Multivalent Binding of Nanocarrier to Endothelial Cells under Shear Flow

Abstract

We investigate the effects of particle size, shear flow, and resistance due to the glycocalyx on the multivalent binding of functionalized nanocarriers (NC) to endothelial cells (ECs). We address the much- debated issue of shear-enhanced binding by computing the binding free-energy landscapes of NC binding to the EC surface when the system is subjected to shear, using a model and simulation methodology based on the Metropolis Monte Carlo approach. The binding affinities calculated based on the free-energy profiles are found to be in excellent agreement with experimental measurements for different-sized NCs. The model suggests that increasing the size of NCs significantly increases the multivalency but only moderately enhances the binding affinities due to the entropy loss associated with bound receptors on the EC surface. A significant prediction of our model is that under flow conditions, the binding free energies of NCs are a nonmonotonic function of the shear force. They show a well-defined minimum at a critical shear value, and thus quantitatively mimic the shear-enhanced binding behavior observed in various experiments. More significantly, our results indicate that the interplay between multivalent binding and shear force can reproduce the shear-enhanced binding phenomenon, which suggests that under certain conditions, this phenomenon can also occur in systems that do not show a catch-bond behavior. In addition, the model also suggests that the impact of the glycocalyx thickness on NC binding affinity is exponential, implying a highly nonlinear effect of the glycocalyx on binding.

Introduction

The dynamic interplay between shear-dominated hydrodynamics and receptor-ligand interactions is well appreciated in the binding of functionalized nanocarriers (NCs) (1), as well as leukocytes (2), platelets (3) and bacteria (4), to cells. A broad range of physical and tunable factors influence binding and engulfment (internalization), including particle size and shape (5–10), and local flow conditions (hydrodynamics) at the site of binding. The latter dictates a range of emergent behaviors such as arrest, rolling, and detachment (6,11–13). The endothelial glycocalyx layer, which usually extends hundreds of nanometers on the cell exterior under in vivo conditions, is also an important determinant of binding (14–18).

From a rational design perspective, inherent physiological conditions such as shear stress, the presence of glycocalyx, expression of targeting receptors (at the site of inflammation, disease, or injury), characteristics of receptor-ligand interactions, and cell membrane mobility have to be synergized with experimentally tunable properties such as carrier size/shape and ligand density. Given the multivariate nature of factors that impact binding, the development of a unified theoretical model could provide an integrated mechanistic view and aid in the optimal experimental design of carriers for attachment to endothelial cells (ECs) (13,19–21). The binding affinity of NC to EC has often been singled out as an important parameter for optimal targeting (22–24). We recently developed a methodology for calculating the absolute binding free energies for antibody functionalized NC binding to the EC surface mediated by intracellular adhesion molecule 1 (ICAM-1) receptors (20). This method enables a direct comparison of the measured binding affinities with those computed in simulations, and the results are in excellent agreement with results obtained from in vitro, in vivo, and atomic force microscopy (AFM) experiments. The remarkable success of this model has motivated investigators to address the next challenge, namely, the development of a model for NC/cell binding to EC under shear, in which context the phenomenon of shear-enhanced binding is widely debated (19,25,26).

Rolling of blood cells, bacteria, and carriers is mediated by intermittent and stochastic engagement and rupture of receptor-ligand bonds (25,27–30). The shear-enhanced binding is characterized by a threshold flow shear rate for initial tethering and stable rolling of adherent cells or carriers. This effect is manifested as a decrease in rolling velocity with an increase in shear rate for rates below the threshold shear value, and an increase in rolling velocity with increasing shear above the threshold value. The initial decrease is counterintuitive because the dissociation rate of receptor-ligand bonds increases exponentially with increasing applied force based on the Bell model (see Section S1 in the Supporting Material). To explain this phenomenon, the concept of catch bonds (31), which prolong the lifetime of receptor-ligand attachment upon application of a tensile force, is invoked. Catch bonds in different systems were directly observed in recent AFM experiments (25,32–36), and subsequently shear-enhanced binding was commonly attributed to the formation of catch bonds (26,27,32,37). Indeed, a number of conceptual two-pathway or two-state models (38–44) have been proposed and successfully implemented to reproduce the experimental data for shear-enhanced binding. However, using adhesion dynamics simulations, Beste and Hammer (19) demonstrated that exact knowledge of the catch-bond kinetics is not necessary, and only two phenomenological parameters—f_cr (critical force) and ɛ (kinetic efficiency)—are sufficient to reproduce the experimental data of leukocyte adhesion. Recently, using adhesion dynamics simulations and flow-chamber experiments, Whitfield et al. (45) showed that shear-stabilized rolling of Escherichia coli may be due to an increased number of bonds resulting from the fimbrial deformation, whereas the stationary (or firm) adhesion may be due to the catch bonds.

Here, we show that a second mechanism involving an interplay among multivalency, shear flow, and bond compliance (of the receptor-ligand bond) may also lead to shear-enhanced binding that is independent of catch bonds. By treating the stochastic nature of receptor-ligand interactions from a quasi-equilibrium perspective, we provide a free-energy landscape description of the adhering NC under flow by extending our recent methodology (20) to compute the free-energy landscapes and binding affinities of NC. We investigate three important factors that influence the binding affinity of NC to EC surfaces: 1), the size of the NCs; 2), shear flow; and 3), the effect of the glycocalyx. Our results suggest that as a parallel to rolling velocities, the equilibrium binding affinity can also serve as an interesting quantity in the study of shear-enhanced binding.

Theory and Model

Following the framework described by Woo and Roux (46), we developed a computational model (20) to calculate the absolute binding affinities between functionalized NCs and the EC surface based on the potential of mean force (PMF). Our model is based on the Metropolis Monte Carlo (MC) method and the weighted histogram analysis method (WHAM) (47), which provides a unique advantage by enabling us to directly compare our results with a variety of experimental measurements, as we showed in a previous study (20). We provide the computational details of our model in the Supporting Material (see Section S1, Section S2, Section S3, and Section S4 for methods, and Section S5 and Section S6 for parameter estimation and sensitivity), and present only a brief summary of those details here.

The main extension of our earlier model is the inclusion of shear flow. As described in Section S3, we introduce a one-dimensional shear flow having shear rate S. The flow-induced drag F_x and torque T_y are calculated by solving the steady-state Stokes equation for shear flow past a sphere (NC) near a surface (cell), with additional no-slip boundary conditions and inlet flow condition of u = Sz, where u is the velocity and z is the distance away from the surface.

Binding between an antibody-coated NC and ICAM-1-expressing EC surface is simulated using the Metropolis MC method. During each MC step, we randomly choose one of the following four types of movement: bond formation/breaking, NC translation, NC rotation, and ICAM-1 translation (diffusive cells), with probabilities of 50%, $\text{0}.25\ast {\mathit{N}}_{\mathit{a}}/{\mathit{N}}_{\mathit{t}}$, $\text{0}.25\ast {\mathit{N}}_{\mathit{a}}/{\mathit{N}}_{\mathit{t}}$, and $\text{0}.5\ast {\mathit{N}}_{\mathit{ICAM}\text{-}1}/{\mathit{N}}_{\mathit{t}}$, respectively. Here N_t = N_a + N_ICAM–1, where N_a is the number of antibodies and ${\mathit{N}}_{\mathit{ICAM}-1}$ is the number of ICAM-1 molecules. Based on the particular MC move, the new system energy is computed and the movement is accepted according to Metropolis criteria. The step sizes for NC translation/rotation and ICAM-1 translation are adaptively updated to ensure an MC acceptance rate of 50%. Because the ICAM-1 flexural movement is highly orientation-dependent, we implement a configurational-biased sampling technique (48) in our model to improve the sampling efficiency of the configurations of flexural movement (see Section S2). To account for the effects of hydrodynamics from flow on the MC moves, in similarity to Pierres et al. (49), we approximate the continuous expressions of F_x(z) and T_y(z) (see Eqs. S7 and S8, and Fig. S2) by fitting the measured discrete points. The force and torque introduce additional energy change during each translational and rotational movement of the sphere, and therefore influence the Metropolis acceptance criteria.

In the calculation of binding affinities, we define the reaction coordinate z as the vertical distance between the center of the NC and cell surface. The binding affinity (or association constant) can be expressed as (20):

$$K_a=\frac{1}{\left[\mathbf{L}\right]}\times T_1\times T_2\times T_3.$$ (1)

Here, [L] is the NC concentration, and the term T₁ accounts for the entropy loss from the bounded receptors

$$T_1=\frac{A_{R,b}^{\left(1\right)}\times A_{R,b}^{\left(2\right)}\times \mathrm{\dots }\times A_{R,b}^{\left(N_b\right)}}{A_{R,ub}^{\left(1\right)}\times A_{R,ub}^{\left(2\right)}\times \mathrm{\dots }\times A_{R,ub}^{\left(N_b\right)}},$$ (2)

$A_{R,b}^{\left(n\right)}$ is the accessible surface area available to the nth receptor in the bound state, and $A_{R,ub}^{\left(n\right)}$ is the corresponding quantity in the unbound state. The term T₂ is associated with the NC rotational entropy loss upon binding,

$$T_2=\frac{\left(N_{ab}/N_b\right)\Delta \omega }{8{\pi }^2},$$ (3)

where N_ab is the number of antibodies (ligands) per NC (antibody surface density), and N_b is the total number of bonds in the equilibrium state; hence, $N_{ab}/N_b$ denotes the multiplicity of the NC. Δω is the rotational volume of the NC in the bound state, which is estimated from the root mean-squared deviations of Euler angles as described previously (50). The term T₃ accounts for NC translational entropy loss:

$$T_3=\frac{A_{NC,b}\int e^{-\beta W\left(z\right)}dz}{A_{NC,ub}{l}_z},$$ (4)

where A_{NC, b} is the area accessible for the translation of the NC in the bound state, A_NC,ub and A_NC,ubl_z are the area and volume accessible to the NC in the unbound state, and W(z) is the PMF (20,46). The NC concentration can be expressed as $\left[\text{L}\right]=\text{1}/\left({\mathit{A}}_{\mathit{NC}}\mathit{,}{}_{\mathit{ub}}{\mathit{l}}_{\mathit{z}}\right)$ to yield the expression for K_a in Eq. 1.

Results

Effects of particle size

The PMFs are calculated for binding of NCs of diameters 100 and 200 nm, respectively (see Fig. 1, a and b). In both cases, the antibody surface coverage, σ_s, is kept constant at ∼74%, a value employed in experiments. (We note that the PMF for single receptor-ligand bonds in the absence of the NC was also computed and found to be in agreement with experimental results; see Section S5 and Fig. S3.) As is evident from Fig. 1 a, for the NC of size 100 nm, three firm bonds are observed in the equilibrium state of NC bound to the cell surface (at z ∼83 nm, indicated by the arrow; see also the multivalency in Fig. 1 c), with a characteristic free energy well of ∼−32 k_BT. The spatially averaged distribution of bound ICAM-1s relative to the center of NC forms an annulus pattern (inset in Fig. 1 a), consistent with previous reports (20,51). Combining the PMF profile and annulus distribution, and using Eq. 1, the binding affinity is calculated as K_a = 5.9 × 10¹⁰ nm³. The corresponding dissociation constant K_d = 1/K_a = 28.0 pM, in agreement with experimental measurements (22).

Figure 1.

PMF profiles of (a) 100 nm NC and (b) 200 nm NC, and (c and d) the corresponding multivalency. The antibody surface coverage σ_s ∼74% and the temperature T = 27°C. The arrows in panels a and b indicate the equilibrium state. The average spatial bond distributions relative to the NC center are shown in the insets of a and b.

For the larger NC shown in Fig. 1 b, the characteristic free-energy change at the equilibrium state is ∼−42 k_BT at z ∼132.7 nm. As shown in Fig. 1 d, the number of bonds at equilibrium lies between 5 and 6, indicating five firm bonds and one transitional (dynamic) bond. The binding affinity calculated from Eq. 1 with N_b = 5 yields K_a = 2.5 × 10¹¹ nm³ and K_d = 6.6 pM. (Here, we set $A_{R,ub}^{\left(1\right)}$ = $A_{R,ub}^{\left(2\right)}$ = … = $A_{R,ub}^{\left(5\right)}$ = πr_o², $A_{R,b}^{\left(1\right)}$ = πr_o², $A_{R,b}^{\left(2\right)}$ = π(r_o² – r_i²), $A_{R,b}^{\left(3\right)}$ = $A_{R,b}^{\left(4\right)}$ = $A_{R,b}^{\left(5\right)}$ = A_{NC, b} = (r_o – r_i)², where r_o = 18.5 nm and r_i = 14.8 nm are the outer and inner radii, respectively, of the annulus distribution (see inset in Fig. 1 b).

Our results imply that the change in binding affinity is only moderate with increasing size of spherical NCs, even though the multivalency and the PMF increase significantly. This modest enhancement of binding is due to the fact that the entropy loss of bound receptors also increases with the increasing multivalency. The binding affinities calculated from our model are found to be in excellent agreement with those measured in experiments (22,52) (see Table 1).

Table 1.

Binding affinity values: comparison of model predictions with experiments

	100 nm	200 nm	1 μm
Experiments	77 pM (22)	N/A	1.6 pM (52)
Model predictions	28.0 pM	6.6 pM	N/A

Effect of shear flow

To investigate the effect of shear flow on the binding of 200 nm NCs (σ_s = 74%) to EC surface, we compute the PMF profiles at different shear forces (F_x); the shear force is related to shear rate by Eq. S7 in Section S3. We note that the drag force scales as the square, and the torque scales as the cube of the particle diameter at fixed shear rate close to surface. Here we state our results in terms of the drag force and torque acting on the particle (and therefore the force transmitted to the individual binding complex). For NC of 200 nm diameter, such forces and torques are small under normal physiological flow rates. However, the force and the torque terms are significant for NCs of larger (∼μm) diameters. Instead of investigating NCs of larger diameters under physiological flow conditions, we choose to investigate NCs of smaller diameters under higher than physiological flow rates because the latter are computationally more tractable and lead to much better control of statistical error in our MC simulations. In so doing, we make use of the scaling (of force and torque with NC diameter) described above and report our results in terms of drag force (in the range of tens of piconewtons) to enable a comparison across two NC diameters.

Fig. 2 a shows the PMF profiles at shear force from 0 to ∼50 pN. We note that the maximum force is still less than the rupture force of 300 pN as previously measured by AFM (20). Evidently, the flow tends to lower the free energy until a threshold shear force is reached, beyond which there is an increase in the free energy. Based on the PMF profiles, we calculate the binding affinities K_a for each flow rate and plot the dissociation constant K_d = 1/K_a as a function of the shear force on the NC in Fig. 2 b. The results clearly show a biphasic effect: below a threshold shear force, an increase in force actually promotes NC binding by reducing K_d, whereas above the threshold, an increase in shear force decreases binding by promoting NC detachment. In Fig. 2 b, we also demonstrate that the observation of flow-enhanced binding is relatively insensitive to the particle size (we also carried out simulations for two different NC sizes: 100 and 200 nm) if we plot the results as a function of drag force, and the emergent shear-enhanced binding behavior is manifested in a similar fashion.

Figure 2.

Effect of shear flow on binding between NC and EC surface. The antibody surface coverage is 74%. (a) The PMF profiles under different shear force for 200 nm NCs. (b) Dissociation constants calculated based on a as a function of the shear force on NC introduced by flow (different symbols represent different sizes). (c) Experimental measurement of the rolling velocities of microspheres at different tether forces (26); different symbols represent different sizes (1 and 3 μm) and different fluid medium viscosities. As discussed in the main text, the model and the experiments do not have a direct correspondence. We note that the definitions of shear force on NC in our model and tether force in Yago et al. (26) are not identical. Moreover, our model parameters are chosen to mimic interactions between R6.5 and ICAM-1, whereas the interactions in the study by Yago et al. (26) were between L-selectin and P-selectin glycoprotein ligand-1.

We note that the flow-enhanced binding described above is strictly an equilibrium phenomenon. Even though the flow perturbs thermodynamic equilibrium, we made the tacit assumption that the shear flow is steady and acts as an external field on the quiescent system, which from a quasi-equilibrium (or linear-response) perspective justifies our use of the equilibrium definition of the binding affinity or association constant. The shear-enhanced binding reported here is distinct from the shear-enhanced rolling reported by Yago et al. (26); see Fig. 2 c, which shows the experimentally determined rolling velocities of NCs at various tether forces. In particular, the rolling velocity shows a biphasic trend, initially decreasing with increasing tether force, and subsequently, above a threshold tether force, the rolling velocity increases with increasing force. A related phenomenon of shear stabilized rolling behavior was also reported by Whitfield et al. (45), as discussed briefly in the Introduction. The rolling velocity of NC is mediated by the stochastic nature of receptor-ligand bond formation and rupture; hence, the rolling velocity is governed by the underlying free energy landscape of adhesion. In particular, the average slip velocity is governed by the inherent timescale for NC motion, the free-energy barrier for bond rupture, as well as the length scale over which the partially adhered NC pivots before reengaging the full extent of the multivalent interactions. Hence, the same free-energy landscape that governs NC binding affinity can be employed to relate to NC rolling behavior. However, the crucial difference is that the reaction coordinate for NC rolling is, in general, distinct from the reaction coordinate z we employed to compute PMF. The reaction coordinate for rolling behavior can be quite complex because of the multivalent interaction, but the x or y coordinate may be a better surrogate to resolve the free-energy barriers for rolling than the z coordinate we employed to compute the binding affinity. Although we did not comprehensively pursue rolling in this study, we note that PMF along x or y can indeed resolve free-energy barriers to rolling. From such a free-energy landscape, one can employ kinetic MC simulations to extract rolling rates and velocities.

To summarize, the NC rolling velocity and NC binding affinity are both governed by the underlying free-energy landscape. They are similar in that both are governed by an Arrhenius form involving the free-energy landscape. However, the crucial difference is that rolling can be viewed as connecting two bound states of NC, and the binding affinity is defined by connecting the bound with the unbound state of NC. Hence, the underlying reaction coordinates are distinct. It is still notable that the shear-threshold behavior we have computed in Fig. 2 b looks remarkably similar in form to the experimental measurements of rolling velocities in Fig. 2 c. In particular, it appears that the shear threshold (reported in terms of the drag force) for the NC binding is similar to the threshold for the rolling behavior. A more thorough investigation of the exact relationship between these distinct phenomena could be a promising avenue for future study.

To further understand the underlying mechanism of this shear-enhanced binding, we calculate the average receptor flexure angle θ and receptor-ligand bond length d (see Fig. S1 for definitions) at the equilibrium position of the bound NC. We plot the results as a function of shear force on the NC in Fig. 3. Whereas the flexure angle increases monotonically with an increasing shear force (Fig. 3 a), the equilibrium bond length (Fig. 3 b) first decreases to a minimum and then increases with increasing shear force. The trend in the binding affinity mirrors the trend in the equilibrium bond length. At equilibrium, there are always some weaker bonds present in multivalent interactions. Our results (Fig. 3) imply that below the threshold value for shear, the force and torque generated from flow tend to shift the NC conformation to a more stable state by making the weaker bonds stronger; above the threshold, the hydrodynamic force is strong to rupture bonds and reduce the multivalency (see Fig. 3, c and d). We note that our model also predicts shear-induced detachment at low σ_s, consistent with experimental results (see Fig. S7 and Fig. S8, and associated Supporting Material). These features clearly demonstrate that the interplay among shear force, multivalent interactions, and bond compliance induces the shear-enhanced binding in our model.

Figure 3.

Average receptor bending angle θ (a) and bond length d (b) as a function of shear force on NC at equilibrium state. (c and d) Evolution of multivalency before and after the threshold (F_s = 2.5 pN and 50.5 pN).

Effect of the glycocalyx

Estimates from experiments (53–56) indicate that the glycocalyx layer extends 100 nm to several hundreds of nanometers over the cell surface. When the NC approaches the cell surface, the resistance due to the glycocalyx layer generally includes a combination of the osmotic pressure, electrostatic repulsion, steric repulsion between NC and the glycoprotein chains, and the entropic forces due to conformational restrictions imposed on the confined glycoprotein chains, which is too complicated to be accounted for in a model. Therefore, we employ a phenomenological model in which we lump all the above effects into a single mechanical resistance by neglecting the detailed microstructures. We account for the normal resistance of the glycocalyx by adding a simple harmonic potential 1/2k_glyxH² per unit differential area, where H = z – h is the penetration depth of the NC into the glycocalyx, h is the thickness of the glycocalyx, and k_glyx is the glycocalyx stiffness (21). The harmonic nature of the glycocalyx potential has its basis in polymer physics, where the entropic elasticity of polymer chains has been shown to assume a harmonic potential (57).

Experimental data for 100 nm NC binding to EC in vivo (17) suggest a factor of 500-fold reduction in the binding affinity in the presence of the glycocalyx compared with that in the absence of the glycocalyx. Following Agrawal and Radhakrishnan (21), and assuming a glycocalyx height of 100 nm (which can be regarded as the lower bound for glycocalyx thickness in vivo), we model the effect of the glycocalyx on NC binding. The PMF profiles for different values of the glycocalyx thickness h are computed as $\mathit{W}{\left(\mathit{z}\right)}_{\mathit{glyx},\mathit{h}}=W{\left(z\right)}_{glyx,h=0}+\int 1/2k_{glyx}{H}^{2}dA$, where A is the surface area of NC immersed in the glycocalyx. Fig. 4 a depicts the PMF profiles of NC binding in the presence of glycocalyx (of varying thicknesses), based on which the binding affinities were also computed.

Figure 4.

(a) Effect of glycocalyx height on the PMF profiles for 100 nm NCs; the glycocalyx stiffness is assumed constant. (b) The effect of glycocalyx height on binding affinity depends on the model of glycocalyx stiffness. The solid lines correspond to a constant glycocalyx stiffness k_glyx (black, 100 nm; gray, 200 nm NCs), whereas for the dotted line it is assumed that the glycocalyx stiffness k_glyx is linearly decreasing from the cell surface to the maximum height. This is meant to mimic the variable density of glycocalyx proteins, which decreases farther from the cell surface, and highlight the fact that the model for k_glyx does not affect the exponential dependence.

To investigate how the thickness of the glycocalyx influences binding in the absence of flow, we report K_{a, 0}/K_{a, h} (i.e., the ratio of the NC binding affinity association constant without glycocalyx to that with glycocalyx) as a function of glycocalyx thickness h in Fig. 4 b. It is evident that the dependence on h is exponential, implying that whereas a 100-nm-thick glycocalyx can affect binding by a factor of 500, reducing the thickness even down to the ∼55–70 nm range only lowers binding by twofold. Hence, the exponential dependence of K_{a, 0}/K_{a, h} versus h could rationalize the large differences between in vitro and in vivo experiments of glycocalyx-mediated NC binding, namely, that binding decreases by a modest twofold in vitro (see Fig. S6) and by several hundred-fold in vivo (17). Such a drastic difference can arise even if the thickness in the glycocalyx changes by 30–50 nm between in vitro and in vivo conditions. Fig. 4 b also depicts a similar exponential dependence of K_{a, 0}/K_{a, h} on h for NCs of diameter 200 nm. We note that we have not presented modeling results for the case in which glycocalyx and shear flow are present simultaneously. Such a situation would be a better representation of the hydrodynamic conditions in vivo. However, we can extend our model to adapt to this situation by following the procedure outlined in Section S8 to resolve the flow inside the glycocalyx layer. This avenue will be pursued in future studies.

Discussion

Using a methodology to compute the absolute free energies of functionalized NC binding to EC, we explored the effects of the NC particle size, shear flow, and glycocalyx on NC binding. Although the results of our earlier model (with flow absent) are in excellent agreement with in vitro, in vivo, and AFM experiments (20), we have now demonstrated that our model also predicts results that are in excellent agreement with experiments in a variety of scenarios that involve flow. We show that increasing particle size only moderately enhances the binding affinities despite showing an increase in multivalent interactions. This is because an increase in enthalpic interactions with increasing multivalency is countered by a significant decrease in the translational entropy of bound receptors. The entropic contributions collectively sum to a significant free-energy penalty under large multivalency. By examining the PMFs at different shear rates, we observe a nonmonotonic trend for binding as a function of shear force, with the dissociation constant showing a characteristic minimum at a critical shear force. This shear-enhanced binding behavior arises in our model solely because of the interplay among multivalency, shear flow, and bond compliance, and their collective influence on the free-energy landscape. Below the threshold shear value, the flow enhances binding by lowering the absolute free energy of the bound state. A decrease in the enthalpy of receptor-ligand bonds for a given multivalency results from a reduction in the average bond length d toward the optimal (equilibrium) value; above the threshold shear, the binding is reduced due to a decrease in multivalency. The shear-enhanced binding behavior is different from shear-enhanced rolling of NCs. Yet, our calculated trend of binding affinity versus shear force mirrors the experimentally measured trend of rolling velocity versus shear force (26). Our results suggest that mechanisms other than those involving catch bonds may also (under certain conditions) lead to shear-enhanced binding. Finally, we show that the glycocalyx can effectively reduce the binding between NC and cell surface. However, the effect of increasing glycocalyx thickness h on the association constant (K_{a, h}) of NC adhesion is exponential, suggesting that an orders-of-magnitude decrease in binding affinity is only achieved when the glycocalyx thickness h is large enough (for h ∼100 nm, the decrease in binding affinity is 500-fold for a 100 nm NC). Thus, even for glycocalyx thickness h in the range of 60 nm (or 2/3 of the typical 100 nm value), the decrease in binding affinity is only a few-fold for 100 nm NCs. Hence, our model can rationalize the contrasting behavior of NC binding mediated by the glycocalyx as observed in in vitro and in vivo experiments.

Several predictions arise from our study, which provide exciting opportunities for future experiments: 1) We have shown that the shear-enhanced binding behavior does not necessarily need to be an attribute of a specialized protein undergoing conformational change when subjected to shear; rather, it could be an emergent property of multivalency, leading to the prediction that a variety of receptor-ligand complexes can sustain this behavior amid multivalent interactions. 2) Given that bond stiffness (compliance) is an important factor, point mutations in the ligand that alter the rupture force but not the binding affinity will have a significant effect on flow-enhanced binding. Similarly, engineering multivalency in functionalized NC with the use of soft polymeric tethers (rather than attaching the antibodies directly to the carrier) will produce a dramatically different behavior in shear-enhanced binding that does not involve catch bonds. 3) The multivalent interactions that are central to our model can be directly imaged through super-resolution imaging experiments. 4) The nonlinear effect of glycocalyx thickness on binding landscapes predicted by our model can also be directly tested in vitro on grafted polymer surfaces. The model predictions can also be exploited for rational reengineering of functionalized NCs for targeted drug delivery.