An Analytical Model for Determining Two-Dimensional Receptor-Ligand Kinetics

Abstract

Cell-cell adhesive interactions play a pivotal role in major pathophysiological vascular processes, such as inflammation, infection, thrombosis, and cancer metastasis, and are regulated by hemodynamic forces generated by blood flow. Cell adhesion is mediated by the binding of receptors to ligands, which are both anchored on two-dimensional (2-D) membranes of apposing cells. Biophysical assays have been developed to determine the unstressed (no-force) 2-D affinity but fail to disclose its dependence on force. Here we develop an analytical model to estimate the 2-D kinetics of diverse receptor-ligand pairs as a function of force, including antibody-antigen, vascular selectin-ligand, and bacterial adhesin-ligand interactions. The model can account for multiple bond interactions necessary to mediate adhesion and resist detachment amid high hemodynamic forces. Using this model, we provide a generalized biophysical interpretation of the counterintuitive force-induced stabilization of cell rolling observed by a select subset of receptor-ligand pairs with specific intrinsic kinetic properties. This study enables us to understand how single-molecule and multibond biophysics modulate the macroscopic cell behavior in diverse pathophysiological processes.

Introduction

Although several techniques have been developed to study receptor-ligand binding kinetics, most of them, such as radio-immunoassays and surface plasmon resonance, require at least one molecule present in solution, which limits their application to the measurement of three-dimensional binding constants. However, cell-cell adhesion is mediated by binding of receptors to ligands, which are both anchored on two-dimensional (2-D) membranes of apposing cells. Not only the binding mechanism is different but also their affinities have different units (M⁻¹ in three dimensions and μm² in two dimensions). Even though three-dimensional kinetic rates have been extensively reported in the literature, they are inadequate to describe the 2-D receptor-ligand binding kinetics (1).

Sophisticated biophysical assays can determine the unstressed 2-D affinity (1,2) of receptor-ligand interactions but fail to disclose its dependence on force. Yet, receptor-mediated cell adhesion in diverse (patho)physiological processes occurring within the vasculature such as inflammation, infection, thrombosis, and metastasis, is regulated by hemodynamic forces generated by blood flow including shear stress (3). On the one hand, fluid shear stress induces collisions between free-flowing cells and the vessel wall, thereby increasing the encounter rate between membrane-bound receptors and their ligands (4). On the other hand, fluid shear shortens the intercellular contact duration, and exerts forces that tend to disrupt the receptor-ligand bonds responsible for cell adhesion (5,6). Cell-cell interactions, therefore, depend on the balance between the dispersive hydrodynamic forces and the adhesive forces generated by the interactions of receptor-ligand pairs. If we could determine the 2-D receptor-ligand kinetics as a function of hydrodynamic force, we would then be able to characterize and/or predict the macroscopic receptor-mediated cell adhesion process in the vasculature. To date, flow-based adhesion assays in conjunction with straightforward mathematical calculations have been used to determine the off-rate, k_off, of selectin-ligand pairs under physiologically relevant flow conditions (7–9). Although the 2-D binding affinity, A_cK_a, and its dependence on applied force can be estimated under fixed geometries (10,11), a full understanding of this dependence is still missing for cell rolling behavior observed under shear flow.

Here we developed an analytical model to estimate for the first time (to our knowledge) the 2-D receptor-ligand kinetics as function of force. We applied this model to characterize the interactions of diverse biomolecular pairs pertinent to the pathophysiological processes of inflammation and infection; these include antibody-antigen, bacterial adhesin-ligand, and E-, P-, or L-selectin-ligand pairs. The model fits successfully to experimental data acquired at both low and elevated receptor-ligand site densities by properly accounting for single and multiple bond interactions, respectively. Using the 2-D affinity values extracted from our model, we accurately predict and explain the counterintuitive force-induced stabilization of cell rolling, in which cell velocity goes through a local minimum while monotonically increasing the shear stress (9).

This novel model overcomes current limitations of studying 2-D kinetics under physiologically relevant shear flow conditions, and advances our knowledge of how single-molecule and multibond biophysics modulate the macroscopic cell behavior in diverse pathophysiological processes pertinent to health and disease.

Methods

Estimation of tether force from wall shear stress

The bond-tether force (f) was estimated by balancing forces and torques on the bound cell assuming static equilibrium. The bond-tether force per unit wall shear stress (f/τ) has been determined to be 273 and 125 pN/(dyn/cm²) for HL-60 cells and neutrophils, respectively (9,12). Using the dimensional analysis proposed by Yago et al. (9), we determined f/τ to be 330 pN/(dyn/cm²) for sLe^x-coated beads. The shear-to-force conversion factors are listed in Table 1.

Table 1.

Model parameters used in calculations

Cell	Receptor	mr, μm−2	R, μm	h, μm	f/τ, pN/dyn-cm2
Rat basophilic leukemia	Anti-DNP IgE	18∗	7.5∗	ND	ND
HL-60	PSGL-1	72†	6.3‡	0.3§	273‡
sLex-coated bead	sLex	90¶	5¶	0.02‖	330∗∗
Neutrophil	L-selectin	251∗∗	4.25∗∗	0.3§	125∗∗

ND, not determined.

^∗Swift et al. (20).

^†Ushiyama et al. (26).

^‡Chen et al. (12).

^§Shao et al. (15).

^¶Brunk and Hammer (24).

^‖Chang and Hammer (14).

^∗∗Yago et al. (9).

Calculation of the cell or microsphere hydrodynamic velocity

The numerical data of the nondimensional translational velocity (U_hd/zγ) and angular velocity (Ω/0.5γ) given by Goldman et al. (13) were fitted to a mathematical function, and yielded excellent fittings (R² > 0.99), as shown in Fig. S1 in the Supporting Material. For the sLe^x-coated microsphere, the gap distance h between the surface of the microsphere and flow chamber wall is assumed to be the equilibrium bond length between sLe^x-E-selectin, which is reported to be 0.02 μm (14). For the HL-60 cell and neutrophil, h is assumed to be the length of an unstressed microvillus, which is equal to 0.3 μm (15). Assuming the viscosity of testing buffer equals to that of water (1 cP at 20°C), the translational hydrodynamic velocities per unit wall shear stress (U_hd/τ) are estimated using Fig. S1 to be 234, 419, and 317 (μm/s)/(dyn/cm²) for the microsphere, HL-60 cell, and neutrophil, respectively.

Shear-induced on-rate

2-D on-rates for P- and L-selectin binding to PSGL-1 were modeled using the shear-induced on-rate equations given by (4,16)

$$k_{+}=\{\begin{array}{cc}\frac{2\pi D}{\ln \left(\raisebox{1ex}{$b$}\!\left/ \!\raisebox{-1ex}{$a$}\right.\right)}\left(\frac{a^2{k}_{in}}{8D+a^2{k}_{in}}\right)& Pe<<1\\ 2DPe\left(\frac{8a^2{k}_{in}}{3\pi DPe+8a^2{k}_{in}}\right)& Pe>>1,\end{array}$$ (1)

where D is the relative diffusion coefficient, which equals to the sum of the surface self-diffusivities for the target receptor and ligand (4), a is the radius of the reactive circle around the receptor, b is half of the mean distance between the ligand molecules on substrate, k_in is the intrinsic reaction rate of a ligand-receptor pair within the reactive circle, Péclet number is defined by Pe = V_rev·a/D, and V_rev is the relative velocity between cell surface and flow chamber wall calculated by V_rev = U_hd−RΩ. By using the data given in Fig. S1, the relative velocity-per-unit-wall-shear stress (V_rev/τ) is estimated to be 213 and 165 (μm/s)/(dyn/cm²) for HL-60 cell and neutrophil, respectively.

Calculation of the bond off-rate

The DNP-anti-DNP IgE and E-selectin-sLe^x interactions are assumed to be slip-bonds (14). The force dependence of the off-rate of slip-bonds is described by the Bell model (5),

$$k_{off}=k_{off}^0\exp \left(\frac{x_{\beta }f}{k_{\text{B}}T}\right),$$ (2)

where k_off⁰ is the unstressed dissociation rate, x_β is the reactive compliance, f is the tether force on the bond, and k_BT is the Boltzmann constant multiplied by temperature.

The biomolecular interactions of P- and L-selectin with PSGL-1 exhibit a catch-slip bond transition, and are thus modeled using the two-pathway off-rate model given by (17)

$$k_{off}=\frac{{\varphi }_0{k}_{1rup}+\exp \left(\frac{f}{f_{12}}\right)\left[k_{2rup}\exp \left(\frac{x_{\beta }f}{k_{\text{B}}T}\right)\right]}{{\varphi }_0{k}_{1rup}+\exp \left(\frac{f}{f_{12}}\right)},$$ (3)

where k_1rup and k_2rup are the off-rates for pathway 1 and 2; ϕ₀ = exp(ΔE₂₁/k_BT), which is the equilibrium constant between the two states at zero force; and f₁₂ is a force scale that governs the force span for the occupancy ratio of the two states. The model parameters were obtained by fitting the published experimental data (7,18,19) to Eq. 3 using Levenberg-Marquardt nonlinear least-square minimization in MATLAB (The MathWorks, Natick, MA). The best fitting to P-selectin-PSGL-1 and L-selectin-PSGL-1 interactions is plotted in Fig. S2, and their dissociation parameters are listed in the Table 2.

Table 2.

Optimization of model parameters

	Selectin-ligand
Parameter	P-selectin-PSGL-1	L-selectin-PSGL-1
Catch-slip off-rate
k1,rup, s−1	2.85	125.4
k2,rup, s−1	1.17	2.27
xβ, Å	0.33	0.58
ΔE21, pN·nm	4.456 kBT	0.692 kBT
f12, pN	3.56	15
Shear-induced on-rate
D, μm2·s−1	0.05	0.05
a, nm	0.74	2.4
kin, s−1	3.2 × 104	8.89 × 105

Calculation of the apparent multibond off-rate

In this work, we considered two different scenarios for multibond dissociation:

Case 1

All n-bonds break up simultaneously. In this case, f is the tether force, which is assumed to be equally distributed among a small number of n-bonds, and the off-rate defined by Eq. 4 represents an apparent off-rate for cell-substrate binding:

$$k_{off,n}=k_{off}^0\exp \left(\frac{x_{\beta }f}{n{k}_{\text{B}}T}\right).$$ (4)

Case 2

The n-bonds break up one-by-one in a stepwise fashion. At each dissociation state, the hydrodynamic force is assumed to be redistributed among the remaining bond(s). By using the Bell model as an example, the off-rate for the n^th bond breaking up is k_off,n = k_off⁰exp(x_βf/nk_BT); the off-rate for the next (n-1)^th bond breaking up is k_off,n-1 = k_off⁰exp[x_βf/(n−1)k_BT]. When the last bond breaks, the cell will detach. Therefore, the total lifetime for n-bond dissociation is estimated by the summation of lifetimes at each dissociation step:

$$t_{total}=t_n+t_{n-1}+\cdots +t_1=\sum _{i=1}^n\frac{1}{k_{off,i}}.$$ (5)

The apparent off-rate for n-bonds is thus given by

$$k_{off,n}=\frac{1}{\sum _{i=1}^n\frac{1}{k_{off,i}}}=\frac{1}{\sum _{i=1}^n\frac{1}{k_{off}^0\exp \left(x_{\beta }f/i{k}_{\text{B}}T\right)}}.$$ (6)

Equations 4 and 6 were applied for the E-selectin-sLe^x interactions, which are assumed to follow the slip-bond dissociation pathway. For the catch-slip-bond dissociation, modified equations, which incorporate the two-pathway model (Eq. 3), were derived for Cases 1 and 2 described above.

Results and Discussion

Estimation of 2-D receptor-ligand kinetics from adhesion data

We developed an analytical model to estimate the 2-D kinetics of receptor-ligand binding mediated by a small number of bonds as a function of the fluid shear stress based on the reversible kinetics of a biomolecular interaction,

$$N_u+S\underset{k_{off}}{\overset{k_{+}}{\rightleftharpoons }}B,$$ (7)

where N_u is the number of unbound cells at time t, B is the cumulative number of cells bound up to time t, N_T = N_u + B is the total number of cells, S is the ligand site density (m_l) on the substrate, and k₊ and k_off are the 2-D on- (μm²·s⁻¹) and off-rates (s⁻¹), respectively. The asymptotic solution of fraction of interacting cells is

$$\frac{\underset{x\to \infty }{\lim }B\left(x\right)}{N_T}=\frac{N_b}{N_T}=\frac{k_{+}{m}_l}{k_{+}{m}_l+k_{off}},$$ (8)

where the time variable, t, is substituted by x = t·U_cell to yield an expression in terms of rolling distance x and cell rolling velocity U_cell. Here we assume that receptors are uniformly distributed on the tips of cell microvilli or the surface of the microsphere at density m_r; hence the number of available receptors inside the effective contact area A_c between the cell surface and the substrate is equal to A_cm_r. The overall 2-D on-rate, k₊, is given by the product of the single bond on rate, k_on, and the number of available receptors (A_cm_r) inside the contact area. The value k₊ encompasses both the encounter and reaction rates of the biomolecular interaction(s) (4). The off-rate is calculated using the Bell model (5) for slip-bonds or the two-pathway model (17) for catch-slip bonds (see Methods).

Antibody-antigen binding

We first applied our model to characterize the 2-D affinity of rat basophilic leukemia (RBL) cells preincubated with anti-dinitrophenyl (DNP) IgE clones SPE-7 or H1 26.82 and perfused over DNP-coated substrates under shear (20). By plugging the Bell model into Eq. 8, the fraction of bound cells is given by

$$\begin{array}{c}\frac{N_b}{N_T}=\frac{A_c{m}_r{m}_l{k}_{on}}{A_c{m}_r{m}_l{k}_{on}+k_{off}^0\exp \left(x_{\beta }f/n{k}_{\text{B}}T\right)}\\ =\frac{1}{1+\frac{1}{m_r{m}_l{A}_c{K}_a^0}\exp \left(\tau /{\tau }_{\beta }\right)}\end{array},$$ (9)

where A_cK_a⁰ = A_ck_on/k_off⁰ is the unstressed 2-D affinity, and τ_β = nk_BT/x_βC represents the characteristic shear stress scale, and C is the tether force per unit shear stress (see Methods). Because the Bell model parameters are not known for the DNP-anti-DNP binding, they are lumped into the two fitting parameters A_cK_a⁰ and τ_β as shown in Eq. 9b. Because the number of bonds (n) cannot be explicitly determined, the use of Bell model essentially estimates the apparent dissociation rate by assuming that all bonds break up simultaneously.

The optimized values of A_cK_a⁰ and τ_β were obtained by fitting data on the fraction of bound RBL cells to Eq. 9b. The fitting successfully traces experimental data, especially at high (R² > 0.96) relative to low (R² > 0.58) DNP site densities (Fig. 1 A). The estimated unstressed 2-D affinity A_cK_a⁰ of H1 26.82 (178 × 10⁻⁴ μm⁴) is 32-fold higher than that of SPE-7 (5.56 × 10⁻⁴ μm⁴). Interestingly, their respective three-dimensional affinity constants also exhibit a 30-fold difference (140 × 10⁶ vs. 4.8 × 10⁶ M⁻¹) (20). The higher 2-D affinity of H1 26.82 rather than SPE-7 clone for DNP is responsible for its increased shear-dependent binding (see Fig. 1 A and Fig. S3). The characteristic shear stress scales are similar for both clones (0.17 and 0.22 dyn/cm²), as expected from inspecting the decay (the falling slope) of bound cells (see Fig. 1 A, and Fig. S3).

Figure 1.

Estimation of 2-D kinetics of antibody-antigen (slip-bond) and P-selectin-ligand (catch-slip bond) binding under shear. (A) Data represent the fraction of bound rat basophilic leukemia (RBL) cells preincubated with anti-dinitrophenyl (DNP) IgE clones SPE-7 (red symbols) or H1 26.82 (green circles) and perfused over DNP-coated substrates (1900 sites/μm²) at prescribed wall shear stresses (20). (Blue symbols) Experimental data for the SPE-7 clone and 280 DNP sites/μm². (B) Data represent the fraction of PSGL-1-expressing human promyelocytic leukemia HL-60 cells rolling on surfaces coated with different site densities (m_l) of P-selectin as a function of wall shear stress (21). Experimental data (symbols) are normalized by N_T = 183 cells/mm² (Fig. S4). (Dashed line) Model output at m_l = 25 sites/μm². (C) Estimation of the average 2-D association rate, k₊ (symbols), as a function of wall shear stress from the N_b/N_T experimental data shown in panel B. (Solid line) Best fitting with the kinetic parameters listed in Table 2.

P-selectin binding to P-selectin glycoprotein ligand-1

We next applied our model to extract the 2-D kinetics of P-selectin binding to P-selectin glycoprotein ligand-1 (PSGL-1), which exhibits a catch-slip transition described by the two-pathway dissociation model (17). The dissociation model parameters were obtained by fitting data (8,19) describing the dependence of off-rate on tether force (Fig. S2 A) to the two-pathway model (Table 2). The 2-D on-rate k₊ was calculated by applying experimental data (21) on the fraction of human promyelocytic leukemia HL-60 cells tethered on P-selectin under shear (Fig. 1 B) to Eq. 8. Given that multibond interactions could occur at the high P-selectin site density and high shear regime, the 2-D on-rate k₊ was estimated from the adhesion data reported at the lower P-selectin site densities (21). Because only two nonzero points were reported for HL-60 cell adhesion to P-selectin at 50 sites/μm² (21), the 2-D on-rate was calculated by using data at both 50 and 140 sites/μm².

Fig. 1 C reveals that the 2-D on-rate is enhanced by shear stress in the low shear regime. In light of this observation, we applied the shear induced on-rate model (4) to obtain the kinetic parameters a and k_in, which correspond to the radius of reactive circle around the cell receptor and the intrinsic reaction rate, respectively (Table 2). When a ligand molecule is within a distance a from a cell receptor, a successful encounter occurs between the receptor and the ligand, which react with a k_in rate (4). The estimated k_in is within the range of 10⁴–10⁵ s⁻¹ previously estimated for other selectin-ligand pairs using computer simulations (4,22). Plugging the calculated kinetic parameters into Eq. 8, we computed the fraction of rolling HL-60 cells at prescribed site densities as a function of shear stress. As shown in Fig. 1 B, the model fits successfully (R² > 0.87) to the experimental data at higher site densities (140 and 375 sites/μm²), and captures the shear threshold phenomenon (Fig. 1 B) in which the number of rolling cells first increases and then decreases while monotonically increasing the shear (23). The lack of good fitting at m_l = 50 sites/μm² is attributed in part to the overestimation of the P-selectin site density, because the model (dashed line) fits better to the data when a lower input value of 25 sites/μm² is assumed (Fig. 1 B).

Of note, the model accurately predicts HL-60 cell binding to P-selectin at the site density of 375 sites/μm² in the lower shear regime (τ ≤ 1 dyn/cm²), suggesting that the majority of these adhesive events is mediated by a single-bond in this region. At higher shear stress (τ ≥ 2 dyn/cm²), our model underestimates the percentage of interacting cells. This is presumably because large hydrodynamic forces induce membrane deformation that increases the cell contact area, and thus the number of receptor-ligand bonds.

Estimation of 2-D receptor-ligand kinetics from rolling velocity data

We also developed an analytical model to estimate the 2-D kinetics of receptor-ligand binding from rolling velocity data. The average cell rolling velocity (U_cell) is estimated by taking the arithmetic mean of the instantaneous velocity over a sufficiently long period of time. A rolling cell exhibits two modes of motion: either moving at hydrodynamic velocity (U_hd) or being bound and thus having zero velocity (U_bound = 0). These two modes of motion (free state versus bound) are governed by the 2-D kinetic constants. The duration time of a cell in the bound state, t_b, represents the bond(s) lifetime (1/k_off). The duration time of a cell in free motion is the time between the rupture of an existing tether bond and the formation of a new one, which is the reciprocal of k₊m_l. Thus, the average cell rolling velocity is given by (see text and Fig. S6)

$$U_{cell}=U_{hd}\left(1-\frac{k_{+}{m}_l}{k_{+}{m}_l+k_{off}}\right),$$ (10)

where U_hd represents the hydrodynamic velocity of a noninteracting cell calculated by the Goldman model (13) as described in the Methods.

Single- and multibond E-selectin-sialyl Lewis^x interactions

We first applied this model to characterize the 2-D kinetics of sialyl Lewis^x (sLe^x)-coated beads rolling on E-selectin under shear (24). Assuming that all n-bonds break up simultaneously (Case 1), the average rolling velocity is provided by

$$U_{cell}=U_{hd}\left[1-\frac{1}{1+\frac{1}{m_r{m}_l{A}_c{K}_a^0}\exp \left(\raisebox{1ex}{$x_{\beta }f$}\!\left/ \!\raisebox{-1ex}{$n{k}_{\text{B}}T$}\right.\right)}\right].$$ (11)

By fitting the experimental data (24) to Eq. 11, for a reported x_β = 0.25 Å (14), the optimized values for the two fitting parameters (A_cK_a⁰ = 3.5 × 10⁻⁴ μm⁴ and n = 5) were obtained. Fig. 2 A shows that the model successfully (R² > 0.99) traces all experimental data. For comparison, cell rolling mediated by a single bond is also plotted, and apparently deviates from measured data (Fig. 2 A). Using the estimated affinity constant (A_cK_a⁰ = 3.5 × 10⁻⁴ μm⁴), we determined that the number of bonds mediating bead rolling increases with increasing the E-selectin site density (Fig. 2 B and inset).

Figure 2.

Analytical model explains the multibond-mediated adhesion of sLe^x-coated microspheres to E-selectin in shear flow. (A) Microsphere rolling velocity (black symbols) as a function of wall shear stress at a fixed selectin and ligand site density (24), m_r × m_l = 90 × 3600 sites/μm⁴. (Blue curves) Theoretical calculations of rolling velocity mediated by either 1-bond or 5-bonds that break up simultaneously. (Red curve) Calculations in which the dissociation of multiple bonds (n = 3) occurs in a stepwise fashion. (B) Microsphere rolling velocity (black symbols) at a shear stress level of 1 dyn/cm² as a function of E-selectin site density (m_l) (24).(Gray dotted line) Rolling velocity at constant number of bond (5-bonds). (Blue solid line) Rolling velocity mediated by a variable numbers of bonds (n) that break up simultaneously. (Red solid line) Calculations of rolling velocity in which the dissociation of bonds occurs in a stepwise fashion. (Inset) Corresponding number of bonds (n) for Cases 1 and 2 at different E-selectin site densities.

Assuming that n-bonds break up in a stepwise manner (Case 2), the average rolling is provided by

$$U_{cell}=U_{hd}\left[1-\frac{1}{1+\frac{1}{m_r{m}_l{A}_c{K}_a^0\left(\sum _{i=1}^n\frac{1}{\exp \left(x_{\beta }f/i{k}_{\text{B}}T\right)}\right)}}\right].$$ (12)

The model successfully traces the experimental data (R² > 0.83 and 0.98 for Fig. 2, A and B, respectively). Our analysis reveals that the number of bonds (n) required to mediate rolling of sLe^x-bearing beads on E-selectin is smaller when bonds are assumed to break one by one (Case 2) rather than all of them simultaneously (Case 1) (Fig. 2 B and inset). This is attributed to the longer overall bond lifetime, as shown in Eq. 5.

Transition from single- to multiple-bond L-selectin-PSGL-1-dependent rolling at elevated shear stresses

We next determined the 2-D kinetics of L-selectin-PSGL-1 binding from rolling velocity data. The dissociation model parameters for single bonds were determined by fitting data (18) describing the dependence of off-rate on tether force (see Fig. S2 B) to the two-pathway model (Table 2). The 2-D on-rate k₊ for a single bond was estimated by applying data (9) (Fig. 3 A) to the neutrophil rolling velocity on immobilized PSGL-1 to Eq. 10. Fig. 3 B shows that the 2-D on-rate has two regimes with a transition occurring at ∼0.7 dyn/cm². Because L-selectin-PSGL-1 behaves as a slip-bond at high shear, its off-rate increases exponentially with shear stress, thereby destabilizing rolling (25).

Figure 3.

Model explains the stabilization of neutrophil rolling by a shear-dependent increase in the number of L-selectin-PSGL-1 bonds. (A) Fitting of the single- versus multibond mathematical models (color curves) to rolling velocity data (black symbols) of L-selectin-expressing neutrophils on PSGL-1 (m_l = 145 sites/μm²) in shear flow (9). (Dotted line) Hydrodynamic velocity (U_hd) of a cell at a distance h = 0.3 μm from the surface. Multiple bonds were assumed to dissociate either entirely simultaneously (n-bonds; Case 1) or in a stepwise fashion (Case 2). (B) Estimation of the 2-D association rate constant k₊ for L-selectin-PSGL-1 bond. Symbols represent rolling velocity data (see A) manipulated using Eq. 10. (Solid line) Best fitting (R² = 0.8) to the data in the low shear regime (τ < 0.7 dyn/cm²). (C) Estimation of the average number of bonds between a neutrophil and a PSGL-1-coated substrate as a function of wall shear stress. Multiple bonds were assumed to break up either simultaneously (black columns) or one by one (gray columns).

To compensate for the rapid bond dissociation, neutrophils have an automatic braking system that stabilizes rolling by a shear-dependent increase in the number of L-selectin bonds (25). Below the shear threshold, neutrophil rolling is mediated by a single bond (22). We thus extracted the 2-D kinetic parameters, a and k_in (Table 2), from the rolling velocity data at low shear (Fig. 3 A) (9), and found them to be similar to those obtained from numerical simulations (4). Our analysis reveals an excellent fitting (R² > 0.9) to rolling data for a single L-selectin-PSGL-1 bond up to 0.7 dyn/cm² (Fig. 3 A). Above this shear threshold, the number of bonds progressively increases with shear stress to stabilize rolling (Fig. 3, A and C). Interestingly, both multibond off-rate models (Cases 1 and 2) fit well to the experimental data (Fig. 3 A), and are in good agreement with each other in terms of the number of bonds needed to mediate neutrophil rolling at different wall shear stress levels (Fig. 3 C).

2-D affinity of P- and L-selectin binding to PSGL-1 as a function of force

The 2-D affinity was determined as a function of the tether force for both selectin-ligand pairs by dividing their respective values of k₊ (Figs. 1 C and 3 B) by those of k_off (Table 2) and m_r (Table 1) (9,26). The 2-D affinity for P- and L-selectin binding to PSGL-1 reaches a maximum value at a tether force of 47.9 pN (corresponding to τ = 0.38 dyn/cm²) and 90 pN (τ = 0.72 dyn/cm²), respectively (Fig. 4 A), which correspond to their reported shear threshold values (21). The profile of our estimated 2-D affinities is remarkably similar to the neutrophil tethering rates on L- and P-selectin (27). Although L-selectin binds less efficiently than P-selectin to PSGL-1 at low shear, its 2-D affinity overtakes that of P-selectin-PSGL-1 at ≥0.5 dyn/cm² (Fig. 4 A).

Figure 4.

Model explains the shear optimum in cell rolling velocity on vascular selectins. (A) The 2-D affinity constants A_cK_a of P- versus L-selectin binding to PSGL-1 as a function of tether force (bottom abscissa). The shear stress (top abscissa) is estimated for neutrophil rolling (see f/τ in Table 1). For P-selectin-PSGL-1 binding, the 2-D affinity increases from A_cK_a⁰ = 1.61 × 10⁻⁵μm⁴ at static conditions to the maximum affinity A_cK_a^max = 1.58 × 10⁻⁴μm⁴ at 47.9 pN tether force, corresponding to a stress level of τ_peak = 0.38 dyn/cm². For L-selectin-PSGL-1 binding, the A_cK_a⁰ is equal to 0.46 × 10⁻⁵μm⁴, and A_cK_a^max = 2.47 × 10⁻⁴μm⁴ occurs at 90 pN, corresponding to τ_peak = 0.72 dyn/cm². The theoretical prediction of the local minimum velocity for L-selectin- (B) or P-selectin- (C) dependent rolling on PSGL-1-coated substrates under shear based on Eq. 13 and the 2-D affinity constants A_cK_a estimated in panel A. The local minimum velocity exists at the intersection points (black circles) between the black line depicting the slope of affinity and colored lines depicting the RHS of Eq. 13. (D) Average rolling velocity of neutrophils on L-selectin or P-selectin at the conditions shown in panels B and C where the local minimum velocity exists.

A recent study using the thermal fluctuation assay measured the unstressed 2-D on- and off-rates of selectin-ligand interactions (28). Dividing the values of A_ck_on⁰ by k_off⁰, reported in Chen et al. (28), the unstressed 2-D affinity constants for P- and L-selectin binding to PSGL-1 are 4 × 10⁻⁵ and 0.6 × 10⁻⁵ μm⁴, respectively, which are in very good agreement with our estimated values at static (no flow) conditions (1.61 × 10⁻⁵ and 0.46 × 10⁻⁵ μm⁴, respectively).

Theoretical prediction of shear optimum in L- but not P-selectin-dependent rolling

PSGL-1-dependent rolling velocity on L-selectin goes through a local minimum while monotonically increasing the hydrodynamic shear (9,18), which has been suggested to represent the transition from catch- to slip-bond dissociation (9,29). This so-called shear optimum in rolling velocity has not been observed for P-selectin-PSGL-1, although this receptor-ligand pair exhibits the catch-slip dissociation (19). The stationary point in rolling velocity is obtained by taking the derivative of Eq. 10 with respect to τ and setting dU_cell/dτ = 0,

$${\left[\frac{d\left(A_c{K}_a\right)}{d\tau }=\frac{1+m_r{m}_l{A}_c{K}_a}{m_r{m}_l\tau }\right]}_{\tau ={\tau }_c},$$ (13)

where τ_c is the shear stress at the stationary point. Because the right-hand side (RHS) of Eq. 13 is always positive, the stationary point can only exist if d(A_cK_a)/dτ > 0 (see the graphical interpretation in Fig. S5). We applied Eq. 13 to analyze the 2-D affinity of P- and L-selectin binding to PSGL-1 at different selectin-ligand site densities m_rm_l. Zero, one, or two intersection points emerge from plotting the left-hand side (LHS) and RHS of Eq. 13 (Fig. 4, B and C). The stationary point does not exist in cell rolling velocity in the absence of an intersection point (see Fig. 4, B and C, and Fig. S5). In the case of one intersection point (gray symbol), the cell exhibits a plateau in rolling velocity (Fig. 4, B and C, and Fig. S5).

A local minimum in rolling velocity is detected only in the presence of two intersection points, and for a specific range of receptor-ligand site densities. Our analysis reveals that the shear optimum theoretically exists for both P- and L-selectin binding to PSGL-1 (Fig. 4, B and C). A site density of m_rm_l > 5400 sites/μm⁴ is sufficient to display the local minimum in L-selectin-dependent rolling velocity (Fig. 4 B), which is of the same order of magnitude as the value recently reported in numerical simulations (29). In contrast to the L-selectin-PSGL-1 pair, an approximately five-fold higher receptor-ligand site density (m_rm_l > 27,000 sites/μm⁴) is required for the P-selectin-PSGL-1 bond to exhibit the shear optimum (Fig. 4 C).

To validate our conclusions, we calculated the neutrophil rolling velocities for prescribed selectin-ligand site densities. Fig. 4 D reveals the existence of a shear optimum for L-selectin-PSGL-1 binding. In contrast, P-selectin-PSGL-1-dependent neutrophil rolling shows an insignificant decrease in rolling velocity near the shear threshold. As shown in Fig. 4 C, the shear stresses corresponding to the local velocity maximum (white symbol) and minimum (black symbol) are of similar magnitude for the P-selectin-PSGL-1 pair, thereby exerting an insignificant effect on rolling velocity. This is further masked by the high receptor-ligand site density that supports slow rolling. According to the expression of Eq. 13, the existence of an evident local minimum in rolling velocity of a receptor-ligand pair necessitates:

• 1.a large value of τ_peak, and
• 2.a large (A_cK_a^max–A_cK_a⁰)/τ_peak (see Fig. 4 and Fig. S5).

These criteria define whether a specific receptor-ligand pair exhibiting catch-bond kinetics is capable of mediating force-induced stabilization of cell rolling. In Fig. 4, comparing panel B to panel C, we conclude that the absence of a shear optimum for P-selectin-PSGL-1-mediated rolling is due to a small τ_peak in 2-D affinity curve.

Shear optimum in erythrocyte rolling on wild-type and mutated bacterial adhesin

Recent experimental studies showed that erythrocyte binding to the Escherichia coli colonization factor antigen I fibrial adhesin CfaE exhibits the shear threshold phenomenon (30). Moreover, a local minimum in erythrocyte rolling velocity on immobilized fimbriae harboring CfaE adhesins is observed at elevated shear stresses (30). We applied our mathematical model to analyze the rolling of erythrocytes on wild-type and mutated CfaE adhesin (30), and predict the presence or absence of a shear optimum in erythrocyte rolling velocity. Because the 2-D affinity constant A_cK_a and receptor-ligand site densities m_rm_l are not known, we rewrote Eq. 13 in terms of N_b/N_T and τ (see text in the Supporting Material):

$${\left[\frac{d\left(\frac{N_b}{N_T}\right)}{d\tau }=\frac{1-\frac{N_b}{N_T}}{\tau }\right]}_{\tau ={\tau }_c}.$$ (14)

Two intersection points emerge from plotting the LHS and RHS of Eq. 14 using experimental data obtained from bovine erythrocyte rolling on wild-type CfaE fimbriae under shear (30), suggesting the existence of a local minimum in rolling velocity (Fig. 5 A). The shear stress predicted by the model (marked by an arrow in Fig. 5 A) for the local velocity minimum is in an excellent agreement with experimental data (Fig. 5 A, inset). Even though erythrocyte binding to G168D-mutated CfaE fimbriae displays the shear threshold phenomenon (30), no intersection points emerge from plotting the LHS and RHS of Eq. 14 (Fig. 5 B), and as such no local velocity minimum is observed by both our model and experimental data (Fig. 5 B, inset) (30). Collectively, we have demonstrated that the catch-bond kinetics is necessary but not sufficient for the force-induced stabilization of cell rolling, and also provided a comprehensive biophysical interpretation for this phenomenon.

Figure 5.

Prediction of the shear optimum in cell rolling velocity on wild-type and mutated bacterial adhesins. The theoretical prediction of the presence and absence of a shear optimum for bovine erythrocyte rolling on wild-type (A) or G168D-mutated CfaE fimbriae (B), respectively. (Black and orange curves) LHS and RHS of Eq. 14, respectively. Our analysis accurately predicts the optimal shear stress level (marked by an arrow in A) for the local minimum in the erythrocyte rolling velocity on wild-type CfaE fimbriae. It also correctly reveals that the shear optimum does not exist for erythrocyte rolling on G168D-mutated CfaE fimbriae (B). (Insets) Experimental data of the erythrocyte rolling velocity as a function of shear stress.

Concluding Remarks

In this study, we developed an analytical tool to estimate, for the first time to our knowledge, the 2-D kinetic parameters of diverse receptor-ligand pairs such as antibody-antigen, vascular selectin-ligand, and bacterial adhesin-ligand, as a function of physiological shear stresses from flow adhesion assays. Our method is applicable to receptor-ligand pairs exhibiting slip or catch-to-slip dissociation bond kinetics. Using this model, we can also account for multiple bond interactions at elevated shear stresses or receptor-ligand site densities, which represent a natural mechanism for cells to mediate adhesion and assist detachment amid high hemodynamic forces.

Our analytical model provides a comprehensive mechanistic interpretation for the shear optimum in rolling velocity, which is observed only by a subset of bonds exhibiting the catch-slip behavior with specific intrinsic kinetic and micromechanical properties. Our analysis reveals that the catch-bond kinetics is necessary but not sufficient for the shear optimum in cell rolling. A shear optimum is detected only when the increase of the 2-D binding affinity with shear stress, which is characteristic of the catch-bond kinetics, is substantial and occurs over a relatively wide range of shear stresses. The proposed model enables us to predict and explain why a shear optimum is detected for L- but not P-selectin-dependent leukocyte rolling. Although L-selectin-PSGL-1 binding fulfills the aforementioned criteria, the affinity of P-selectin-PSGL-1 bond increases over a narrow range of shear stress (i.e., a small τ_peak value). This analysis also predicted the presence and absence of a shear optimum in bacterial adhesion that has been experimentally observed by Tschesnokova et al. (30).

Using the shear-induced on-rate model (16), we found that the intrinsic reaction rate (k_in) and encounter radius (a) of L-selectin-PSGL-1 binding are higher than that of P-selectin-PSGL-1 (Table 2), which suggests that the kinetics of PSGL-1 binding to L-selectin is faster than that to P-selectin (31). Although our estimated k_in and a values of L-selectin-PSGL-1 are in excellent agreement with previously reported ones obtained by simulations (4), the physical significance of these parameters should be further examined. Of note, our model is capable of incorporating different on- and off-rate kinetics equations for analyzing cell adhesion data.

Even though we herein demonstrate that our proposed model fits successfully to a wide array of published cell adhesion data, there are certain key assumptions that have been made. Because the on-rate and affinity of receptor-ligand interactions are determined by measuring the fraction of cell binding events or the average rolling velocity from flow-based adhesion assays, we assume that cell adhesion is mediated by a small number of bonds. Furthermore, the hydrodynamic force is assumed to be equally distributed among this small number of stressed bonds. The multiple bond off-rate was calculated by assuming that the receptor-ligand bonds break up either entirely simultaneously, or one-by-one in a stepwise fashion.

The former scenario, thus, provides an apparent cell-substrate off-rate constant. The latter may be more appropriate for modeling the dissociation of a relatively large number of stressed bonds under low shear stress levels, because simultaneous dissociation of all bonds seems to be unrealistic under these conditions. Nevertheless, the two scenarios provide very similar results when a small number of bonds with relatively fast kinetics is required for cell-substrate interactions. Based on these assumptions, we are able to keep the virtue of simplicity in our model, which can readily be applied for extracting the 2-D receptor-ligand binding kinetics from flow-based adhesion assays.

Taken together, our model can be applied to diverse receptor-ligand complexes and offers scientific insights on fundamental mechanisms of cell adhesion. We anticipate that use of this model in microfluidic lab-on-a-chip assays (32,33) will enhance our knowledge of how single-molecule and multiple-bond biophysics influence the macroscopic cell behavior in diverse pathophysiological interactions occurring in the vasculature.