Biophysical Journal Volume 101 July 2011 319–326 319

Multivalent Binding of Nanocarrier to Endothelial Cells under Shear Flow

Jin Liu,† Neeraj J. Agrawal,‡ Andres Calderon,§ Portonovo S. Ayyaswamy,{ David M. Eckmann,§

and Ravi Radhakrishnan†*†Department of Bioengineering, ‡Department of Chemical and Biomolecular Engineering, §Department of Anesthesiology and Critical Care,and {Department of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, Pennsylvania

ABSTRACT We investigate the effects of particle size, shear flow, and resistance due to the glycocalyx on the multivalentbinding of functionalized nanocarriers (NC) to endothelial cells (ECs). We address the much- debated issue of shear-enhancedbinding by computing the binding free-energy landscapes of NC binding to the EC surface when the system is subjected toshear, using a model and simulation methodology based on the Metropolis Monte Carlo approach. The binding affinities calcu-lated 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 moderatelyenhances the binding affinities due to the entropy loss associated with bound receptors on the EC surface. A significant predic-tion 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 behaviorobserved in various experiments. More significantly, our results indicate that the interplay between multivalent binding and shearforce can reproduce the shear-enhanced binding phenomenon, which suggests that under certain conditions, this phenomenoncan also occur in systems that do not show a catch-bond behavior. In addition, the model also suggests that the impact of theglycocalyx thickness on NC binding affinity is exponential, implying a highly nonlinear effect of the glycocalyx on binding.

INTRODUCTION

The dynamic interplay between shear-dominated hydrody-namics and receptor-ligand interactions is well appreciatedin the binding of functionalized nanocarriers (NCs) (1), aswell as leukocytes (2), platelets (3) and bacteria (4), to cells.A broad range of physical and tunable factors influencebinding and engulfment (internalization), including particlesize and shape (5–10), and local flow conditions (hydrody-namics) at the site of binding. The latter dictates a rangeof emergent behaviors such as arrest, rolling, and detach-ment (6,11–13). The endothelial glycocalyx layer, whichusually extends hundreds of nanometers on the cell exteriorunder in vivo conditions, is also an important determinant ofbinding (14–18).

From a rational design perspective, inherent physiologicalconditions such as shear stress, the presence of glycocalyx,expression of targeting receptors (at the site of inflammation,disease, or injury), characteristics of receptor-ligand interac-tions, and cell membrane mobility have to be synergizedwith experimentally tunable properties such as carrier size/shape and ligand density. Given the multivariate nature offactors that impact binding, the development of a unifiedtheoretical model could provide an integrated mechanisticview and aid in the optimal experimental design of carriersfor attachment to endothelial cells (ECs) (13,19–21). Thebinding affinity of NC to EC has often been singled out asan important parameter for optimal targeting (22–24). Werecently developed a methodology for calculating the abso-

Submitted January 29, 2011, and accepted for publication May 23, 2011.

*Correspondence: rradhak@seas.upenn.edu

Editor: Denis Wirtz.

� 2011 by the Biophysical Society

0006-3495/11/07/0319/8 $2.00

lute binding free energies for antibody functionalized NCbinding to the EC surface mediated by intracellular adhesionmolecule 1 (ICAM-1) receptors (20). This method enablesa direct comparison of the measured binding affinities withthose computed in simulations, and the results are in excel-lent agreement with results obtained from in vitro, in vivo,and atomic force microscopy (AFM) experiments. Theremarkable success of this model has motivated investiga-tors to address the next challenge, namely, the developmentof a model for NC/cell binding to EC under shear, in whichcontext the phenomenon of shear-enhanced binding iswidely debated (19,25,26).

Rolling of blood cells, bacteria, and carriers is mediatedby intermittent and stochastic engagement and rupture ofreceptor-ligand bonds (25,27–30). The shear-enhancedbinding is characterized by a threshold flow shear rate forinitial tethering and stable rolling of adherent cells orcarriers. This effect is manifested as a decrease in rollingvelocity with an increase in shear rate for rates below thethreshold shear value, and an increase in rolling velocitywith increasing shear above the threshold value. The initialdecrease is counterintuitive because the dissociation rate ofreceptor-ligand bonds increases exponentially withincreasing applied force based on the Bell model (seeSection S1 in the Supporting Material). To explain thisphenomenon, the concept of catch bonds (31), whichprolong the lifetime of receptor-ligand attachment uponapplication of a tensile force, is invoked. Catch bonds indifferent systems were directly observed in recent AFMexperiments (25,32–36), and subsequently shear-enhancedbinding was commonly attributed to the formation of catch

doi: 10.1016/j.bpj.2011.05.063

320 Liu et al.

bonds (26,27,32,37). Indeed, a number of conceptual two-pathway or two-state models (38–44) have been proposedand successfully implemented to reproduce the experi-mental data for shear-enhanced binding. However, usingadhesion dynamics simulations, Beste and Hammer (19)demonstrated that exact knowledge of the catch-bondkinetics is not necessary, and only two phenomenologicalparameters—fcr (critical force) and 3 (kinetic efficiency)—are sufficient to reproduce the experimental data of leuko-cyte adhesion. Recently, using adhesion dynamics simula-tions and flow-chamber experiments, Whitfield et al. (45)showed that shear-stabilized rolling of Escherichia colimay be due to an increased number of bonds resultingfrom the fimbrial deformation, whereas the stationary (orfirm) adhesion may be due to the catch bonds.

Here, we show that a second mechanism involving aninterplay among multivalency, shear flow, and bond com-pliance (of the receptor-ligand bond) may also lead toshear-enhanced binding that is independent of catch bonds.By treating the stochastic nature of receptor-ligand interac-tions from a quasi-equilibrium perspective, we provide afree-energy landscape description of the adhering NC underflow by extending our recent methodology (20) to computethe free-energy landscapes and binding affinities of NC. Weinvestigate three important factors that influence the bindingaffinity of NC to EC surfaces: 1), the size of the NCs; 2),shear flow; and 3), the effect of the glycocalyx. Our resultssuggest that as a parallel to rolling velocities, the equilib-rium binding affinity can also serve as an interesting quan-tity 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 theabsolute binding affinities between functionalized NCsand 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 byenabling us to directly compare our results with a varietyof experimental measurements, as we showed in a previousstudy (20). We provide the computational details of ourmodel in the Supporting Material (see Section S1, SectionS2, Section S3, and Section S4 for methods, and SectionS5 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 inclusionof shear flow. As described in Section S3, we introduce aone-dimensional shear flow having shear rate S. The flow-induced drag Fx and torque Ty are calculated by solvingthe steady-state Stokes equation for shear flow past a sphere(NC) near a surface (cell), with additional no-slip boundaryconditions and inlet flow condition of u ¼ Sz, where u is thevelocity and z is the distance away from the surface.

Biophysical Journal 101(2) 319–326

Binding between an antibody-coated NC and ICAM-1-expressing EC surface is simulated using the MetropolisMC method. During each MC step, we randomly chooseone of the following four types of movement: bond forma-tion/breaking, NC translation, NC rotation, and ICAM-1translation (diffusive cells), with probabilities of 50%,0:25�Na=Nt, 0:25�Na=Nt , and 0:5�NICAM-1=Nt, respec-tively. Here Nt ¼ Na þ NICAM–1, where Na is the numberof antibodies and NICAM�1 is the number of ICAM-1 mole-cules. Based on the particular MC move, the new systemenergy is computed and the movement is accepted accord-ing to Metropolis criteria. The step sizes for NC transla-tion/rotation and ICAM-1 translation are adaptivelyupdated to ensure an MC acceptance rate of 50%. Becausethe ICAM-1 flexural movement is highly orientation-depen-dent, we implement a configurational-biased sampling tech-nique (48) in our model to improve the sampling efficiencyof the configurations of flexural movement (see Section S2).To account for the effects of hydrodynamics from flow onthe MCmoves, in similarity to Pierres et al. (49), we approx-imate the continuous expressions of Fx(z) and Ty(z) (see Eqs.S7 and S8, and Fig. S2) by fitting the measured discretepoints. The force and torque introduce additional energychange during each translational and rotational movementof the sphere, and therefore influence the Metropolis accep-tance criteria.

In the calculation of binding affinities, we define the reac-tion coordinate z as the vertical distance between the centerof the NC and cell surface. The binding affinity (or associ-ation constant) can be expressed as (20):

Ka ¼ 1

½L� � T1 � T2 � T3: (1)

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

T1 ¼ Að1ÞR;b � A

ð2ÞR;b � . � A

ðNbÞR;b

Að1ÞR;ub � A

ð2ÞR;ub � . � A

ðNbÞR;ub

; (2)

AðnÞR;b is the accessible surface area available to the nth

receptor in the bound state, and AðnÞR;ub is the corresponding

quantity in the unbound state. The term T2 is associatedwith the NC rotational entropy loss upon binding,

T2 ¼ ðNab=NbÞDu8p2

; (3)

where Nab is the number of antibodies (ligands) per NC(antibody surface density), and Nb is the total number ofbonds in the equilibrium state; hence, Nab=Nb denotes themultiplicity of the NC. Du is the rotational volume of theNC in the bound state, which is estimated from the rootmean-squared deviations of Euler angles as described previ-ously (50). The term T3 accounts for NC translationalentropy loss:

Nanocarrier Binding to Endothelial Cells 321

T3 ¼ ANC;b

Re�bWðzÞdz

ANC;ublz; (4)

where ANC, b is the area accessible for the translation of theNC in the bound state, ANC,ub and ANC,ublz are the area andvolume accessible to the NC in the unbound state, and W(z)is the PMF (20,46). The NC concentration can be expressedas ½L� ¼ 1=ðANC; ub lzÞ to yield the expression for Ka inEq. 1.

RESULTS

Effects of particle size

The PMFs are calculated for binding of NCs of diameters100 and 200 nm, respectively (see Fig. 1, a and b). Inboth cases, the antibody surface coverage, s_s, is keptconstant at ~74%, a value employed in experiments. (Wenote that the PMF for single receptor-ligand bonds in theabsence of the NC was also computed and found to be inagreement with experimental results; see Section S5 andFig. S3.) As is evident from Fig. 1 a, for the NC of size100 nm, three firm bonds are observed in the equilibriumstate of NC bound to the cell surface (at z ~83 nm, indicatedby the arrow; see also the multivalency in Fig. 1 c), witha characteristic free energy well of ~�32 kBT. The spatiallyaveraged distribution of bound ICAM-1s relative to thecenter of NC forms an annulus pattern (inset in Fig. 1 a),consistent with previous reports (20,51). Combining thePMF profile and annulus distribution, and using Eq. 1, thebinding affinity is calculated as Ka ¼ 5.9 � 1010 nm3. The

z (nm)

PM

F(k

BT

)

132 133 134 135

-40

-30

-20

-10

0 b

z (nm)

PM

F(k

BT

)

83 84 85

-30

-20

-10

0 a

z (nm)

mul

tival

ency

132 133 134 1350

2

4

6 d

z (nm)

mul

tival

ency

83 84 850

1

2

3

4c

x (nm)

y(n

m)

-50 0 50-50

0

50

x (nm)

ynm

-50 0 50-50

0

50

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

ss ~74% and the temperature T ¼ 27�C. The arrows in panels a and b indi-

cate the equilibrium state. The average spatial bond distributions relative to

the NC center are shown in the insets of a and b.

corresponding dissociation constant Kd ¼ 1/Ka ¼ 28.0pM, in agreement with experimental measurements (22).

For the larger NC shown in Fig. 1 b, the characteristic free-energy change at the equilibrium state is ~�42 kBT at z~132.7 nm.As shown in Fig. 1 d, the number of bonds at equi-librium lies between 5 and 6, indicating five firm bonds andone transitional (dynamic) bond. The binding affinity calcu-lated from Eq. 1 with Nb¼ 5 yields Ka¼ 2.5� 1011 nm3 and

Kd¼ 6.6 pM. (Here, we setAð1ÞR;ub ¼A

ð2ÞR;ub ¼.¼A

ð5ÞR;ub ¼pro

2,

Að1ÞR;b ¼ pro

2, Að2ÞR;b ¼ p(ro

2 – ri2), A

ð3ÞR;b ¼ A

ð4ÞR;b ¼ A

ð5ÞR;b ¼ ANC, b¼

(ro – ri)2, where ro ¼ 18.5 nm and ri ¼ 14.8 nm are the outer

and inner radii, respectively, of the annulus distribution (seeinset in Fig. 1 b).

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

Effect of shear flow

To investigate the effect of shear flow on the binding of200 nm NCs (ss ¼ 74%) to EC surface, we compute thePMF profiles at different shear forces (Fx); the shear forceis related to shear rate by Eq. S7 in Section S3. We notethat the drag force scales as the square, and the torque scalesas the cube of the particle diameter at fixed shear rate closeto surface. Here we state our results in terms of the dragforce and torque acting on the particle (and therefore theforce transmitted to the individual binding complex). ForNC of 200 nm diameter, such forces and torques are smallunder normal physiological flow rates. However, the forceand the torque terms are significant for NCs of larger(~mm) diameters. Instead of investigating NCs of largerdiameters under physiological flow conditions, we chooseto investigate NCs of smaller diameters under higher thanphysiological flow rates because the latter are computation-ally more tractable and lead to much better control of statis-tical error in our MC simulations. In so doing, we make useof 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 comparisonacross 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

TABLE 1 Binding affinity values: comparison of model

predictions with experiments

100 nm 200 nm 1 mm

Experiments 77 pM (22) N/A 1.6 pM (52)

Model predictions 28.0 pM 6.6 pM N/A

Biophysical Journal 101(2) 319–326

c

rolli

ng

velo

city

(m

/s)

40

0

10

20

30

z (nm)

PM

F (

kT

)

133 134-60

-40

-20

0

F =0 pNF =1.5 pNF =2.5 pNF =5.1 pNF =15.2 pNF =25.3 pNF =37.9 pNF =50.5 pN

a

shear force (pN)

K(n

m)

0 20 40 60 8010101010101010

b

0 50 100 150 200 250 300

Tether force (pN)

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 mm) 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.

322 Liu et al.

the rupture force of 300 pN as previously measured by AFM(20). Evidently, the flow tends to lower the free energy untila threshold shear force is reached, beyond which there is anincrease in the free energy. Based on the PMF profiles, wecalculate the binding affinities Ka for each flow rate andplot the dissociation constant Kd ¼ 1/Ka as a function ofthe shear force on the NC in Fig. 2 b. The results clearlyshow a biphasic effect: below a threshold shear force, anincrease in force actually promotes NC binding by reducingKd, whereas above the threshold, an increase in shear forcedecreases binding by promoting NC detachment. In Fig. 2 b,we also demonstrate that the observation of flow-enhancedbinding is relatively insensitive to the particle size (wealso carried out simulations for two different NC sizes:100 and 200 nm) if we plot the results as a function ofdrag force, and the emergent shear-enhanced bindingbehavior is manifested in a similar fashion.

We note that the flow-enhanced binding described aboveis strictly an equilibrium phenomenon. Even though the flowperturbs thermodynamic equilibrium, we made the tacitassumption that the shear flow is steady and acts as anexternal field on the quiescent system, which from aquasi-equilibrium (or linear-response) perspective justifiesour use of the equilibrium definition of the binding affinityor association constant. The shear-enhanced binding re-ported here is distinct from the shear-enhanced rollingreported by Yago et al. (26); see Fig. 2 c, which shows theexperimentally determined rolling velocities of NCs atvarious tether forces. In particular, the rolling velocityshows a biphasic trend, initially decreasing with increasingtether force, and subsequently, above a threshold tetherforce, the rolling velocity increases with increasing force.

Biophysical Journal 101(2) 319–326

A related phenomenon of shear stabilized rolling behaviorwas also reported by Whitfield et al. (45), as discussedbriefly in the Introduction. The rolling velocity of NC ismediated by the stochastic nature of receptor-ligand bondformation and rupture; hence, the rolling velocity is gov-erned by the underlying free energy landscape of adhesion.In particular, the average slip velocity is governed by theinherent timescale for NC motion, the free-energy barrierfor bond rupture, as well as the length scale over whichthe partially adhered NC pivots before reengaging the fullextent of the multivalent interactions. Hence, the samefree-energy landscape that governs NC binding affinitycan be employed to relate to NC rolling behavior. However,the crucial difference is that the reaction coordinate for NCrolling is, in general, distinct from the reaction coordinate zwe employed to compute PMF. The reaction coordinate forrolling behavior can be quite complex because of the multi-valent interaction, but the x or y coordinate may be a bettersurrogate to resolve the free-energy barriers for rolling thanthe z coordinate we employed to compute the bindingaffinity. Although we did not comprehensively pursue roll-ing in this study, we note that PMF along x or y can indeedresolve free-energy barriers to rolling. From such a free-energy landscape, one can employ kinetic MC simulationsto extract rolling rates and velocities.

To summarize, the NC rolling velocity and NC bindingaffinity are both governed by the underlying free-energylandscape. They are similar in that both are governed by anArrhenius form involving the free-energy landscape.However, the crucial difference is that rolling can be viewedas connecting two bound states of NC, and the bindingaffinity is defined by connecting the bound with the unboundstate of NC. Hence, the underlying reaction coordinates aredistinct. It is still notable that the shear-threshold behaviorwe have computed in Fig. 2 b looks remarkably similar inform to the experimental measurements of rolling velocitiesin Fig. 2 c. In particular, it appears that the shear threshold (re-ported in terms of the drag force) for theNCbinding is similarto the threshold for the rolling behavior. A more thoroughinvestigation of the exact relationship between these distinctphenomena could be a promising avenue for future study.

To further understand the underlying mechanism of thisshear-enhanced binding, we calculate the average receptorflexure angle q and receptor-ligand bond length d (seeFig. S1 for definitions) at the equilibrium position of thebound NC. We plot the results as a function of shear forceon the NC in Fig. 3. Whereas the flexure angle increasesmonotonically with an increasing shear force (Fig. 3 a),the equilibrium bond length (Fig. 3 b) first decreases to aminimum and then increases with increasing shear force.The trend in the binding affinity mirrors the trend in theequilibrium bond length. At equilibrium, there are alwayssome weaker bonds present in multivalent interactions.Our results (Fig. 3) imply that below the threshold valuefor shear, the force and torque generated from flow tend to

z (nm)

mul

itval

ency

132 133 134 1350

2

4

6

8c

z (nm)

mul

itval

ency

132 133 134 1350

2

4

6

8d

d(n

m)

0 50 1000.11

0.12

0.13

0.14

0.15b

θ(o )

0 50 1002

4

6

8a

shear force (pN) shear force (pN)

FIGURE 3 Average receptor bending angle q (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 (Fs ¼ 2.5 pN and 50.5 pN).

Nanocarrier Binding to Endothelial Cells 323

shift the NC conformation to a more stable state by makingthe weaker bonds stronger; above the threshold, the hydro-dynamic force is strong to rupture bonds and reduce themultivalency (see Fig. 3, c and d). We note that our modelalso predicts shear-induced detachment at low ss, consistentwith experimental results (see Fig. S7 and Fig. S8, and asso-ciated Supporting Material). These features clearly demon-strate that the interplay among shear force, multivalentinteractions, and bond compliance induces the shear-enhanced binding in our model.

h (nm)

Ka,

0/K

a,h

50 1000

200

400

600b

h (nm)

Ka,

0/K

a,h

60 800

5

10

15

20

z (nm)

PM

F(k

BT

)

82 83 84 85

-30

-20

-10

0h = 0 nmh = 50 nmh = 60 nmh = 75 nmh = 100 nm

a

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 glycoca-

lyx stiffness. The solid lines correspond to a constant glycocalyx stiffness

kglyx (black, 100 nm; gray, 200 nm NCs), whereas for the dotted line it is

assumed that the glycocalyx stiffness kglyx 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 kglyx does not affect the

exponential dependence.

Effect of the glycocalyx

Estimates from experiments (53–56) indicate that the glyco-calyx layer extends 100 nm to several hundreds of nanome-ters over the cell surface. When the NC approaches the cellsurface, the resistance due to the glycocalyx layer generallyincludes a combination of the osmotic pressure, electrostaticrepulsion, steric repulsion between NC and the glycoproteinchains, and the entropic forces due to conformationalrestrictions imposed on the confined glycoprotein chains,which is too complicated to be accounted for in a model.Therefore, we employ a phenomenological model in whichwe lump all the above effects into a single mechanicalresistance by neglecting the detailed microstructures. Weaccount for the normal resistance of the glycocalyx by add-ing a simple harmonic potential 1/2kglyxH

2 per unit differen-tial area, where H ¼ z – h is the penetration depth of the NCinto the glycocalyx, h is the thickness of the glycocalyx, andkglyx is the glycocalyx stiffness (21). The harmonic nature ofthe glycocalyx potential has its basis in polymer physics,where the entropic elasticity of polymer chains has beenshown 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 bindingaffinity in the presence of the glycocalyx compared withthat in the absence of the glycocalyx. Following Agrawaland Radhakrishnan (21), and assuming a glycocalyx heightof 100 nm (which can be regarded as the lower bound forglycocalyx thickness in vivo), we model the effect of theglycocalyx on NC binding. The PMF profiles for differentvalues of the glycocalyx thickness h are computed asWðzÞglyx; h ¼ WðzÞglyx;h¼0 þ

R1=2kglyxH

2dA, where A isthe surface area of NC immersed in the glycocalyx. Fig. 4a depicts the PMF profiles of NC binding in the presenceof glycocalyx (of varying thicknesses), based on which thebinding affinities were also computed.

To investigate how the thickness of the glycocalyx influ-ences binding in the absence of flow, we report Ka, 0/Ka, h

(i.e., the ratio of the NC binding affinity association constantwithout glycocalyx to that with glycocalyx) as a function ofglycocalyx thickness h in Fig. 4 b. It is evident that thedependence on h is exponential, implying that whereas a100-nm-thick glycocalyx can affect binding by a factor of500, reducing the thickness even down to the ~55–70 nmrange only lowers binding by twofold. Hence, the exponen-tial dependence of Ka, 0/Ka, h versus h could rationalize thelarge differences between in vitro and in vivo experiments ofglycocalyx-mediated NC binding, namely, that bindingdecreases by a modest twofold in vitro (see Fig. S6) andby several hundred-fold in vivo (17). Such a drastic differ-ence can arise even if the thickness in the glycocalyxchanges by 30–50 nm between in vitro and in vivo condi-tions. Fig. 4 b also depicts a similar exponential dependenceof Ka, 0/Ka, h on h for NCs of diameter 200 nm. We note thatwe have not presented modeling results for the case in whichglycocalyx and shear flow are present simultaneously. Sucha situation would be a better representation of the hydrody-namic conditions in vivo. However, we can extend our

Biophysical Journal 101(2) 319–326

324 Liu et al.

model to adapt to this situation by following the procedureoutlined in Section S8 to resolve the flow inside the glyco-calyx layer. This avenue will be pursued in future studies.

DISCUSSION

Using a methodology to compute the absolute free energiesof functionalized NC binding to EC, we explored the effectsof the NC particle size, shear flow, and glycocalyx on NCbinding. Although the results of our earlier model (withflow absent) are in excellent agreement with in vitro,in vivo, and AFM experiments (20), we have now demon-strated that our model also predicts results that are in excel-lent agreement with experiments in a variety of scenariosthat involve flow. We show that increasing particle sizeonly moderately enhances the binding affinities despiteshowing an increase in multivalent interactions. This isbecause an increase in enthalpic interactions with increasingmultivalency is countered by a significant decrease in thetranslational entropy of bound receptors. The entropic con-tributions collectively sum to a significant free-energypenalty under large multivalency. By examining the PMFsat different shear rates, we observe a nonmonotonic trendfor binding as a function of shear force, with the dissociationconstant showing a characteristic minimum at a criticalshear force. This shear-enhanced binding behavior arisesin our model solely because of the interplay among multiva-lency, shear flow, and bond compliance, and their collectiveinfluence on the free-energy landscape. Below the thresholdshear value, the flow enhances binding by lowering theabsolute free energy of the bound state. A decrease in theenthalpy of receptor-ligand bonds for a given multivalencyresults from a reduction in the average bond length d towardthe optimal (equilibrium) value; above the thresholdshear, the binding is reduced due to a decrease in multiva-lency. The shear-enhanced binding behavior is differentfrom shear-enhanced rolling of NCs. Yet, our calculatedtrend of binding affinity versus shear force mirrors theexperimentally measured trend of rolling velocity versusshear force (26). Our results suggest that mechanisms otherthan those involving catch bonds may also (under certainconditions) lead to shear-enhanced binding. Finally, weshow that the glycocalyx can effectively reduce the bindingbetween NC and cell surface. However, the effect ofincreasing glycocalyx thickness h on the association con-stant (Ka, h) of NC adhesion is exponential, suggestingthat an orders-of-magnitude decrease in binding affinity isonly achieved when the glycocalyx thickness h is largeenough (for h ~100 nm, the decrease in binding affinity is500-fold for a 100 nm NC). Thus, even for glycocalyx thick-ness h in the range of 60 nm (or 2/3 of the typical 100 nmvalue), the decrease in binding affinity is only a few-foldfor 100 nm NCs. Hence, our model can rationalize the con-trasting behavior of NC binding mediated by the glycocalyxas observed in in vitro and in vivo experiments.

Biophysical Journal 101(2) 319–326

Several predictions arise from our study, which provideexciting opportunities for future experiments: 1) We haveshown that the shear-enhanced binding behavior does notnecessarily need to be an attribute of a specialized proteinundergoing conformational change when subjected to shear;rather, it could be an emergent property of multivalency,leading to the prediction that a variety of receptor-ligandcomplexes can sustain this behavior amid multivalent inter-actions. 2) Given that bond stiffness (compliance) is animportant factor, point mutations in the ligand that alter therupture force but not the binding affinity will have a signifi-cant effect on flow-enhanced binding. Similarly, engineeringmultivalency in functionalized NC with the use of soft poly-meric tethers (rather than attaching the antibodies directly tothe carrier) will produce a dramatically different behavior inshear-enhanced binding that does not involve catch bonds. 3)Themultivalent interactions that are central to our model canbe directly imaged through super-resolution imaging exper-iments. 4) The nonlinear effect of glycocalyx thickness onbinding landscapes predicted by our model can also bedirectly tested in vitro on grafted polymer surfaces. Themodel predictions can also be exploited for rational reengin-eering of functionalized NCs for targeted drug delivery.

SUPPORTING MATERIAL

Eight sections, nine figures, and references are available at http://www.

biophysj.org/biophysj/supplemental/S0006-3495(11)00668-0.

Note added in proof: We were notified of a relevant experimental paper (58)

that supports our proposed prediction in this article, namely that nonmono-

tonic behavior in binding affinity as a function of shear rate can provide

strong evidence of shear enhanced binding.

We thank Drs. D. A. Hammer, V. Muzykantov, T. N. Swaminathan, and B.

Uma for helpful discussions.

This work was supported by the National Institutes of Health (grants R01-

EB006818 and R01-HL087036) and the National Science Foundation

(grant CBET-0853389). Computational resources were provided in part

by the National Partnership for Advanced Computational Infrastructure

under grant No. MCB060006.

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