Mechanical Model for Catch-Bond-Mediated Cell Adhesion in Shear Flow

International Journal of Molecular Sciences

Article Mechanical Model for Catch-Bond-Mediated Cell Adhesion in Shear Flow

Long Li 1,2,* , Wei Kang 1 and Jizeng Wang 1,*

1 Key Laboratory of Mechanics on Disaster and Environment in Western China, Ministry of Education, College of Civil Engineering and Mechanics, Lanzhou University, Lanzhou 730000, China; [email protected] 2 PULS Group, Institute for Theoretical Physics, FAU Erlangen-Nürnberg, 91058 Erlangen, Germany * Correspondence: [email protected] (L.L.); [email protected] (J.W.)

Received: 5 December 2019; Accepted: 13 January 2020; Published: 16 January 2020

Abstract: Catch bond, whose lifetime increases with applied tensile force, can often mediate rolling adhesion of cells in a hydrodynamic environment. However, the mechanical mechanism governing the kinetics of rolling adhesion of cells through catch-bond under shear flow is not yet clear. In this study, a mechanical model is proposed for catch-bond-mediated cell adhesion in shear flow. The stochastic reaction of bond formation and dissociation is described as a Markovian process, whereas the dynamic motion of cells follows classical analytical mechanics. The steady state of cells significantly depends on the shear rate of flow. The upper and lower critical shear rates required for cell detachment and attachment are extracted, respectively. When the shear rate increases from the lower threshold to the upper threshold, cell rolling became slower and more regular, implying the flow-enhanced adhesion phenomenon. Our results suggest that this flow-enhanced stability of rolling adhesion is attributed to the competition between stochastic reactions of bonds and dynamics of cell rolling, instead of force lengthening the lifetime of catch bonds, thereby challenging the current view in understanding the mechanism behind this flow-enhanced adhesion phenomenon. Moreover, the loading history of flow defining bistability of cell adhesion in shear flow is predicted. These theoretical predictions are verified by Monte Carlo simulations and are related to the experimental observations reported in literature.

Keywords: catch bond; rolling adhesion; Markovian process; shear rate; ﬂow-enhanced adhesion; bistability

1. Introduction Cell adhering to one another or their extracellular matrix under hydrodynamic flow is a crucial issue in many basic physiological and pathological processes, such as immune response [1,2], tumor metastasis [3], and targeted delivery of therapeutics to tissues [4]. For example, in leukocyte extravasation, the circulating leukocytes are recruited to the damaged or inflamed tissues with blood flow. In this case, the successful arrest and attachment of rolling leukocytes to vascular endothelium require interaction between membrane receptors and their ligands on the endothelial surface to sufficiently withstand the applied force from blood flow. Referring to this ability of specific recognition of rolling leukocytes, artificial nanocarriers with adhesive properties of leukocytes can be treated as potential platforms for targeted drug delivery in blood circulation [5]. Therefore, fully understanding the rolling adhesion of cells in a fluid environment will contribute greatly to fundamental biological processes and biomedical applications. Cell adhesion is essentially mediated by the binding of receptors and ligand molecules. In principle, the formation and dissociation of receptor-ligand bonds are reversible and involve stochastic chemical reactions. Particularly, for traditional receptor-ligand bonds (slip bonds), the lifetime decreases

Int. J. Mol. Sci. 2020, 21, 584; doi:10.3390/ijms21020584 www.mdpi.com/journal/ijms Int. J. Mol. Sci. 2019, 20, x FOR PEER REVIEW 2 of 16 Int. J. Mol. Sci. 2020, 21, 584 2 of 16 lifetime decreases with tensile force applied to the bond [6]. To explain the mechanism of this bond dissociation process under loading force, the theoretical framework describing the kinetic reactions with tensile force applied to the bond [6]. To explain the mechanism of this bond dissociation between receptors and ligands was initially established by Bell [7,8]. Nevertheless, there is a process under loading force, the theoretical framework describing the kinetic reactions between counterintuitive observation that one type of bonds (so-called catch bonds) can live longer at higher receptors and ligands was initially established by Bell [7,8]. Nevertheless, there is a counterintuitive tensile force [9,10]. Subsequently, two-pathway models of catch bond have been proposed to describe observation that one type of bonds (so-called catch bonds) can live longer at higher tensile force [9,10]. the kinetic reaction of catch-bond dissociation [11–13]. In addition to the chemical reactions of Subsequently, two-pathway models of catch bond have been proposed to describe the kinetic receptor–ligand bonds, the effect of other physical and mechanical factors, such as loading condition reaction of catch-bond dissociation [11–13]. In addition to the chemical reactions of receptor–ligand [14–17], deformation of substrate [18–25], membrane fluctuation [26–29], multiple bonds [30,31], bonds, the effect of other physical and mechanical factors, such as loading condition [14–17], cluster size [32], and cytoskeletal contraction [33,34] on cell or membrane adhesion have been deformation of substrate [18–25], membrane fluctuation [26–29], multiple bonds [30,31], cluster size [32], discussed extensively. However, these studies focus on the occurrence of adhesion in a relatively and cytoskeletal contraction [33,34] on cell or membrane adhesion have been discussed extensively. quiescent environment. However, these studies focus on the occurrence of adhesion in a relatively quiescent environment. Although plenty of experimental and theoretical efforts [35–45] have been conducted to Although plenty of experimental and theoretical efforts [35–45] have been conducted to elucidate elucidate the kinetic and stochastic behaviors of rolling adhesion via slip bonds in response to stimuli the kinetic and stochastic behaviors of rolling adhesion via slip bonds in response to stimuli from shear from shear flow, relatively few studies have investigated catch-bond-mediated adhesion dynamics flow, relatively few studies have investigated catch-bond-mediated adhesion dynamics under shear under shear flow. The adhesion of leukocytes through selectin–ligand bond (a typical kind of catch flow. The adhesion of leukocytes through selectin–ligand bond (a typical kind of catch bond) has been bond) has been experimentally proposed to exhibit the shear threshold effect, in which increasing experimentallylevels of shear rate proposed stabilize to exhibitleukocyte the rolling shear threshold under flow eff ect,until in the which threshold increasing of shear levels rate of shear[9,46–48]. rate stabilizeA number leukocyte of adhesive rolling dynamics under flow simulations until the [49–51] threshold have of suggested shear rate [that9,46 –the48 ].catch A number behavior of adhesiveof bonds dynamicsmay result simulations in this shear [49–51 threshold] have suggested phenomenon. that the catchHowever, behavior this of suggestion bonds may resultwas not in this directly shear thresholdconfirmed. phenomenon. A theoretical model However, that this can suggestionextract the wasdynamics not directly of cell confirmed.rolling along A substrate theoretical via model catch thatbonds can in extract shear the flow dynamics and examine of cell rolling the mechanical along substrate mechanism via catch underlying bonds in shear such flow shear and threshold examine thephenomenon mechanical is mechanismneeded. In the underlying present paper, such shear we extend threshold our previous phenomenon adhesion is needed. model In to the establish present a paper,mechanical we extend description our previous based adhesionon energy model conservation to establish in acell mechanical adhesion descriptionby taking into based account on energy the conservationstochastic reaction in cell of adhesion catch bonds by taking as well into as accounthydrodynamic the stochastic impact reactionfrom shear of catchflow. bonds as well as hydrodynamic impact from shear flow. 2. Theoretical Model 2. Theoretical Model Figure 1 schematically illustrates a rolling cell in adhesive interaction with an elastic substrate Figure1 schematically illustrates a rolling cell in adhesive interaction with an elastic substrate via catch-bond cluster under laminar flow of shear rate . To describe the problem conveniently, we via catch-bond cluster under laminar flow of shear rate γ. To describe the problem conveniently, we treat the cell as a rigid circular cylinder of radius R (two-dimensional problem). Parameters a and b treat the cell as a rigid circular cylinder of radius R (two-dimensional problem). Parameters a and represent the distances from the right and left contact edges to the apex of contact, respectively. The b represent the distances from the right and left contact edges to the apex of contact, respectively. catch bonds are represented by thermalized harmonic springs with elastic stiffness kLR and rest The catch bonds are represented by thermalized harmonic springs with elastic stiffness kLR and rest length l . The flow shear stress imposes a propulsive force F and a driving torque M to the rolling length lb.b The flow shear stress imposes a propulsive force F and a driving torque M to the rolling cell. Thecell. cellTheis cell then is then dragged dragged by the by tether the tether to substrate to substrate through through bonds bonds due due to trailing to trailing edge edge separation. separation.

Figure 1. Schematic of rolling adhesion of cell to substrate under shear ﬂow.flow.

Int. J. Mol. Sci. 2020, 21, 584 3 of 16

2.1. Lifetime of Catch Bonds We ﬁrst consider the adhesion of cell and substrate under interface separation to examine the lifetime of catch bonds near the trailing edge. Hence, the stochastic process of the adhesion cluster of catch bond can be assumed as a Markov process, as governed by the master equation [52]:

dpn = (n + 1)koﬀpn+1 + [Nt (n 1)]konpn 1 [nkoﬀ + (Nt n)kon]pn, (1) dt − − − − − where pn is the probability that n bonds are closed at time t. The total number of bonds near the trailing edge is denoted as Nt. Considering the dissociation process of a single catch bond as a Brownian virtual particle escaping out of the potential well via two alternative pathways, as shown in Figure2, Evans et al. [11] proposed a two-pathway model to determine the dissociation rate of a catch bond by

c h s i Φ0k + exp( f / fcs) k exp f / fβ koﬀ = , (2) Φ0 + exp( f / fcs) where kc and ks are the spontaneous dissociation rates in the absence of a loading force f through the catch and slip pathways, respectively; and fβ k T/∆xs with the width from the bottom to the slip ≡ B barrier of the potential well, ∆xs, and fcs k T/∆xcs with the interval between two barriers, ∆xcs, are ≡ B the two characteristic force scales, respectively. In an unstressed condition, Φ = exp[(∆Es ∆Ec)/k T] 0 − B is the equilibrium constant between the two pathways by Boltzmann distribution with two related energy barriers, ∆Es and ∆Ec. For typical catch bonds, the catch barrier ∆Ec is lower than the slip barrier ∆Es. However, in the limit of ∆Ec ∆Es, Equation (2) can completely be reduced to the Bell rate that describes a slip bond breaking [7,8], indicating the rupture of bond only through the slip pathway. Given the intersurface separation between cell and substrate, δ, the loading force experienced by each closed catch bond, f, can also be expressed as f = k (δ l l ). The related total force, ft, has a LR − b − bind = PNt form of ft f n=0 npn. The sum term in the above equation is the average number of closed bonds. In Equation (1), the association rate, kon, of bonds can be given by [53]

0  2  kon  kLR(δ lb lbind)  kon = exp − −  (3) Z − 2kBT 

0 where lbind is the reaction radius around the binding site, kon is the spontaneous association rate, and the partition function Z can be determined by [19] r   r   r  1 πkBT   kLR   kLR  Z = erf(δ lb)  + erflb  (4) lbind 2kLR   − 2kBT   2kBT 

With the error function, erf ( ). · The lifetime of catch-bond cluster can be defined by an average timescale to reach rupture of all bonds, starting from a cluster consisting of Nt bonds. Therefore, we consider such stochastic process of bond cluster as a transition state problem, as illustrated in Figure3. We can then solve the master equation, Equation (1), by using conventional numerical methods under reflecting and absorbing boundary conditions at the states of n = 0 and n = Nt, respectively, as shown in Figure3. Subsequently, on the basis of the concept of mean first passage time [54], the probability density of lifetime of catch bonds, P(t), evolving with time can be obtained by

Nt 1 X− dpn(t) P(t) = (5) − dt n=0 Int.Int. J.J. Mol.Mol. Sci.Sci. 2019,, 20,, xx FORFOR PEERPEER REVIEWREVIEW 4 4 of of 16 16

Int. J. Mol. Sci. 2020, 21, 584 4 of 16 − () N t−11 dpn () t Pt()=− t n (5) Pt()=−n=0 (5) n=0 dt The mean lifetime, τ, can hence be expressed by τ The mean lifetime, τ ,, cancan hencehence bebe expressedexpressed byby Z∞ ∞ ∞ ττ== PttdtP()(t)tdt.. (6) (6) = 0 Pttdt() . (6) 0 0

AfterAfter addressingaddressing the the lifetime lifetime of catch-bondof catch-bond cluster cluster near thenear trailing the trailing edge, the edge, tether the bonds tether inducing bonds inducinginducingeﬀective works effectiveeffective of adhesionworksworks ofof adhesion willadhesion be predicted willwill bebe predictedpredicted in the following inin thethe followingfollowing subsection. subsection.subsection.

FigureFigure 2.2. SchematicSchematic of of conceptual conceptual energy energy landscape landscape of two-pathway of two-pathway model model [11,12 [11,12].]. A Brownian A Brownian virtual virtualparticle particle can either can escape either over escape the catchover barrierthe catch or overbarrier the or slip over barrier, the slip starting barrier, from starting the bottom from of the the bottompotential of well. the potential well.

Figure 3. Kinetic network describing the evolution of the number of closed bonds. Figure 3. Kinetic network describing thethe evolutionevolution ofof thethe numbernumber ofof closedclosed bonds.bonds. 2.2. Effective Works of Adhesion for Cell Adhesion 2.2. Effective Works of Adhesion for Cell Adhesion 2.2. EffectiveAs the cellWorks rolls of Adhesion along the for surface Cell Adhesion of the substrate, the works of adhesion via bonds resist to its rollingAs and the shouldcell rolls be along consumed the surfac by thee shearof the flow.substrate, These the works works have of significantadhesion via relationships bonds resist with to theits rollingevolution and of should closed be bonds. consumed Specifically, by the withshear the flow rolling.. TheseThese of theworksworks cell, havehave the catch significantsignificant bonds relationshipsrelationships would be initially withwith thetheformed evolutionevolution near the ofof leadingclosedclosed bonds.bonds. edge, then Specifically,Specifically, maintain withwith stochastic thethe rollingrolling association ofof thethe and cell,cell, dissociation thethe catchcatch bondsbonds reactions wouldwould within bebe initiallyinitiallythe contact formedformed region, nearnear and thethe would leadingleading be edge,edge, totally thenthen broken maintamainta ininin the stochasticstochastic end during associationassociation trailing edge andand dissociationdissociation separation. reactionsreactions withinAfter the averagingcontact region, the master and woul equationd be (Equationtotally broken (1)) over in the time end [55 du],ring the Bell trailing equation edge [ 7separation.,8] governing the evolutionAfter averaging of the average the master density equation of closed (Equation bond, ξ, can(1)) be over yielded time as [55], the Bell equation [7,8] ξ governing the evolution of the average density of closed bond, ξ ,, cancan bebe yieldedyielded asas dξ ξ = konξall (kon + koff)ξ (7) ddtξ =−+ξξ−() =−+kkkonξξ all() on off (7) dt on all on off (7) With the density of total bonds, ξdtall. When the cell surface approachesξ to the substrate in the vicinity outside the leading edge, the With the density of total bonds, ξ all . With the density of total bonds, all . a intersurface separation is assumed to evolve through δ(t) = δ0 vct. Here, vc denotes the linear When the cell surface approaches to the substrate in the vicinity− R outside the leading edge, the speed of the cell center during rolling, and the initial separation betweena two approaching surfaces, intersurface separation is assumed to evolve through δδ()=−a . Here, v denotes the linear intersurfaceδ0, meets δ0 separationlb. Once is the assumed closed bond to evolve density through near the δδ() leadingtvttvt=−0c edge,. ξHere,LD(t) ,v hasc denotes been obtained the linear by 0cR c solving the governing equation, Equation (7), under initial condition, ξLD(0)= 0, the adhesive traction, speed of the cell center during rolling, and the initialinitial separationseparation betweenbetween twotwo approachingapproaching surfaces,surfaces, referringδ toδ the mean force per unit contacting area, can be given by ξ () δ0,, meetsmeets δ0bll .. OnceOnce thethe closedclosed bondbond densitydensity nearnear thethe leadingleading edge,edge, ξ LD ()tt ,, hashas beenbeen obtainedobtained 0 0b LD a ξ by solving the governing equation,σLD(t) =EquationkLRξLD ((7),t) δ under0 v cinitialt lb condition,lbind ξ ()0=0, the adhesive(8) by solving the governing equation, Equation (7), under− R initial− −condition, LD 0=0, the adhesive traction,traction, referringreferring toto thethe meanmean forceforce perper unitunit contactingcontacting area,area, cancan bebe givengiven byby

Int. J. Mol. Sci. 2020, 21, 584 5 of 16

The last right bracket term stands for the deformation of each closed bond. By considering a/R 0.05 [35], the relevant advancing work of adhesion, wa, can be determined by ≈

Z ts wa 0.05vc σLD(t)dt (9) ≈ 0

In Equation (9), ts represents the time that the intersurface separation starts from initial δ0 to the value at the leading edge. Considering the fact that the bonds are at their unstressed length in the contact region, we can obtain ts = (δ l l )/(0.05vc). 0 − b − bind During trailing edge separation with speed of vcb/R (in the normal direction), the elastic deformations of elongated bonds also contribute to the works of adhesion between cell and substrate. Taking t = 0 as the moment of time when cell and substrate surfaces near the trailing edge start to b separate, the associated separation can be given as a function of time, δ(t) = lb + lbind + R vct. Here, we assume b/R 0.05. The closed bond density as a function of time near the trailing edge, ξ (t), ≈ TR can be obtained by solving the governing equation (Equation (7)) under the initial condition [35]:

Here, the initial value ξTR(0) accounts for the evolution of closed bond density from the onset of the attachment near the leading edge to the start of the detachment at the trailing edge (see our previous work [35] for details). The adhesive force per unit area and the eﬀective receding work of adhesion can be deﬁned by

Z tc σTR(t) 0.05vctkLRξTR(t) and wr 0.05vc σTR(t)dt (11) ≈ ≈ 0 where tc means the cutoﬀ time when most (99.9999% considered here) bonds are broken. As the cell velocity, vc, tends to zero, the association and dissociation reactions of bonds reach equilibrium in QS QS every moment, and hence there is wr = wa = w with the value, w , in a quasi-static process. The eﬀective advancing and receding works of adhesion can be numerically calculated from Equations (9) and (11). On the basis of the calculated works of adhesion, the kinetics of cell rolling will be addressed by taking into account energy conservation in cell rolling.

2.3. Kinetics of Cell Rolling in Shear Flow According to the basic fluid mechanics [56], the propulsive force and driving torque per unit thickness imposed on a stationary rigid cylinder are 4πηRγ and 2πηR2γ, respectively, with the viscosity of flow, η. Here, following our previous work [35], we assume the hydrodynamic loadings exerted on a rolling cell as 2 F(t) = 4πηR[γ vc(t)/R] and M(t) = 2πηR [γ vc(t)/R] (12) − − The work done by the shear flow from time 0 to t can be readily expressed as

Z t Z t U1(t) = F(t)vc(t)dt + M(t)ω(t)dt (13) 0 0 where ω(t) = vc(t)/R is the angular velocity of rolling cell, as shown in Figure1. Additionally, the kinetic energy of cell is expressed by

1 U (t) = J ω2(t) (14) 2 2 0 Int. J. Mol. Sci. 2020, 21, 584 6 of 16

2 where J0 = 3mR /2 denotes the rotational inertia of the cell concerning the apex of contact and m = πρR2 is the cell mass with its material density, ρ. By considering the equilibrium conditions for such an adhesion system, the resistant torque caused by adhesion with the substrate can be expressed by [57]

Mr(t) = R[wr(t) wa(t)] (15) − The energy dissipation due to the hysteresis of cell adhesion can thus be given by

Z t U3(t) = Mr(t)ω(t)dt (16) 0

According to energy conservation, a rolling cell with initial kinetic energy, U2(0), should have

U1(t) + U2(0) = U2(t) + U3(t) (17)

After substituting Equations (12–16) into Equation (17), taking the derivative with respect to time yields J0 dvc(t) = M(t) + F(t)R [wr(t) wa(t)]R (18) R dt − − Equation (18) indeed describes the torque equilibrium of cell. The left term is the inertia moment. The right-hand side stands for the resultant torque applied on the cell. Inserting Equation (12) into Equation (18) by normalization manipulations gives the dynamic governing equation of cell rolling as dvc t h i 2h i = 4π γ vc t wr t wa t (19) dt − − 3 − ηvc(t) ηt ηR wr wa With vc t , t , γ γ, wr , and wa . For a given initial cell speed, this ≡ wQS ≡ m ≡ wQS ≡ wQS ≡ wQS governing equation can be numerically solved by the classic Runge-Kutta method. Hereafter, we take the representative values of system parameters as shown in Table1.

Table 1. Model parameters.

Quantity Symbol Value Reference Cell radius R 4.25 µm [58,59] Cell density ρ 1.1 g/cm3 [60] Flow viscosity η 0.001 Pa s [58] ·2 Density of total bonds ξall 140 µm− [7,58] Rest length of bond lb 11 nm [19] Reacting radius lbind 1 nm [61] Bond stiﬀness kLR 0.5 pN/nm [62] 0 1 Spontaneous association rate kon 25 s− [45,63] s 1 Spontaneous dissociation rate of slip barrier k 1 s− [61] c 1 Spontaneous dissociation rate of catch barrier k 15 s− [64]

3. Results and Discussion

3.1. Lifetime of Catch-Bond Cluster To examine the catch-bond behavior, we begin by considering the relationship between lifetime of bond cluster and interfacial force during interface separation. Figure4 plots the predicted probability distribution of cluster lifetime as a function of time under different interfacial forces for catch and slip bonds. As time goes on, the probability distribution of lifetime increases first, then decreases, and finally tends to zero for both catch and slip bonds. Likewise, the relevant Monte Carlo simulations Int. J. Mol. Sci. 2019, 20, x FOR PEER REVIEW 7 of 16

Int. J. Mol. Sci. 2019, 20, x FOR PEER REVIEW 7 of 16 3. Results and Discussions 3. Results and Discussions 3.1. Lifetime of Catch-Bond Cluster 3.1. LifetimeTo examine of Catch-Bond the catch-bond Cluster behavior, we begin by considering the relationship between lifetime of bond cluster and interfacial force during interface separation. Figure 4 plots the predicted To examine the catch-bond behavior, we begin by considering the relationship between lifetime probability distribution of cluster lifetime as a function of time under different interfacial forces for of bond cluster and interfacial force during interface separation. Figure 4 plots the predicted catch and slip bonds. As time goes on, the probability distribution of lifetime increases first, then probability distribution of cluster lifetime as a function of time under different interfacial forces for Int.decreases, J. Mol. Sci. and2020, finally21, 584 tends to zero for both catch and slip bonds. Likewise, the relevant Monte7 Carlo of 16 catch and slip bonds. As time goes on, the probability distribution of lifetime increases first, then simulations (see Appendix) are performed to validate the theoretical model. Comparison with our decreases, and finally tends to zero for both catch and slip bonds. Likewise, the relevant Monte Carlo numerical results from Equation (5) shows excellent agreement (no fitting parameters). (seesimulations Appendix (seeA) areAppendix) performed are toperformed validate the to validat theoreticale the model. theoretical Comparison model. Comparison with our numerical with our resultsnumerical from results Equation from (5) Equation shows excellent (5) shows agreement excellent (no agreement ﬁtting parameters). (no fitting parameters).

Figure 4. Probability density of lifetime of (a) catch-bond and (b) slip-bond clusters as a function time for =10 and different interfacial forces (lines: theoretical predictions from Equation (5); symbols: FigureFigure 4. 4.Probability Probability density density of of lifetime lifetime of of ( a()a) catch-bond catch-bond and and (b (b)) slip-bond slip-bond clusters clusters as as a a function function time time forMonteN = Carlo10 and simulations). diﬀerent interfacial forces (lines: theoretical predictions from Equation (5); symbols: for t =10 and different interfacial forces (lines: theoretical predictions from Equation (5); symbols: MonteMonte Carlo Carlo simulations). simulations). The evolution of mean lifetime of bond clusters with the change of interfacial forces for catch andThe slip evolution bonds is of plotted mean lifetimein Figure of bond5. The clusters mean withlifetime the changeof catch-bond of interfacial cluster forces increases for catch up andto a The evolution of mean lifetime of bond clusters with the change of interfacial forces for catch slipmaximum bonds is before plotted decreasing in Figure5 with. The tensile mean lifetimeforce exerte of catch-bondd on the bonds. cluster This increases finding up indicates to a maximum that the and slip bonds is plotted in Figure 5. The mean lifetime of catch-bond cluster increases up to a beforecatch decreasingbonds are initially with tensile strengthened force exerted by an on increase the bonds. in force. This ﬁnding As expected, indicates the that mean the lifetime catch bonds of cluster are maximum before decreasing with tensile force exerted on the bonds. This finding indicates that the initiallymonotonously strengthened decreases by an and increase eventually in force. diminishes As expected, with respect the mean to the lifetime force of for cluster slip bonds. monotonously However, catch bonds are initially strengthened by an increase in force. As expected, the mean lifetime of cluster decreasesalthough andno catch eventually behavior diminishes is observed with at respectall for s tolipthe bonds, force the for slip slip bonds bonds. have However, much longer although lifetime no monotonously decreases and eventually diminishes with respect to the force for slip bonds. However, than that of catch bonds when the interfacial force is lower than 5 pN. As the slip barrier is stronger catchalthough behavior no catch is observed behavior at is all observed for slip bonds,at all for the slip slip bonds, bonds the have slip much bonds longer have lifetimemuch longer than thatlifetime of than the catch barrier, the slip bonds are more stable under low force. This distinction of force- catchthan bondsthat of when catch the bonds interfacial when the force interfacial is lower thanforce 5 is pN. lower As thethan slip 5 pN. barrier As the is stronger slip barrier than is the stronger catch dependent lifetime of single catch and slip bond is also observed in previous single-molecule force barrier,than the the catch slip bonds barrier, are the more slip stable bonds under are lowmore force. stable This under distinction low force. of force-dependent This distinction lifetime of force- of measurements [65]. singledependent catch andlifetime slip bondof single is also catch observed and slip in bond previous is also single-molecule observed in previous force measurements single-molecule [65]. force measurements [65].

Figure 5. Mean lifetime of bond clusters as a function of interfacial force with N = 10 for (a) catch Figure 5. Mean lifetime of bond clusters as a function of interfacial force with t =10 for (a) catch andand (b ()b slip) slip bonds. bonds. The The theoretical theoretical results results (solid (solid curves) curves) are are in in good good agreement agreemen witht with the the simulations simulations Figure 5. Mean lifetime of bond clusters as a function of interfacial force with =10 for (a) catch (discrete(discrete symbols). symbols). and (b) slip bonds. The theoretical results (solid curves) are in good agreement with the simulations To(discrete understand symbols). the intrinsic mechanism resulting in these two diﬀerent lifetimes between catch and slip bonds, Figure6 shows a comparison of the dissociation rate between the two bonds as a function of the tensile force. There exists a critical tensile force corresponding to a minimum value of dissociation rate for catch bond. A single catch bond seems to maintain the most stable formation under this optimal tensile force. For a low tensile force, thermally induced escape of adhesion molecule from the weak catch barrier can easily make the catch bond break. If the tensile force is large, the force reducing the slip barrier height also destabilizes the catch bond, although the force simultaneously strengthens the catch barrier. Both low and large tensile forces result in short lifetime of catch bond. However, diﬀerent from the catch bond, the dissociation rate of the slip bond monotonously increases Int. J. Mol. Sci. 2019, 20, x FOR PEER REVIEW 8 of 16

To understand the intrinsic mechanism resulting in these two different lifetimes between catch and slip bonds, Figure 6 shows a comparison of the dissociation rate between the two bonds as a function of the tensile force. There exists a critical tensile force corresponding to a minimum value of dissociation rate for catch bond. A single catch bond seems to maintain the most stable formation under this optimal tensile force. For a low tensile force, thermally induced escape of adhesion molecule from the weak catch barrier can easily make the catch bond break. If the tensile force is Int.large, J. Mol. the Sci. force2020, 21reducing, 584 the slip barrier height also destabilizes the catch bond, although the8 offorce 16 simultaneously strengthens the catch barrier. Both low and large tensile forces result in short lifetime of catch bond. However, different from the catch bond, the dissociation rate of the slip bond asmonotonously the tensile force increases increases as the owing tensile to the force gradually increases decreased owing to slip the barriergradually height decreased by force, slip thereby barrier leadingheight toby a force, force-reduced thereby leading lifetime. to a force-reduced lifetime.

FigureFigure 6. 6.Dissociation Dissociation rate rate of of catch catch (black (black line line) and) and slip slip (red (red cycles cycles) bonds) bonds versus versus tensile tensile force. force. The red The cyclesred cycles represent represent the theoretically the theoretically predicted predicted results re forsults slip for bond slip frombond thefrom Bell the model Bell model [7]. [7].

OnceOnce the the force force increasing increasing the the catch catch bonds’ bonds’ lifetime lifetime (a (a characteristic characteristic feature feature of of catch catch bonds) bonds) has has beenbeen investigated, investigated, the the works works of of adhesion adhesion via via bonds bonds resisting resisting to to the the cell cell motion motion in in shear shear flow flow will will be be shownshown in in the the following. following. 3.2. Works of Adhesion 3.2. Works of Adhesion In Figure7, for a cell rolling along the substrate, we plot the variation of normalized closed bond In Figure 7, for a cell rolling along the substrate, we plot the variation of normalized closed bond density near the leading and trailing edges with the change of intersurface separation between the cell density near the leading and trailing edges with the change of intersurface separation between the surface and substrate for different speeds of cell rolling. The closed bond density and adhesion traction cell surface and substrate for different speeds of cell rolling. The closed bond density and adhesion show distinct dependences on the speed of cell rolling. During leading edge approach, the closed bond traction show distinct dependences on the speed of cell rolling. During leading edge approach, the density monotonously increases as the change of separation decreases for vc = 1 and 10 µm/s. This is closed bond density monotonously increases as the change of separation decreases for = because of the domination of bond association in leading edge approach. However, in the limit of 1 and 10 μm/s. This is because of the domination of bond association in leading edge approach. vc 0 (quasi-static condition) with decreasing change of separation, the closed bond density, ξLD, However,→ in the limit of →0 (quasi-static condition) with decreasing change of separation, the increases first, then reaches its maximum, and finally maintains within a relatively high range, as closed bond density, , increases first, then reaches its maximum, and finally maintains within a shown in Figure7a. For the trailing edge separation, Figure7b shows that the closed bond density first relatively high range, as shown in Figure 7a. For the trailing edge separation, Figure 7b shows that increases and then decreases to zero as the separation increases, implying a bond dissociation-induced the closed bond density first increases and then decreases to zero as the separation increases, fracture of cell-substrate interface. implying a bond dissociation-induced fracture of cell-substrate interface.

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FigureFigure 7. 7.Normalized Normalized closed closed bond bond density density and and adhesion adhesion traction traction as as a functiona function of of the the change change of of intersurfaceintersurface separation separation (∆ (δΔ=−−δδ= δ lll l ) for) ( fora) leading (a) leading edge approach edge approach and (b) and trailing (b) trailingedge separation edge −b b −bindbind separationprocesses processes by taking by into taking account into the account different the dispeedsﬀerent of speeds cell rolling. of cell Curves rolling. are Curves theoretical are theoretical results and resultsdiscrete and symbols discrete symbolsstand for standresults for from results Monte from Carlo Monte simulations. Carlo simulations.

InIn addition addition to to closed closed bond bond density, density, Figure Figure7 also 7 also shows shows the the evolution evolution of relatedof related adhesion adhesion traction traction withwith the the change change of intersurfaceof intersurface separation separation occurring occurring near near the the leading leading and and trailing trailing edges. edges. A criticalA critical separationseparation associated associated with with a maximuma maximum value value of of adhesion adhesion traction traction for for both both leading leading approach approach and and trailingtrailing separation separation processes processes can be can seen. be This seen. is aThis result is of a the result competition of the betweencompetition bond between deformation bond anddeformation closed bond and density. closed bond density. AfterAfter integrating integrating the the obtained obtained adhesion adhesion traction traction with with respect respect to to the the change change of of intersurface intersurface µ separation,separation, Figure Figure8 shows 8 shows the the work work of adhesionof adhesion as aas function a function of cellof cell speed speed ranging ranging from from 0 0m µm/s/s to to µ 10001000m µm/s./s. As As the the cell cell speed speed increases, increases, the the advancing advancing work work of of adhesion adhesion induced induced by by leading leading edge edge QS approach, wa, monotonously decreases from an initial value, w , because there is not enough time approach, , monotonously decreases from an initial value, , because there is not enough time forfor bond bond formation formation at at the the high high speed speed of of approaching. approaching. However, However, for for the the receding receding work work of of adhesion, adhesion, wr, Figure8 shows a maximum wr at a critical cell speed of v∗ ∗7.5 µm/s. This optimized work of , Figure 8 shows a maximum at a critical cell speed of ≈v ≈7.5 µm/s. This optimized work of adhesion can be explained by the competition between the stochastic reaction of molecular bonds adhesion can be explained by the competition between the stochastic reaction of molecular bonds and elongation of each closed bond. For a low cell speed, the stochastic reaction of molecular bonds and elongation of each closed bond. For a low cell speed, the stochastic reaction of molecular bonds dominates the evolution of closed bond density. Therefore, the closed bond under a large elongation dominates the evolution of closed bond density. Therefore, the closed bond under a large elongation cannot survive. When the cell speed is high, a small density of closed bond causes a low receding cannot survive. When the cell speed is high, a small density of closed bond causes a low receding work of adhesion as there is not enough time to form bonds. The receding work of adhesion coupling work of adhesion as there is not enough time to form bonds. The receding work of adhesion coupling closed bond density and bonds’ elongation would reach a maximum value at a critical cell speed. closed bond density and bonds’ elongation would reach a maximum value at a critical cell speed.

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FigureFigure 8. 8.Work Work of of adhesionadhesion asas aa function of cell speed. speed. The The inset inset isis a apartial partial enlarged enlarged drawing drawing for for the µ µ v thecell cell speed speed ranging ranging from from 0 0µm/sm/s to to 100 100 µm/s.m/s. WhenWhen c →00, thethe workwork ofof adhesion adhesion in in quasi-static quasi-static QS 2 → conditioncondition can can be be calculated calculated as asw ==0.54 0.54µJ μJ/m/m .. 3.3. Kinetics and Shear Threshold of Cell Rolling 3.3. Kinetics and Shear Threshold of Cell Rolling Once the advancing and receding works of adhesion have been calculated, as shown in Figure8, Once the advancing and receding works of adhesion have been calculated, as shown in Figure the kinetics of cell rolling can be described on the basis of Equation (19). We consider the steady 8, the kinetics of cell rolling can be described on the basis of Equation (19). We consider the steady state of cell rolling in a shear flow by setting dv /dt = 0. Subsequently, Figure9 shows a comparison state of cell rolling in a shear flow by setting c ̅ ⁄̅ =0. Subsequently, Figure 9 shows a comparison between theoretical predictions and experimental results by Yago et al. [46] for steady speed of a cell as between theoretical predictions and experimental results by Yago et al. [46] for steady speed of a cell a function of shear rate in the cell rolling process regulated by catch bonds. Two theoretically predicted as a function of shear rate in the cell rolling process regulated by catch bonds. Two theoretically critical shear rates for detachment and attachment can be seen for cell adhesion. Beyond the upper predicted critical shear rates for detachment and attachment can be seen for cell adhesion. Beyond threshold, the shear flow keeps a cell rolling instead of firm adhesion states. Nevertheless, for the the upper threshold, the shear flow keeps a cell rolling instead of firm adhesion states. Nevertheless, flow with a smaller shear rate than the lower threshold, such rolling adhesion of cell cannot happen. for the flow with a smaller shear rate than the lower threshold, such rolling adhesion of cell cannot Particularly, with the shear rate ranging between these two critical shear rates, the cell is either at firm happen. Particularly, with the shear rate ranging between these two critical shear rates, the cell is or at rolling adhesion that relies on the dynamic shear rate loading path of flow from initial zero shear either at firm or at rolling adhesion that relies on the dynamic shear rate loading path of flow from rate (see AppendixB). This predicted bistable behavior of cell adhesion in shear flow is consistent initial zero shear rate (see Appendix). This predicted bistable behavior of cell adhesion in shear flow with the previous experimental observations on leukocyte, circulating tumor cell, and red blood cell is consistent with the previous experimental observations on leukocyte, circulating tumor cell, and adhesion in shear flow [40,45,66,67]. Meanwhile, as the shear rate increases, the theoretically predicted red blood cell adhesion in shear flow [40,45,66,67]. Meanwhile, as the shear rate increases, the steady cell speed is asymptotically close to the steady speed of cell free rolling, in which there is no theoretically predicted steady cell speed is asymptotically close to the steady speed of cell free rolling, adhesion between cell and substrate. This is because the faster the cell speed, the smaller the number in which there is no adhesion between cell and substrate. This is because the faster the cell speed, the of bonds formed during cell rolling. smaller the number of bonds formed during cell rolling.

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ST FigureFigure 9. 9.Steady Steady speed speed of cell,of cell,vc , as, a as function a function of shear of shear rate rate of flow, of flow,γ. The γ. superscriptsThe superscripts c and c s and in the s in legendthe legend mark themark catch-bond- the catch-bond- and slip-bond-mediated and slip-bond-medi cellated adhesion, cell adhesion, respectively. respectively. The blue The solid blue and solid greenand dash–dotgreen dash–dot lines denote lines thedenote firm the adhesion firm adhesion state where state the where cellexhibits the cell aexhibits slow motion. a slowThe motion. purple The dashpurple and greendash and dot linesgreen represent dot lines the represent rolling adhesion the rolling state adhesion where the state cell where rolls along the cell the substraterolls along at athe relativelysubstrate high at a speed. relatively The high black speed. dash–dot The lineblack means dash–dot the non-adhesion line means the rolling non-adhesion state, where rolling the cellstate, freelywhere rolls the without cell freely adhesion rolls without interaction adhesion between interact cellion and between substrate. cell The and red substrate. circles are The experimental red circles are resultsexperimental of neutrophils results rolling of neutrophils on P-selectin rolling glycoprotein on P-selectin ligand-1 glycoprotein (PSGL-1)-coated ligand-1 substrate (PSGL-1)-coated under shear flowsubstrate by Yago under et al. shear [46], inflow which by Yago the P-selectin–PSGL-1 et al. [46], in which complex the P-selectin–PSGL-1 shows a typical complex catch behavior shows a under typical 1 1 tensilecatch force. behavior The under inset is tensile a partially force. enlarged The inset drawing is a partially for the enlarged shear rate drawing ranging fo fromr the 45shear s− rateto 105 ranging s− andfrom shows 45 s bistable−1 to 105 processess−1 and shows of cell bistable adhesion processes in a shear of cell flow. adhesion in a shear flow.

Furthermore,Furthermore, Figure Figure9 shows 9 shows that that the the cell cell steady steady speed speed decreases decreases as as the the shear shear rate rate increases increases from from lowerlower to to upper upper thresholds, thresholds, indicating indicating the the ability ability of cells of cells to withstand to withstand high high shear shear stress stress better better than low than shearlow stress.shear stress. The cell The rolling cell rolling begins begins to speed to upspeed as long up as as long the unceasingly as the unceasingly increased increased shear rate shear is larger rate is thanlarger the than upper the thresholds. upper thresholds. Such flow-enhanced Such flow-enh cellanced adhesion cell phenomenonadhesion phenomenon has also been has observed also been inobserved experiments in experiments [46,67–69], as[46,67–69], shown in as Figure shown9. Here,in Figure we predict9. Here, the we upper predict threshold the upper of shearthreshold rate of 1 1 forshear detachment rate for detachment as 99.2 s− , as which 99.2 iss−1 close, which to is the close experimentally to the experimentally observed observed value range value from range 70 sfrom− 1 to70 100 s− s1− to[46 100]. The s−1 quantitative[46]. The quantitative differences betweendifferences theoretical between predictions theoretical (2D) predictions and experimental (2D) and observationsexperimental (3D) observations are attributed (3D) toare the attributed distinction to the in distinction 2D and 3D in situations 2D and 3D of situations cell rolling of adhesion, cell rolling especiallyadhesion, for especially large shear for rate.large shear rate. Additionally,Additionally, Figure Figure9 shows 9 shows that that the the cell cell steady steady speed speed corresponding corresponding to theto the upper upper threshold threshold of of ∗ shearshear rate rate is is 8 µ8 mµm/s,/s, which which is is close close to tov ∗,v a, criticala critical cell cell speed speed associated associated with with a maximuma maximum receding receding workwork of of adhesion, adhesion, as as shown shown in in Figure Figure8. This8. This result result suggests suggests that that the the flow-enhanced flow-enhanced cell cell adhesion adhesion phenomenonphenomenon can can be be explained explained by by the the maximum maximum receding receding work work of of adhesion adhesion withstanding withstanding cell cell rolling. rolling. Nevertheless,Nevertheless, extensive extensive eff effortsorts have have been been made made to to treat treat the the catch catch behavior behavior of of bond bond as as the the underlying underlying mechanismmechanism of of this this flow-enhanced flow-enhanced adhesion adhesion phenomenon phenomenon [10 [10,69–71].,69–71]. To To identify identify whether whether the the catch catch behaviorbehavior of bondof bond dominates dominates the flow-enhanced the flow-enhanced cell adhesion cell adhesion phenomenon, phenomenon, we also plot we thealso predicted plot the variationpredicted of steadyvariation cell speedof steady with cell the shearspeed rate with for slip-bond-mediatedthe shear rate for slip-bond-mediated cell rolling adhesion, cell as shown rolling inadhesion, Figure8. The as shown slip-bond-mediated in Figure 8. The cell slip-bond-medi adhesion clearlyated shows cell aadhesion similar flow-enhanced clearly shows cella similar adhesion flow- phenomenon,enhanced cell even adhesion though phenomenon, the slip bond doeseven notthough exhibit the the slip catch bond behavior does not at exhibit all, as shown the catch in Figure behavior5. Thisat all, finding as shown directly in indicatesFigure 5. thatThis thefinding shear directly threshold indicates phenomenon that the of shear cell rolling threshold is regulated phenomenon by the of energycell rolling competition is regulated rather by than the theenergy catch compet behaviorition of rather bond. than the catch behavior of bond.

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4. Conclusions 4. Conclusions We have developed a mechanical model integrating hydrodynamic impact and stochastic processWe haveof formation developed and a mechanicaldissociation model of bonds integrating to investigate hydrodynamic the dynamics impact of and catch-bond-mediated stochastic process ofcell formation adhesion andin shear dissociation flow. Through of bonds energy to investigate conservation, the the dynamics kinetic equation of catch-bond-mediated governing cell rolling cell adhesionunder shear in shear flow flow. is obtained. Through energyResults conservation, showed that thethe kinetic shear equationrate of flow governing significantly cell rolling affects under the shearsteady flow speed is obtained. of cell rolling. Results Irrespective showed that of thethe shearcatch ratebond of behavior flow significantly of force-increased affects the lifetime, steady speedshear- offlow-stabilized cell rolling. Irrespective cell adhesion of the occurred, catch bond which behavior can be ofexplained force-increased by the dynamic lifetime, shear-flow-stabilizedcompetition between cellkinetics adhesion of receptor-ligand occurred, which reactions can be and explained cell rolling. by the Two dynamic critical competition shear rates are between also identified kinetics of to receptor-liganddefine the distinct reactions steady andstates cell of rolling.cell adhesion, Two criticalincluding shear firm, rates rolling, are alsoand identifiedbistable adhesion. to define These the distinctfindings steady not only states help of cellus adhesion,understand including the mechanical firm, rolling, mechanism and bistable of catch-bond-mediated adhesion. These findings rolling notadhesion only help of cells us understand in response theto hydrodynamic mechanical mechanism impact but of also catch-bond-mediated can be useful in designing rolling adhesion targeted oftherapy cells in in response biomedical to hydrodynamic applications. impact but also can be useful in designing targeted therapy in biomedical applications. Author Contributions: L.L. and J.W. contributed to the method design and wrote the manuscript. L.L. and W.K. Authorperformed Contributions: the numericalL.L. calculation and J.W. contributed and data analysis. to the method L.L. and design J.W. andsuperv wroteised the the manuscript. project. All L.L. authors and W.K.have performedread and agreed the numerical to the publis calculationhed version and dataof the analysis. manuscript. L.L. and J.W. supervised the project. All authors have read and agreed to the published version of the manuscript. Funding: This work was supported by the National Natural Science Foundation of China (11602099 and Funding: This work was supported by the National Natural Science Foundation of China (11602099 and 11925204), the11925204), China Scholarship the China CouncilScholarship (CSC, Council File No. (CSC, 201806185038), File No. 201806185038), and the 111 Project and the (B14044). 111 Project The (B14044). APC was The funded APC bywas the funded National by Naturalthe National Science Natural Foundation Science of Foundation China (11925204). of China (11925204). ConflictsConflicts of of Interest: Interest:The The authors authors declare declare no no conflict conflict of of interest. interest.

AppendixAppendix A. A. Monte Monte Carlo Carlo Simulations Simulations TheThe stochasticstochastic processesprocesses ofof adhesionadhesion bondsbonds describeddescribed inin EquationsEquations (1)(1) andand (7)(7) cancan alsoalso bebe numericallynumerically obtainedobtained through through Monte Monte Carlo Carlo simulations. simulations. In light Inlight of the of first the reaction first reaction method method derived derivedfrom the from Gillespie the Gillespie algorithm algorithm [72,73], [72 ,73the], theMont Montee Carlo Carlo simulations simulations can can be be proposed proposed usingusing thethe followingfollowing steps: steps: First, for each binding site, a set of independent random numbers, ς , that are uniformly distributed First, for each binding site, a set of independent random numbers,i , that are uniformly 0 overdistributed [0, 1] is generated.over [0, 1] Then,is generated. the normalized Then, th reactione normalized rate at eachreaction site israteλi =at keachoff/k onsitefor is closed = bonds⁄ n ς o 0 1 i andfor λclosedi = kon bonds/kon for and open = bonds.⁄ The for incremental open bonds. reaction The incremental time can be definedreaction as timedt = can0 bemin definedln λ as. kon − i The = reactionmin site −ln inducing. The reaction the incremental site inducing reaction the incremen time is recorded.tal reaction The time bond is recorded. state at that The site bond is changed from closed to open or from open to closed. Repeating the above procedure yields the state at that site is changed from closed to open or from open to closed. Repeating the above evolution of closed bond density. The mean value of bond number or density can be determined by procedure yields the evolution of closed bond density. The mean value of bond number or density taking the average over 10,000 different simulation trajectories. can be determined by taking the average over 10,000 different simulation trajectories. While calculating the evolution of closed bond density, as shown in Figure7, we take the total While calculating the evolution of closed bond density, as shown in Figure 7, we take2 the total number of reaction sites as 140 corresponding to the total bond density of ξall = 140 µm− . number of reaction sites as 140 corresponding to the total bond density of = 140 μm . Appendix B. Effect of Flow’s Shear Rate History on Steady Speed of Cell Appendix B. Effect of Flow’s Shear Rate History on Steady Speed of Cell

Figure A1. Cont.

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