectin (up to 20 pN per bond) [123,161].

Evans and colleagues (2004) proposed a two-pathway model to describe the catch-slip bond behavior [46]. The two pathways for bond dissociation originate from two possible bound state configurations. Pathway 1, which occurs with a bond dissociation rate k_1rup, is very fast and dominates at low forces, whereas pathway 2 with a bond dissociation rate k_2rup is slower and dominates in the high shear regime. It is assumed that k_1rup for the fast pathway is constant [46]. For the slow pathway, bond dissociation follows the Bell model (Eq. 8), and can be defined as $k_{2rup}=k_2^0$ exp(γF_b/κ_BT), where $k_2^0$ is the unstressed reverse reaction rate for that pathway and F_b is the force on the bond (Eq. 2) [15]. The selectin-ligand bond dissociates along the dominant pathway, which is determined by the occupancy ratio of the two states (S₁/S₂). Under the application of force, the energy of each state shifts to cause a change in the occupancy ratio of the two bound states as shown in Eqs. 13 and 14 [46]:

$$\frac{S_1}{S_2}={\Phi }_{\mathrm{o}}\text{exp}\left[\text{-}\frac{F_{\mathrm{b}}}{f_{12}}\right]$$ (13)

$${\Phi }_{\mathrm{o}}=\text{exp}\left[\frac{\Delta E_{21}}{{\kappa }_{\mathrm{B}}\mathrm{T}}\right]$$ (14)

where Φ₀ is the initial occupancy ratio under static conditions, f₁₂ is the force scale at which the occupancy ratio (S₁/S₂) changes 2.7-fold, and ΔE₂₁ represents the difference in energy between the two states which is about 4 - 5κ_BT for P-selectin-PSGL-1 binding [46]. The switch from the low impedance pathway 1 to the high impedance pathway 2 by the application of force is governed by the change in the occupancies of the two bound states, and this determines the transition from catch to slip bond behavior. For bonds exhibiting catch-slip transition, the overall bond dissociation rate can be defined as shown in Eq. 15 [46]:

$$k_r=\frac{{\Phi }_{\mathrm{o}}{k}_{1\text{rup}}+\text{exp}\left(\frac{F_{\mathrm{b}}}{f_{12}}\right)\left[k_2^{\mathrm{o}}\text{exp}\left(\frac{\gamma F_{\mathrm{b}}}{{\kappa }_{\mathrm{B}}\mathrm{T}}\right)\right]}{{\Phi }_{\mathrm{o}}+\text{exp}\left(\frac{F_{\mathrm{b}}}{f_{12}}\right)}$$ (15)

Based on the crystal structure and molecular dynamics simulations of L- or P-selectin bound to sLe^x or PSGL-1-derived glycosulfopeptide, site-directed mutagenesis, Monte Carlo modeling, and flow chamber assays [90-91,121,129,148,181], two models, the ‘sliding-rebinding mechanism’ [121,122,129] and the ‘allosteric model’ [186,206], have been proposed to relate ‘catch-slip’ bond behavior to the tertiary structure of lectin-EGF domains of selectins. The ‘sliding-rebinding’ model emphasizes the angle between the lectin domain and the EGF-like domain of the selectin molecule [121,122]. Molecular dynamics simulations revealed that the structure of P-selectin bound to PSGL-1 has an interdomain angle of 139.3°, which corresponds to the open-angle or extended conformation, whereas that of P-selectin bound to sLe^x has an interdomain angle of 114.6° which corresponds to the closed-angle or bent conformation [122]. When tensile force is applied to the selectin-ligand complex, there is a greater probability for the selectin to exist in the open angle or extended conformation [121]. Taking into account the noncovalent interactions at the atomic level, molecular dynamics simulations of P-selectin unbinding from PSGL-1 showed that the interdomain angle opening allows rotation of the lectin domain, tilts the binding interface and aligns it with the direction of force [122]. This is thought to result in sliding of the ligand along the binding interface causing the preexisting interactions to dissociate and allowing new interactions between the two sides of the interface, thereby bringing the system back to its previously bound state. This prolongs the lifetimes by slowing dissociation, resulting in catch bonds. The PSGL-1 glycan: fucose, sialic acid, galactose, N-acetylglucosamine and eight residues from the PSGL-1 peptide including three sulfated tyrosines were observed to form new interactions most frequently [122]. Once the interdomain angle is fully open, the probability for rebinding becomes maximum. Further increase in applied force will not increase the lifetimes, but rather accelerate the dissociation resulting in the catch-to-slip bond transition [122].

The ‘allosteric model’ was introduced by Waldron and Springer (2009) based on the crystal structures of P-selectin in the absence or presence of ligand [206]. Waldron and Springer (2009) identified a lobe in the lectin domain that reorients upon ligand binding [206]. Reorientation of this lobe induces the movement of a loop (residues 83-89), which approaches the ligand binding site, thereby augmenting the ligand binding interface [206]. To test this hypothesis, Waldron and Springer (2009) mutated alanine 28 to histidine (A28H), which fills up the space and opens the lobe in the lectin domain. Flow-based adhesion assays revealed that cells displaying the A28H mutated P-selectin relative to wild-type P-selectin, exhibited slower rolling on a PSGL-1 substrate at higher shear stresses, although no differences were noted in the low shear regime (τ_w ≤ 0.2 Pa) [206]. The dissociation constant (K_d) measured by surface plasmon resonance was 3- to 4-fold lower for mutated than wild-type P-selectin [206]. This increase in the binding affinity of A28H P-selectin for PSGL-1 provides evidence for an allosteric pathway through the lectin domain, which connects the conformational change at the lectin-EGF domains to the ligand-binding interface [186,206]. A simple allosteric model together with an equilibrium constant K_c describing the ratio of the two unbound conformations was introduced to explain the catch bond behavior of P-selectin-PSGL-1 binding [206]. Increasing the value of K_c favors the higher-affinity extended conformation, thereby resulting in enhancement of the overall binding affinity. In addition to the mechanistic interpretation provided by the ‘sliding-rebinding’ model [121,122], the ‘allosteric model’ suggests that force could act as an allosteric effector in the mechanochemistry of selectin-ligand bindings [186,206]. Similar force-induced conformational changes have been observed in integrins [9,73,175].

Simulation studies suggest that the catch-slip bond kinetics at the nanoscale level are primarily responsible for the shear-threshold phenomenon in which the extent of neutrophil rolling first increases and then decreases while monotonically increasing the wall shear stress [145]. Although shear-induced cell deformation can increase the contact area between the cell and the substrate, thereby enhancing the binding probability, numerical studies suggest that cell deformation plays a rather auxiliary role in the shear threshold phenomenon [145]. This notion is further supported by experimental observations showing that cell-free flow assays using purified selectins and their respective ligands can successfully recapitulate the shear threshold phenomenon [123,219]. In addition to the catch bond behavior, the effect of shear flow on the on-rate has been suggested to contribute to the shear threshold phenomenon [23,25]. Under the influence of a viscous flow and the presence of a vessel wall, the cell translational velocity is always larger than the surface tangential velocity, which results in a relative motion between the cell receptor and the immobilized ligand on the vessel wall [54]. This relative motion allows each cell receptor of a free-flowing cell to potentially react with any counter-receptor present in its reactive zone, thereby increasing the receptor-ligand encounter rate for bond formation [23,25]. In addition to the relative sliding between cell receptors and immobilized ligands on the surface, Zhu and coworkers (2007) proposed two other transport mechanisms for the flow-enhanced bond formation between a free-flowing cell and the substrate: Brownian motion and molecular diffusion [220]. The Brownian motion causes a cell to move with a wavy trajectory perpendicular to the wall thereby altering the gap distance between receptors and immobilized ligands. Binding can only occur when a ligand locates within the reactive zone of a receptor on the free flowing cell [23]; as a result, the Brownian motion of the cell modulates the initial bond formation by inducing fluctuations in gap distance [220]. Although the movement of receptors and ligands is limited due to their anchorage on their respective cell/wall surfaces, receptors and ligands can still undergo rotational diffusion, which orients their binding sites for molecular docking [220]. The molecular diffusivity of selectins can be increased by enhancing the hinge flexibility of the lectin and EGF domains, such as by disrupting the Tyr37-Asn138 hydrogen bond in L-selectin [121]. Compared to wild-type L-selectin, microspheres bearing a more flexible L-selectin (N138G mutated) hinge display a higher tethering rate on PSGL-1-coated surfaces [121]. In addition, the greater hinge flexibility of mutated L-selectin further prolongs bond lifetimes at smaller forces, thereby lowering the shear threshold for rolling from 0.07 to 0.04 Pa [121]. Taken together, experimental and simulation studies collectively show that the origin of shear threshold phenomenon is caused primarily by the mechanical regulation of the association and dissociation rates.

4.1.3. Role of bond clusters in leukocyte rolling

It has been well documented that the spatial patterns of receptor-ligand bonds, in particular bond clusters, are significant factors in leukocyte rolling on P-selectin [56,81,87,152,173]. In neutrophils the bond clustering phenomenon is enabled by local concentrations of neutrophil microvilli visible as cell surface ridges [68,163], concentrations of PSGL-1 molecules (ligands) on microvilli tips [133], and local concentrations of P-selectin molecules (receptors) on the substrate. All three of these conditions are present prior to the beginning of a cell rolling process.

Two modeling studies suggest that the first two factors do not affect leukocyte rolling much. Shao and Xu [173] simulated an adhesion process (by cell-substrate contact, no rolling) for normal cells with ligands concentrated on the microvilli tips and smooth cells (no microvilli) with the same number of ligands distributed randomly on the cell's surface. In both cases the receptors were of the same density and randomly distributed on the substrate. They tested the receptor and ligand densities up to 3000 molecules/μm² and found that the ligand concentration on the microvilli tips did not affect the total number of adhesion events much. Pospieszalska et al. [152] conducted neutrophil rolling simulations by ETMA (Event-Tracking Model of Adhesion) for 941 microvilli per cell and 8 PSGL-1 dimers per microvillus (Case I) or 2,306 microvilli per cell and 4 PSGL-1 dimers per microvillus (Case II). In both cases, P-selectin at a site density of 150 molecules/μm² was distributed randomly on the substrate and the shear stress was 0.05 Pa. Each dimer could form one bond with P-selectin which was assumed to be monomeric [123]. Despite the drastic differences in microvillus and ligand distributions, the rolling for the two cases was similar when taking into consideration different number of PSGL-1 dimers per cell.

In a second simulation in the study by Shao and Xu [173], when the receptors at a fixed average density were randomly distributed on the substrate, or as random clusters of three molecules, or as random clusters of seven molecules, the number of adhesion events decreased with the increase in size of the cluster. Although in vitro rolling studies assume P-selectin to be randomly distributed on the substrate, recent studies have shown that P-selectin is preferentially presented as clusters of dimers in the clathrin-coated pits on EC and this clustering takes place through the interaction of the cytoplasmic domains of dimeric P-selectin with clathrin-coated pits [129,168,170]. Even a random distribution does not imply that there are no P-selectin clusters. For example, assuming that P-selectin at 150 molecules/μm² is randomly distributed on a 100-μm² substrate, there is a 22.4% chance that an area of 0.02 μm² will contain a three-receptor cluster (Pospieszalska and Ley, unpublished data).

Figure 4a shows the positions of the bond receptor bases (places where the receptors are attached to the substrate) at a selected time point generated by ETMA. The plane xy represents the substrate, and the cell translates in the positive x-direction. The red colored circles indicate the receptor bases of load-bearing bonds. Despite a random distribution of receptors, there are three-load-bearing-bond clusters and one two-load-bearing-bond cluster (all four marked by ovals) with each cluster corresponding to a different microvillus. Figure 4b shows the history of the 11 bonds seen in Fig. 4a as load bearing. Each bond is represented by a horizontal segment with three points. The left point marks the time when the bond is formed, the middle point marks the time when the bond becomes load-bearing, and the right point marks the time when the bond dissociates. The bonds form and dissociate at different times, but all except one become load-bearing approximately at the same time. This causes a sudden increase in the total bond force acting on the cell, and the cell suddenly slows down or suddenly stops moving temporarily (the latter occurs in the case illustrated in Fig. 4, data not shown). That is a source, but not exclusively, of the jerky motion seen in cell rolling. Causing sudden reductions in the cell translational velocity and sudden cell stops is the most pronounced role of bond clusters.

Fig. 4.

ETMA simulations reveal bond clusters in the footprint of a rolling neutrophil. (a) Distribution of the PSGL-1-P-selectin bond bases in the footprint of a rolling neutrophil (the cell translates in the positive x-direction). The load-bearing bonds are marked with red circles. Ovals indicate the load-bearing bond clusters and each cluster corresponds to a different microvillus. (b) Lifetime of 11 load-bearing bonds seen in (a). Each bond is represented by a black segment (the no-load time period) and a red segment (the load time period). All bonds but one become load-bearing at the same time (t = 4 s). Unpublished data (Pospieszalska and Ley).

4.1.4. Role of cell-substrate separation distance

The cell-substrate separation distance is defined as the minimum distance between the cell body and the substrate excluding the bendable microvilli (Fig. 5-insert). Unlike bond clusters (discussed in section 4.1.3), the cell-substrate separation is a consequence of cell rolling except during the initial state when the cell is freely floating in the medium. In ETMA simulations [152] conducted for neutrophils rolling on a P-selectin coated substrate (150 molecules/μm²) at a wall shear stress of 0.05 Pa, a freely floating cell in shear flow approaches the substrate because of gravity to form a first PSGL-1-P-selectin bond. The simulations revealed that the cell-substrate distance varies following loading and dissociation of individual bonds during stable rolling. When at least one load-bearing bond was present, the cell was attached to the substrate resulting in a small separation distance with an average value of 0.16 μm [150,152]. This separation distance depends on the component of the total bond force perpendicular to the substrate and the net colloidal force [151], while the total bond force depends on the number of load-bearing bonds and the configuration of the loaded bonds and tethers in the footprint of a rolling neutrophil [150,152,165]. The dissociation of the last load-bearing bond leads to two possible outcomes: if no other bonds (not-loaded) are present, the cell-substrate separation distance will approach its initial value which is the separation distance for a freely flowing sphere near the wall, unless a new bond forms at the very beginning of the dissociation process. If there are other unloaded bonds present, then the increasing separation distance will result in loading of at least one of those bonds forcing the cell to return to the state of at least one load-bearing bond in the footprint. Thus, the gap between the cell and the substrate modulates cell rolling. During stable rolling, only the second possibility is realized. Usually many bonds form and dissociate without ever becoming load-bearing (31 - 44% reported in [152] which is in agreement with [165]). However, some of these bonds may become loaded when the cell-substrate separation suddenly increases, and prevents the cell from escaping from the vicinity of the substrate. An example of stable rolling generated using ETMA is given in Fig. 5 (Pospieszalska and Ley; unpublished data), which shows a scatter plot of the cell-substrate separation distance as a function of the cell translational velocity for Case II mentioned in section 4.1.3. A microvillus length of 0.2 μm and unstressed bond length of 0.07 μm were assumed [152]. Instantaneous changes in the separation distance are seen as individual traces. The trace marked with an asterisk represents the first, gravity driven approach of the cell to the substrate. The almost vertical trace for the high translational velocities represents instantaneous changes in the separation distance when no load-bearing bonds were present and the cell attempted to depart from the substrate.

Fig. 5.

Scatter plot showing cell translational velocity as a function of cell body to substrate seperation distance (distance does not include microvilli-see inset). Instantaneous changes in the separation distance are seen as individual traces. The asterisk indicates the trace representing the first approach of the cell to the substrate. The almost vertical trace for the high translational velocities represents instantaneous changes in the separation distance when no load-bearing bonds were present. The trace discontinuity near velocity zero is due to a discrete process of modeling owing to the time step. Unpublished data (Pospieszalska and Ley).

4.2. Cellular biomechanics

4.2.1. Whole cell deformation

Theoretical [36,37,72,84-86] and experimental [48,49,142,218] studies have shown that both molecular (i.e. kinetic and micromechanical properties of receptor-ligand pairs) and cellular properties such as cell deformation have a significant impact on the dynamics of rolling when cells are subjected to high yet physiologically relevant shear stresses. For instance, deformable polymorphonuclear leukocytes roll significantly slower and smoother than rigid microbeads coated with a similar density of selectin ligand molecules [142]. Bulk cell deformation is predicted to increase the contact area between the cell and substrate [33,49], as was recently confirmed experimentally [191]. The hydrodynamic loading of receptor-ligand bonds is also reduced due to a flattened cell shape [36,37,86].

A 3-D computational model of leukocyte rolling was proposed to simulate rolling of a PSGL-1-decorated deformable cell on a P-selectin-coated surface under flow [72]. In this model, the immersed boundary method (IBM) [146] was used to simulate motion of an elastic capsule near a plane wall in a linear shear field. The IBM was coupled to the Hookean spring model (Eq. 10-12) to simulate the force-dependent kinetics of receptor-ligand interactions [35], while the stochastic behavior of P-selectin-PSGL-1 bond formation and breakage was simulated by the Monte Carlo method [26,60]. The model successfully predicted that for a given shear rate, the extent of cell deformation and the cell-substrate contact area increase with decreasing cell membrane stiffness and thus increasing compliance. Furthermore, the rolling of more compliant cells is relatively smoother and slower compared to cells with stiffer membranes [72] which is in accord with experimental observations [142,218]. The average number of bonds for a cell as well as those for a single microvillus was found to increase with decreasing values of membrane stiffness which translates to increased deformability [72]. Along these lines, bond lifetimes decreased with increasing shear rate and also with increasing membrane stiffness [72]. More recently, this model was extended to incorporate the three different kinetic models of receptor-ligand binding (viz. Bell, Dembo and Evans two-pathway model) and to account for microvillus deformability [145]. These simulations predicted that the catch-slip bond behavior and to a lesser extent bulk cell deformation contribute to the shear-threshold phenomenon observed in the low shear regime. Moreover, cells bearing deformable rather than rigid microvilli rolled slower only at elevated P-selectin site densities and wall shear rates (≥ 400 sec^-1). Another 3D computational model [84-85] of leukocyte rolling which assumes the cell to be a compound visoelastic drop (a viscoelastic nucleus covered with a thick layer of less viscoelastic cytoplasm) has shown that a cell with viscoelastic cytoplasm is more deformable than the one with Newtonian fluid-like cytoplasm when suitable parameters are used. Owing to the increased deformability, the hyrdrodynamic drag force and the torque acting on the rolling cell are significantly reduced for the viscoelastic cell resulting in more stable and smaller rolling velocity. The computational model in this study was based on the volume of fluid method coupled to the Hookean spring model [35] (Eqs. 10-12) for force-dependent receptor-ligand kinetics. Taken together, these computational studies suggest that the interplay of bulk cell deformation (mesoscale), microvillus deformability (microscale) and receptor-ligand binding kinetics (nanoscale) regulates cell rolling especially at elevated yet physiologically relevant shear stresses [145].

Quantitative dynamic footprinting (qDF) microscopy can be used to reconstruct the footprint of rolling neutrophils, which provides 3D locations of the microvilli tips as well as stretched lengths of the P-selectin-PSGL-1 bonds inside the cell-substrate contact area [191]. For a mouse neutrophil rolling on P-selectin at 0.6 Pa, the size of the footprint within 100 nm above the substrate was around 12 μm² which is about 4-fold larger than that previously estimated based on low-resolution in vivo measurements and modeling [33,152,191]. The large size of the footprint seems to be attributable to the cell deformation since a significantly smaller footprint (~3 μm²) was obtained in ETMA simulations by assuming a spherical cell shape (comparison shown in Fig. 6) [150, 152]. In fact, the location of a stretched microvillus was reported as far as 3 μm behind the cell center in the qDF assay, which is longer than the value (<1 μm) estimated by ETMA simulation (Fig. 6) [152,191]. Contrary to ETMA applied to non-deformable cells, the compound viscoelastic model of leukocyte has been shown to have much larger footprints and the footprint size was shown to be inversely related to the cytoplasmic viscosity [84]. These observations support the concept that the membrane deformation as well as cytoplasmic viscoelasticity both play important roles in the dynamics of cell rolling under high shear stress [229]. This may explain the failure of the numerical models [22,88,150] which assume the cell to be a nondeformable sphere, to reproduce stable rolling at high shear stresses commonly observed in vivo [33,49].

Fig. 6.

Whole cell deformation results in a larger footprint at high shear stress [(b) and (c) adapted from Sundd, P. et al. Nat Methods 7, 821-824 (2010) with permission]. (a) A schematic showing the side and bottom views of the footprint of a rolling neutrophil following deformation by the shear forces to a nonspherical shape. Dashed circle shows the nondeformed neutrophil for comparison. Red closed circles are the tips of microvilli in the footprint. (b) Coordinates of the microvilli tips in the footprint of an EGFP-expressing neutrophil rolling on P-selectin substrate extracted from qDF micrographs. P-selectin 20 molecules/μm². Wall shear stress 0.6 Pa. (c) Coordinates of the microvilli tips in the footprint of a non-deformable spherical rolling neutrophil revealed using ETMA simulations. P-selectin 150 molecules/μm². Wall shear stress 0.05 Pa. Color of the data points in (b) and (c) represents the z-coordinates of microvilli tips based on color bars shown on the right. Black arrows denote the direction of rolling.

4.2.2. Microvillus extension

Many living cells have an ability to form tubular surface protrusions in response to a low to moderate pulling force F(F < ~ 92 pN for neutrophils, Pospieszalska and Ley, unpublished data) applied locally to the cell membrane (Fig. 7b). If the force F is greater than the threshold force F_th (a non-zero pulling force below which the separation of the cell membrane from the cytoskeleton cannot occur), and acts for a sufficiently long time, a tether forms on the extension of the protrusion (Fig. 7c) [208]. The protrusion to tether transition is called the crossover, and is generally viewed in the literature as the termination of the protrusion's elongation [45,172,216]. The cellular protrusions and tethers (the latter are discussed in section 4.2.3) have been observed experimentally [16,45,65,71,142,154,155,166,167,172,191,215,216], analyzed theoretically [16,17,69,210], and modeled computationally [22,65,88,145,150,154,167,216,221]. In rolling neutrophils, pulling forces are applied to the tips of microvilli by molecular bonds formed between the microvilli and the substrate (Fig. 2). A protrusion formed at the tip of a microvillus is often described in literature as a microvillus extension (Fig. 7b).

Fig. 7.

Microvillus deformation under tensile force. Schematic showing a neutrophil interacting with a P-selectin coated substrate through PSGL-1 expressed on the microvillus tip. To denote the individual microvillus/tether force acting on the cell, symbol F_mv is used when there is no tether and symbol F_te when a tether is present. (a) Unstressed microvillus for F_mv = 0. (b) Stressed microvillus without tether for 0 < F_mv < F_th. (c) Stressed microvillus with a long membrane tether formed following dissociation of plasma membrane from the cytoskeleton under force F_te = F_mv > F_th. The symbols for biophysical parameters are defined in Table 2. P-selectin and PSGL-1 shown as blue and green, respectively. Actin cytoskeleton shown in red. Tether anchorage point shown as a round structure at the end of tether. Not drawn to scale.

Three models have been proposed for describing microvillus extension material and applied to pulling experiments involving PSGL-1-P-selectin bonds: an elastic model [172] represented by a spring shown in Fig. 8a, a Kelvin-Voigt viscoelastic model [150,216] represented by a spring and a dashpot connected in parallel shown in Fig. 8b, and a viscoelastic model composed of a Kelvin-Voigt unit connected in series with an elastic unit [154,216] shown in Fig. 8c. The parameters σ_mv and κ_mv are the spring constants for the corresponding springs, and η_{eff_mv} is the effective viscosity of the dashpot. The equation under each schematic representation gives the relation between the pulling force, F_mv, and the microvillus extension, L_mv. The extension L_mv under a constant pulling force F_mv is shown in Fig. 8d for F_mv = 45 pN, and the neutrophil parameters σ_mv = 43 pN μm^-1 and η_{eff_mv} = 33 pN s μm^-1 [150,172], and κ_mv = 800 pN μm^-1 [154]. (The biophysical parameters and their symbols are defined in Table 2.)

Fig. 8.

Models for the microvillus extension material used in published PSGL-1/P-selectin pulling experiments. (a) Elastic model represented by a spring. (b) Kelvin-Voigt viscoelastic model represented by a spring and a dashpot connected in parallel. (c) Viscoelastic model composed of a Kelvin-Voigt unit and an elastic unit connected in series. In a-c, σ_mv and κ_mv are the spring constants for the corresponding springs, η_{eff_mv} is the effective viscosity of the dashpot, F_mv is the pulling force, L_mv is the microvillus extension, L̇_mv = dL_mv/dt is the microvillus extension rate, and Ḟ_mv = dF_mv/dt is the loading rate. (d) Microvillus extension under a constant force, F_mv = 45 pN, described by the models a-c for σ_mv = 43 pN/μm, = η_{eff_mv} = 33 pN s/μm, and κ_mv = 800 pN/μm.

As shown in Fig. 8d, the elastic model (Fig. 8a) does not represent the instantaneous changes in the microvillus extension under constant force F_mv, which have been observed experimentally. The Kelvin-Voigt viscoelastic model (Fig. 8b), describing the instantaneous changes in the microvillus extension, is consistent with the instantaneous experimental data of [172,216]. In the experiment of [172], the neutrophil is not completely immobilized, but moves in a micropipette under a suction pressure. The third model (Fig. 8c) is proposed in [216] in the context of a pulling experiment where the neutrophil is held at the tip of the micropipette by a suction pressure. It has been shown [45] that a red blood cell held at the tip of a micropipette by suction pressure responds as a spring (an elastic response) to a pulling force with a spring constant defined by the suction pressure, micropipette radius, cell radius, and the radius of the outer spherical portion of the cell. Therefore, most likely, the additional spring in Fig. 8c, compared to Fig. 8b, does not represent the tether, but rather the elastic response of the neutrophil body reacting to the pulling force similar to the response of red blood cells.

The occurrence of a crossover, which is viewed as a termination of the microvillus extension and beginning of tether formation, seems to depend on the pulling pattern. Generally, it is assumed that the crossover occurs when there is a clear change in the shape of the pulling force as a function of time [45,172,216]. In the experiment of [172], the crossover occurs when the initially increasing tether pulling force becomes constant. In different pulling pattern experiments of [45,65,216], the crossover occurs when the apparently linear pulling force starts arching to approach a plateau force at long times. The apparently linear force is really not linear as it is expressed using an exponential function (Eq. A1 in [216]). The same applies to the extension of the microvillus, as seen in Eq. A2 in [216].

Based on the Kelvin-Voigt model, larger σ_mv and η_{eff_mv} (i.e. less extendable microvilli) result in faster cell rolling (Pospieszalska and Ley, unpublished data) because, overall, the less extendable microvillus results in a larger force (F_mv,1 in Fig. 2) on the bonds decreasing their life time. Thus the ability of microvilli to extend stabilizes the rolling adhesion in the presence of hydrodynamic shear forces by increasing the lifetime of selectin-ligand bonds. Three published cell rolling models consider the contribution of microvillus extension [22,145,152]. However, the role of tether extension in stabilizing neutrophil rolling has not been studied in detail.

4.2.3 Tether formation

As discussed in the previous section, when a bond pulling force F_mv is greater than the threshold force, F_th, and acts for a sufficiently long time, a tether starts to form on the tip of the microvillus. At that point the microvillus is elongated if F_mv is low to moderate. The biological nature of cellular tethers is not fully understood. The transition (a crossover) from microvillus elongation to tether formation is believed to occur when the cell membrane separates from the underlying cytoskeleton (Fig. 7c) and starts flowing around integral proteins embedded in the membrane and bound to the cytoskeleton nodes, and into the tether [16]. Long cellular membrane tethers were observed in vivo [182], studied in flow chambers and micropipette/laser trap experiments in vitro [16,39,45,65,71,140-142,154,155,166,167,172,191,215,216], and modeled in silico [22,88,145,150,216,221]. Recently, qDF microscopy has shown that a neutrophil rolling on P-selectin under high shear stress (0.6-0.8 Pa) forms 3-4 long tethers which can extend up to 16 μm behind the rolling cell (Fig. 9) [191]. The anchorage points of long tethers were visible in the qDF micrographs of rolling neutrophils (Fig. 9a) and the 3D reconstruction of the qDF footprints (Fig. 9b) revealed that the tether anchorage points are held to the substrate through stressed P-selectin-PSGL-1 bonds (L_sep = 125-175 nm).

Fig. 9.

Long tethers facilitate neutrophil rolling at high shear stress [Adapted from Sundd, P. et al. Nat Methods 7, 821-824 (2010) with permission]. (a) qDF micrograph of a DiI-labeled neutrophil rolling on P-selectin substrate processed to saturation to reveal anchorage points of long tethers behind the footprint. (b) 3D reconstruction of the footprint shown in (a) reveals z position of tether anchorage points above the P-selectin substrate which is equal to the length of the stressed P-selectin-PSGL-1 bonds holding the tether anchorage point to the substrate. P-selectin 20 molecules/μm². Wall shear stress 0.8 Pa. Flow is from left to right. Scale bar 10 μm. Arrows mark tether anchorage points.

Several studies led to the formulation of a theoretical framework for the tether phenomenon [16,17,69,210]. These studies resulted in a mathematical description of the tether dynamic equilibrium as a state of non-zero duration characterized by a constant tether pulling force, F_te > 0, and a constant tether extension rate, L̇_te = dL_te/dt > 0, where L_te denotes the tether extension. Constant F_te and L̇_te of a dynamic equilibrium satisfy Eq. 16:

$${\stackrel{.}{L}}_{\text{te}}=[{F_{\text{te}}}^3-F_{\text{te}}{\left(F_{\text{th}}\right)}^2]∕\left[16{\pi }^3{\left(k_{\mathrm{c}}\right)}^2{d}_{\text{int}}\ln (r_c{F}_{\text{te}}∕2\pi k_{\mathrm{c}})\right]$$ (16)

In Eq. 16, F_te > F_th, r_c is the cell radius, k_c is the cell membrane curvature modulus, and d_int is the interfacial drag coefficient (i.e. the surface density of the bound integral proteins multiplied by the surface viscosity). Equation 16 is valid for 0.01 μm/s < L̇_te < L̇_c, where L̇_c is the upper limit for the integral proteins to remain bound to the cytoskeleton (L̇_c ≈ 100 μm/s for neutrophils [17]). Equation 16 describes the relation between F_te and L̇_te exclusively for a tether in a dynamic equilibrium. However, there are pulling processes which demonstrate a crossover and no equilibrium.

So far, material properties of cellular tethers have been treated only phenomenologically. Three propositions for the tether material representation have been suggested in the literature based on agreement with at least one experiment. According to the first study [172], the force on the tether, F_te, can be defined as

$$F_{\text{te}}\left(t\right)=F_{\text{th}}+{\eta }_{\text{eff}\_\text{te}}{\stackrel{.}{L}}_{\text{te}}\left(t\right)$$ (17)

where η_{eff_te} is the tether effective viscosity [22,150]. (Note that the factor preceding L̇_te(t) in Eq. 17 can be also found in literature in the form of 2πη_eff as in [69] where η_eff is referred as the effective viscosity). Equation 17 represents a viscous unit, η_{eff_te}L̇_te(t), being pulled by a force of F_te(t) – F_th (a force smaller than the pulling force), and shows that L̇_te is linearly dependent on F_te which is not true for tethers in dynamic equilibriums (L̇_te in Eq. 16 is a non-linearly increasing function of F_te). A second study [65] suggests that the force, F_te, on a neutrophil tether can be defined as:

$$F_{\text{te}}\left(t\right)=60{\left[{\stackrel{.}{L}}_{1,\text{te}}\left(t\right)\mu {\mathrm{m}}^{-1}\mathrm{s}\right]}^{0.25}=\left[60{\left(\mu {\mathrm{m}}^{-1}\mathrm{s}\right)}^{0.25}{\stackrel{.}{L}}_{1,\text{te}}{\left(t\right)}^{-0.75}\right]{\stackrel{.}{L}}_{1,\text{te}}\left(t\right)=k{L}_{2,\text{te}}\left(t\right)$$ (18)

F_te has the units of pN, L̇_1,te has the units of μm/s, and L_1,te + L_2,te = L_te. After singling out L̇_1,te(t, the authors view their formula as a viscous-like unit of extension L_1,te with its effective viscosity being a function of t, connected in series with an elastic unit of extension L_2,te with spring constant k (not specified). The tether extension is the total extension of both units. When L̇_te = 0, Eq. 18 yields F_te = 0 instead of F_te = F_th ≠ 0. A correction to this discrepancy was suggested [30,171] that requires substituting L̇_1,te(t) in Eq. 18 with L̇_te(t) + a (no elastic unit), where a > 0 is a way that the corrected formula yields F_te = F_th for L̇_te = 0. The authors interpret –a as the tether retraction velocity for F_te = 0. However, preliminary data obtained using qDF microscopy (Sundd, Pospieszalska and Ley; unpublished data) indicates that the retraction velocity may depend on the state of the tether in the pulling process when the pulling force F_te becomes zero. In Pospieszalska et al. [230] we propose to substitute the above three phenomenological formulations with a mathematically derived formula for the tether material.

Published modeling studies on neutrophil rolling on P-selectin that take microvillus deformation (i.e. microvillus extension and tether formation) into consideration use the elastic model in Fig. 8a [22,145] or the Kelvin-Voigt viscoelastic model in Fig. 8b [150] for microvillus extension, and Eq. 17 for tether formation [22,145,150]. All three studies [22,145,150] conclude that microvillus deformation reduces the cell translational velocity. The reduction is up to 35% for a shear rate of 400 s^-1 and P-selectin site density of 150 molecules/μm², as reported in [145]. Lower translational velocities are the result of larger number of bonds and longer bond lifetimes. The modeling work of [88] confirms that the microvillus deformation in rolling neutrophils results in increased P-selectin-PSGL-1 bond life times [88].

Tether elongation seems to depend on the pulling pattern. In the experiment of [172], a tether establishes a dynamic equilibrium already at crossover and stays in that equilibrium as long as the pulling process continues. In different pulling pattern experiments of [65,216] a tether approaches a dynamic equilibrium at long times. In an experiment with a constant tether loading rate the pulling force increases linearly and the tether cannot establish a dynamic equilibrium. A dynamic equilibrium creates favorable conditions for the survival of molecular bonds holding the tether in place, because the force on a bond is no longer increasing. The chance for a molecular bond to survive until a dynamic equilibrium of its tether is reached has not been established. However, the presence of very long tethers observed by qDF microscopy [191], and by other techniques [155,166], suggests that such bonds must exist.

5. Summary and concluding remarks

Neutrophil recruitment to sites of inflammation is mediated by selectin-dependent rolling. P-and E-selectin expressed on EC bind with intermediate affinity and high off-rates to their respective sialo-mucin ligands expressed on neutrophils. Forced bond dissociation in the presence of shear flow is balanced by rapid bond formation to mediate rolling interactions. PSGL-1, which is constitutively clustered on the tips of neutrophil microvilli, serves as the high-affinity ligand for all three selectins. The shearing fluid exerts a shear force and torque on the rolling neutrophil, which is balanced by the bond forces generated in the load-bearing bonds at the rear of the rolling neutrophil. Selectin-ligand molecular interactions have fast unstressed forward and reverse reaction rates. The reverse reaction rate is further modulated by the tensile bond force. At smaller bond forces, the reverse reaction rate may decrease with increasing force to reach a minimum value at an optimum force, and then increases with increasing force. During the first phase, the selectin-ligand bond is thought to behave as a ‘catch bond’ and during the second phase as a ‘slip bond’. The force-dependent receptor-ligand binding kinetics responsible for ‘slip’ bond behavior can be explained using the ‘Bell (1978)’ [15] or ‘Dembo (1988)’ [35] model. The ‘catch-slip’ transitional behavior is captured by Evans’ two-pathway model (2004) [46]. Based on crystal-structure data and molecular dynamics simulations, the ‘catch-slip’ bond behavior may be explained using the ‘sliding-rebinding mechanism’ [121,122] or the ‘allosteric model’ [186,206]. Both models assume an equilibrium between the ‘bent or closed’ and ‘open or extended’ confirmation of the lectin-EGF domain hinge in the selectin molecule. The equilibrium shifts towards the ‘open or extended’ form under applied tensile force, decreasing reverse reaction rate and increasing the lifetime of selectin-ligand bonds.

Neutrophil rolling is known to exhibit a shear threshold phenomenon, which is most pronounced in rolling mediated by L-selectin-PSGL-1 interactions. In in vitro flow chambers, neutrophils do not roll on PSGL-1 coated substrate below a threshold shear stress, and the rolling increases and becomes more stable as the shear stress is increased, reaches a maximum at an optimal shear stress, and then becomes faster and irregular with increasing shear stress. This behavior may be explained by a synergistic effect of the transport-enhanced tethering and ‘catch’ bond-enhanced rolling. Neutrophil rolling is also modulated by the presence of bond clusters in the footprint of a rolling cell. Numerical simulations show that the P-selectin-PSGL-1 bond clusters in the footprint of rolling neutrophils are responsible for the ‘jerky’ rolling motion. Hydrodynamic loading of a bond cluster causes the neutrophil to slow down suddenly or even stop temporarily. Numerical simulations have also shown that the cell-substrate separation distance varies with the dissociation and loading of P-selectin-PSGL-1 bonds and correlates with the cell translational velocity.

Neutrophils rolling at high shear stress (> 0.6 Pa) flatten and create a larger footprint. This mesoscale deformation contributes to the ability of neutrophils to roll at relatively uniform velocities over a wide range of shear stresses by increasing the number of bonds in the footprint and decreasing the hydrodynamic force on the bonds. In addition to whole cell deformation, microvilli deformation (viscoelastic extension and formation of long membrane tethers) further facilitates neutrophil rolling at high shear stress by increasing the length of the moment arm, and thus decreasing the force acting on the bonds required to balance the hydrodynamic torque acting around the cell center. Reduced bond forces result in longer bond lifetimes, and thus more stable rolling at high shear stresses.

Although the biomechanics of leukocyte rolling have been studied extensively over the last two decades, the biomolecular and biophysical mechanisms that allow leukocytes to roll at venular shear stresses larger than 0.6 Pa both in vivo and in vitro are still elusive. Measurements of molecular and cellular parameters from in vitro and in vivo studies have facilitated the creation of several in silico models of leukocyte rolling. However, none of the currently available models predict leukocyte rolling at high venular shear stresses (0.6-3.5 Pa [33]) that are commonly observed in vivo. Quantitative dynamic footprinting used to visualize the footprint of neutrophils rolling in whole blood at high shear stress (τ_w > 0.6 Pa) revealed that the deformation of neutrophils results in a large footprint and formation of 3-4 long membrane tethers at the rear of the rolling neutrophil. These mechanisms may facilitate neutrophil rolling at high shear stresses.

Elucidation of the biophysical and structural mechanisms behind neutrophil rolling at high shear stress requires a theoretical model based on the measurements of qDF microsopy, which can describe whole cell deformation as well as long tether formation at venular shear stresses. Such a model does not exist currently, therefore, the relative contribution of long tethers and the stressed microvilli to neutrophil rolling at high shear stress cannot be estimated. The hydrodynamic shear force and torque acting on a rolling neutrophil are balanced by the forces acting on the long tethers and stressed microvilli. Although total force acting on all the stressed microvilli and the tethers combined can be easily estimated using the force balance with experimental measurement of structural parameters, the individual contribution of a single tether or a stressed microvilli has never been measured experimentally. A combination of qDF with traction force microscopy [158,178] may allow measurements of forces acting on individual microvilli or tethers. Theoretical models that capture both whole cell deformation and long tether formation, along with experimental studies that allow measurement of the microvillus or tether bond force are required for an improved understanding of the biomechanics behind leukocyte rolling at high shear stress.