GLOSSARY

Top Abstract Glossary Introduction Analysis Comparison with experiment Discussion Conclusions Appendix A Appendix B References

a	bond interaction parameter (in nm)
Ac	contact area (in µm2)
Bi	renormalization factor
C	orbit constant (given by Eq. A10)
Cf, Cf	angle factor (= sin21 sin 21), mean value
D	empirical parameter (given by Eq. 19; in µm4 nMq)
fc, fp	two-body collision frequency per particle, capture value (in s1)
F	applied force (Eq. 3)
FN, FS	hydrodynamic normal and tangential forces, respectively (given by Eqs. 1 and 2; in pN)
FN,max, FN,max	maximum normal force in a force cycle, mean value (in pN)
g(x, t), gm(x, t)	probability-generating function, conditioned by assuming there are m bonds initially
G	shear rate (in s1)
h	minimum distance of approach between sphere surfaces (in nm)
Hc, Hp, p	total two-body collision frequency per unit volume, capture value, weighted time average of Hp (in s1 · µl1)
i	number of half-rotations
J(u), Jm(u)	arbitrary integration function, conditioned by assuming there are m bonds initially
kf(n), kf, kfL, kfH, kfM	forward rate coefficient per unit density for the formation of the nth bond, constant value, values derived from fitting data from the low and high shear rate periods as well as combined both periods (in µm2 · s1)
kr, kr(n), kr0	reverse rate coefficient, value for the dissociation of the nth bond, value at zero force (in s1)
kB	Boltzmann constant (= 1.38 × 102 nm · pN · K1)
mr, ml, mmin	number densities of receptors and ligands, minimum value of the two (in µm2)
Mr, Ml, Mb, Mw	molecular species of receptor, ligand and bond, respectively, molecular weight
n, n	number of bonds, mean value at t = t2
N	number of samples or data points
Nf	number of fitting parameters
Ni	number of measurements comprising the ith data point
NS	number density of singlets in particle suspension (in µl1)
pb, pf	probability of bond breakage and formation in time t, respectively (given by Eq. 4)
pn, pn|m	probability of having n bonds (= 0, 1, 2 ...), conditioned by assuming that there are m bonds initially
p+, p	random number obeying a uniform distribution in (0, 1)
P( · )	probability of the argument
Pa	probability of adhesion per two-body collision
q	dimensionless empirical parameter (given by Eq. 19)
r	polar coordinate (in µm)
re	equivalent axis ratio of doublet (= 1.98)
R	sphere radius (in µm)
i	predicted standard error
t, t1, t2, tf	arbitrary time, lifetime of a collision doublet or contact duration of two-body collision, end time of the low shear rate phase, time scale for the formation of an additional bond in a preformed doublet (in s)
T	absolute temperature (in K)
u	argument of arbitrary integration function J(u)
uj	sphere velocity relative to the reference sphere along Xj axis; u1 = u2 = 0, u3 = GX2 (in µm · s1)
x, xi	argument of probability-generating function g(x, t), of function y(xi)
Xj	Cartesian coordinates with origin at the center of the reference sphere in a simple shear field; j = 1 is the vorticity axis, j = 2 is the direction of velocity gradient, and j = 3 is the direction of flow (in µm)
yi, y(xi)	measurement and prediction at xi, respectively

Greek symbols

N, S	force coefficients (Eqs. 1 and 2)
nm	Kronecker delta symbol
,	two-body collision capture efficiency, weighted time average
t	incremental time step
	suspending medium viscosity (in Poise)
1, 1	polar and azimuthal angles of doublet axis with respect to the vorticity axis (X1) as the polar axis
2	polar angle of doublet major axis with respect to X2 axis as the polar axis
i, i, n	standard deviation, predicted value, value of probability distribution of initial bonds
	end point of contact duration at low shear rate (in s)
2, v2	chi square statistic, reduced value (=2/)
	number of degrees of freedom (= N  Nf)
r, l, b	stoichiometric coefficient of receptor, ligand and bond, respectively (given by Eq. 17)

	INTRODUCTION

Top Abstract Glossary Introduction Analysis Comparison with experiment Discussion Conclusions Appendix A Appendix B References

This paper deals with modeling the formation and break-up of receptor-ligand bonds in sheared suspensions, a subject of considerable importance in the circulation, where formation and break-up of blood cell aggregates occur in a variety of physiological and pathological conditions. Thus, nonpathogenic neutrophil aggregation is thought to be important in the vicinity of tissue damage (Hill, 1987). Agonist-induced platelet aggregation plays a key role in the growth of thrombi on vessel walls, and platelets also aggregate with metastatic tumor cells (Honn et al., 1992), which themselves are known to form aggregates (Weiss et al., 1988).

To model the biophysics of cell aggregation in a shear field is challenging because of the coupling between the hydrodynamics of the cell suspension and the chemical kinetics of the receptor-ligand binding, the latter governing the dynamic changes in bond number during the interaction. When two cells first aggregate upon colliding in a shear field, they are likely linked by only a single bond. This bond can either break, leading to break-up of the doublet, or the number of bonds can grow (albeit nonmonotonically) until dissociation equals formation and a dynamic equilibrium bond number is reached. Further changes in the bond number require a change in external conditions (such as hydrodynamic forces) or in the expression and activation of adhesion molecules. Thus, L-selectin shedding and subsequent ₂-integrin activation have been shown to influence neutrophil aggregation in shear flow (Taylor et al., 1996).

The present paper focuses on the effect of hydrodynamic force on the members of an aggregate and hence the bonds holding it together. These forces can be quite complex, even in the simplest of cases. For example, the following expressions have been derived for the normal force (F_N) acting along and the shear force (F_S) acting normal to the major axis of a doublet of rigid spheres (e.g., a doublet of aggregating neutrophils) in simple shear flow (Tha and Goldsmith, 1986):

(1)

(2)

where  is the suspending medium viscosity, G is the shear rate, R is the sphere radius, and ₁ and ₁ are the azimuthal and polar angles describing the orientation of the doublet axis with respect to the vorticity axis X₁ of the shear field (see Fig. 2 below); _N and _S are force coefficients that are weakly dependent on the distance, h, between sphere surfaces (see the Glossary).

How these hydrodynamic forces are distributed among the bonds depends on the distribution of bonds within the contact area. If the cells in the aggregate can flatten and form large contact areas, bonds at the circumference of the contact area will carry most of the stress, whereas those on the inside will be largely unstressed. For small contact areas or low bond densities, however, the situation is different. A small number of bonds (say, three or four) will likely split the force equally, but should one dissociate, those remaining will feel a large increase in the force per bond.

A likely effect of external force on the bonds is to accelerate their dissociation. Twenty years ago, George Bell (1978) formulated such an effect as an exponential force dependence of the reverse rate coefficient:

(3)

where k_r⁰ is the zero-force reverse rate constant, a is the bond interaction parameter, F(t) is the force (for doublet rotation in shear flow, a periodic function of time) shared among n bonds, k_B is the Boltzmann constant, and T is the absolute temperature. Other expressions for the force dependence of kinetic rates have also been proposed (Dembo et al., 1988; Evans et al., 1991). A major effort in the field is to determine the appropriateness of these expressions and the associated parameters (Evans and Ritchie, 1997; Piper et al., 1998); the present paper contributes in this regard.

Theoretical and experimental models are needed to understand aggregation and disaggregation in the presence of such dynamically changing applied forces and bond numbers. In vitro assays are useful in this connection, because it is possible to control shear profile (e.g., using a flow chamber) and biochemistry (e.g., using molecules attached to latex beads). Relevant assays include homotypic neutrophil aggregation in a flow cytometer (Simon et al., 1990), platelet aggregation in tube flow (Bell et al., 1989a,b, 1990), and the break-up of doublets of sphered red cells (Tha et al., 1986; Tees et al., 1993) in Poiseuille flow. More recently, the latter technique was extended to study formation and breakage of doublets of sphered red cells and latex beads in Couette flow in a cone-and-plate rheoscope (Tees et al., 1993; Tees and Goldsmith, 1996; Kwong et al., 1996).

One method for modeling the shear-induced break-up of cell aggregates is to use Monte Carlo simulation to track the number of bonds under stress (Hammer and Apte, 1992; Tees et al., 1993). In this approach, the initial number of bonds, n, cross-linking the cells is randomly chosen from a Poisson distribution with a preset average number of bonds. Time is divided into discrete steps of length t. In each time step, the forces acting on the bonds are calculated. The probability of bond breakage, p_b, is given by (Hammer and Apte, 1992)

(4a)

where Eq. 3 has been used to expand the reverse reaction rate. Furthermore, it can be postulated that the probability of adding a new bond to a doublet is given by

(4b)

where t_f is the time scale for the formation of an additional bond in an existing doublet. A cycle of force calculation, bond formation, and break-up testing is continued until "break-up" for a series of simulated doublets. The resulting time distribution of break-up can be compared to experimental results to determine parameters for models of force dependence of reaction rates. The Monte Carlo simulation was used in the previous studies from one of our laboratories (Tees et al., 1993; Tees and Goldsmith, 1996; Kwong et al., 1996), because it was thought that forces were too complex to be solved using a closed-form computation with a dynamically changing bond number. The present paper demonstrates, however, that it is indeed possible to directly model the time-varying distributions of bonds despite the formidable expressions for the force loading.

This alternative approach is the probabilistic method recently developed by the other of our laboratories (Piper et al., 1998; Piper, 1997; Chesla et al., 1998; Zhu and Chesla, 1997; Long and Zhu, 1997). In contrast to the Monte Carlo approach, which simulates the fate of a series of bonds to generate an ensemble of realizations from which statistics are obtained, the present method solves the corresponding probabilistic variables directly from a set of master equations (McQuarrie, 1963), which describe the binding kinetics of a small number of receptors and ligands:

(5)

Here, p_n is the probability of having n bonds at time t, and m_r and m_l are the respective number densities of receptor and ligand. An assumption employed by Eq. 5, which simplified the original form of McQuarrie's (1963) equations, was that the number of receptors and ligands in the contact area, A_c, greatly exceeds the number of bonds that have nonvanishing probabilities. Consequently, the forward rate coefficient per unit density, k_f⁽ⁿ⁾ (in µm² s¹), appears in Eq. 5 as a lumped per-cell forward rate coefficient, A_cm_rm_lk_f⁽ⁿ⁾ (in s¹). Another modification to McQuarrie's master equations, first introduced by Cozens-Roberts et al. (1990), was to make the reverse rate coefficient, k_r⁽ⁿ⁾, a function of the applied force, F(t), and the number of bonds, n, that shared the force (Eq. 3). Note that k_r in the second term on the right-hand side of Eq. 5 should be different from the one in the last term, as the former was the reverse rate coefficient for the n bond state, whereas the latter was that for the (n + 1) bond state (as indicated by their respective superscripts). In contrast, the forward rate coefficient is assumed to be constant, k_f⁽ⁿ⁾ = k_f, independent of bond number n in the present work. This simplification has been assumed by several groups, including ourselves (Hammer and Lauffenburger, 1987; Tees et al., 1993), although it has not been tested experimentally, and a more general expression may be needed for future work.

McQuarrie's (1963) work has formally established the master equations as a well-founded generalization of the deterministic kinetic equation suitable for small systems (see also Chesla et al., 1998; Zhu et al., 1998). In Appendix B, the Monte Carlo approach is shown to be mathematically equivalent to a finite-difference approximation of the master equations, thereby putting it on an equally rigorous theoretical footing. One incidental, but important, advantage of the master equations is that the computational cost is significantly less than for the Monte Carlo approach. This allowed us to explore the effect of relaxing many of the simplifying assumptions employed in previous work.

In previous work, only doublet break-up at high shear rate was treated by Monte Carlo simulations. In the present paper, by comparison, all three separate time periods in the experiments have been modeled using the same set of master equations. The three time periods are 1) the encounter period of the shear-induced two-body collision during which the first bond may be formed; 2) the low shear rate period (for a given doublet, this period starts after the formation of the first bond) during which the bond number may grow; and 3) the high shear rate period during which the doublet may break up. The analysis of the first phase enabled us to derive an expression for the collision capture efficiency. Previously, this was either assumed a priori as an empirical parameter (Bell et al., 1989a,b, 1990; Huang and Hellums, 1993a-c) or calculated from a deterministic kinetic criterion (Bell, 1981; Tandon and Diamond, 1997). The analysis of the second phase allowed us to introduce the time-averaged probabilities of survival and break-up at low shear rate, both of which can be directly compared to experimental measurements.

Our goal here is to offer a unifying theoretical framework to interpret experiments regardless of the loading regime, receptor system, and assay technique used. Such a framework is a requirement for a full understanding of the biophysics of cell adhesion at the fundamental level.

	ANALYSIS

Top Abstract Glossary Introduction Analysis Comparison with experiment Discussion Conclusions Appendix A Appendix B References

The experiments that were modeled consisted of studies of formation and breakage of doublets. The three types of particles and the two kinds of cross-linking molecules used in the experiments are illustrated in Fig. 1. In both cases of individual as well as population studies, doublets were first allowed to form through collisions of two singlets at low shear rate (~8 s¹) in Couette flow within the rheoscope. They were then subjected to a known higher shear rate (15-145 s¹). In studies of individual break-ups with sphered and swollen red cells (SSRCs) (Tees et al., 1993) carboxyl-modified latexes (CML), and aldehyde/sulfate (A/S) latexes (Tees and Goldsmith, 1996; Kwong et al., 1996), preformed doublets were continuously observed and videotaped until they broke up or were lost from view. In population studies with A/S latexes (Kwong et al., 1996), after being subjected to low shear rate for 30 min, the number of doublets formed per unit volume was counted, and the suspension was then subjected to a high shear rate for known periods of time, after which the number of doublets broken up per unit volume was determined. Accordingly, Eq. 5 was solved for three distinct time periods with appropriate initial and matching conditions to connect the periods.

View larger version (21K): [in this window] [in a new window]   FIGURE 1   Schematic (not to scale) of the three model spherical particles and two cross-linking molecules used in the doublet break-up experiments (Tees et al., 1993; Tees and Goldsmith, 1996; Kwong et al., 1996). (a) Sphered and swollen red cells (SSRCs) expressing blood group B antigen cross-linked by monoclonal anti-B antigen IgM antibody. The estimated contact area (based on an estimated 20-nm separation between the cell surfaces) and site densities of B antigen and IgM antibody are ~0.25 µm2, ~8 × 103 µm2, and ~60 µm2, respectively (Tees et al., 1993). (b) Carboxyl-modified latex (CML) spheres covalently coupled with synthetic blood group B antigen cross-linked by monoclonal anti-B antigen IgM antibody. The estimated contact area and site densities of B antigen and IgM antibody are ~0.25 µm2, ~4 × 105 µm2, and ~34 µm2, respectively (Tees and Goldsmith, 1996). (c) Aldehyde/sulfate (A/S) latex spheres covalently coupled with monoclonal IgG antibody cross-linked by 0.9 nM divalent Gamma Bind G (a recombinant fragment of protein G, Mw = 22,000). The estimated contact area and site density of IgG antibody are ~0.10 µm2 and ~240 µm2, respectively (Kwong et al., 1996). The Gamma Bind G is present at a fourfold excess in solution over [IgG]. The contact areas for both latex beads were based on an estimated 10-nm separation between bead surfaces.

Formation of doublets upon collision

We first treat the process of doublet formation involving two-body collisions between rigid spheres in Couette flow. Following Smoluchowski (1917), we consider a suspension of uniformly dispersed, equal-sized spheres (singlets) of radius R, whose density (number per unit volume) is N_S, and assume that collisions occur after the rectilinear approach of particles to within a distance 2R of their centers. Upon apparent contact, the collision doublet so formed rotates with the angular velocity of a rigid spheroid of axis ratio r_e (= 1.98), until the mirror image of the position of apparent contact is reached, when the doublet separates (Goldsmith and Mason, 1967). Fig. 2 shows that a reference sphere placed at the origin of the flow field (u₃ = GX₂; u₁, u₂ = 0) will collide with all other spheres whose centers pass through a collision disc of radius 2R and origin coincident with the reference sphere. Taking cylindrical polar coordinates X₃ (direction of flow), r, and ₂ with origin at the center of the reference sphere, the elementary number of collisions, df_c(r, ₂), occurring per unit time in an area element of the disc, r dr d₂, can be written as

(6)

where u₃(r, ₂) = Gr sin ₂ is the sphere velocity relative to the reference sphere. Integrating over the entire collision disc (0  r  2R,    ₂  ) yields the two-body collision frequency per particle,

(7a)

Multiplying Eq. 7a by N_S to account for all singlets in the suspension and dividing by 2 to discount counting of the same singlet twice results in the well-known equation (Smoluchowski, 1917) for the total two-body collision frequency per unit volume:

(7b)

Bell (1981; see also Tandon and Diamond, 1997) introduced the probability of adhesion per collision, P_a, which could take a value of either 0 or 1, based on a deterministic criterion assumed a priori, into the right-hand side of Eq. 6 to obtain the fraction of the elementary number of collisions that results in doublet formation per unit time, df_p.

View larger version (14K): [in this window] [in a new window]   FIGURE 2   Coordinate system describing two-body collisions between rigid spheres of radius R in Couette flow. (a) Cylindrical polar coordinates X2, r, 2 showing collision disc of radius 2R drawn about the reference sphere, radius R, in the X1X2 plane normal to the direction of flow along the X3 axis. All spheres whose centers lie on a path having r < R are assumed to collide with the reference sphere. The number of collisions is equal on both sides (+) and () of the X1X2 plane. (b) Spherical polar coordinates, 1, 1 with X1 as the polar and vorticity axis, constructed at the origin of the field of flow.

Our extension of Bell's approach is to solve P_a from Eq. 5 under the condition that there is no bond initially. The result can be expressed by a Poisson distribution (see Appendix A),

(8a)

and

(8b)

The approximation for p_n given in Eq. 8a is valid because of the relatively short lifetime, t₁, of a collision doublet (encounter duration of two singlets in the absence of adhesive bonds to cross-link them) even at low shear rate (average value t₁ = 5/6G  0.3 s; Goldsmith and Mason, 1967) compared to the time scale of bond formation (t_f = (A_cm_rm_lk_f)¹  29 s (cf. values listed in Table 1)). For the same reason, the force dependence of the reverse rate coefficient should have no effect, as the process is dominated by the forward rate constant. Thus k_r is absent from the approximate solutions given by the far right-hand side of Eq. 8. We also assumed a constant A_c, as we believed this was a better approximation than Bell's (1981) assumption that A_c  (2R)²  r², although results were available either way. Multiplying the right-hand side of Eq. 6 with P_a given by Eq. 8b and following the same steps as those used to derive f_c and H_c, the number of doublets formed in unit time (instantaneous two-body collision capture frequency) per particle, f_p, and the corresponding total number per unit volume, H_p, can be obtained as

(9)

Interestingly, under the short contact duration approximation, the result is independent of the shear rate or the encounter duration, as the effects of shear on increasing collision frequency and decreasing encounter duration cancel one another out. Because of this result, there is no need to calculate the encounter duration and the collision swept area, as was done by Tandon and Diamond (1997).

Survival of doublets at low shear rate

The short duration of two-body collisions also ensures that the doublets so formed are most likely initially linked by only one bond, as can be seen from Eq. 8, in that the ratio of the probability of having multiple bonds to that of having only a single bond is equal to P_a (~10², based on the values of t₁ given above and of A_cm_rm_lk_f listed in Table 1). The formation of more bonds and the breakage of existing bonds in these doublets again obey the master equations. Because these occurred at low shear rate, the effect of applied force was neglected. This enabled Eq. 5 to be solved analytically by means of the probability-generating function (see Appendix A),

(10a)

from which the probability p_n of having n bonds at time t (> t₁) in a doublet that had its first bond formed at time t = t₁ can be obtained:

(10b)

Note that the probability-generating function from which Eq. 8a was derived is simply the exponential part on the right-hand side of Eq. 10a (see Appendix A).

It should be noted that the encounter duration t₁ (~0.3 s) is much shorter than the experimental mixing phase [0, t₂] (~30 min), during which the particle suspension was sheared at a low rate. As such, doublet formation first becomes possible during [0, t₁] but continues to occur throughout [t₁, t₂]. To remind us of this fact, the time periods (1) and (2), i.e., those of doublet forming and surviving, are shifted, respectively, from [0, t₁] and [t₁, t₂] for the first doublet to [  t₁, ] and [, t₂] for an arbitrary doublet, where  designates the end point of a collision, which can take any value between t₁ and t₂. On the other hand, preformed doublets may also break even at low shear rate, despite the fact that the forces acting on the bonds that cross-link the two singlets were neglected. As a result, the number of doublets per unit volume measured at the end of the mixing period (i.e., t = t₂) should not simply be the instantaneous collision capture frequency, H_p, multiplied by the time interval, t₂  t₁, but instead be calculated from the doublets that were formed and survived via a convolution integral,

(11)

where the overbar designates the time-averaged value weighted by the (instantaneous) doublet survival probability, 1  p₀. _p, defined by Eq. 11, is referred to as the weighted time average of the two-body collision capture frequency per unit volume.

For the same reason, the efficiency of doublet formation measured at the end of the low shear mixing phase should not simply be the instantaneous efficiency of doublet formation (two-body collision capture efficiency; van de Ven and Mason, 1977),  = H_p/H_c = (/G)A_cm_rm_lk_f, but should rather be its weighted (again by the doublet survival probability) time-averaged value, , calculated from

(12)

to discount doublets that had spontaneously broken up. Physically, / (= _p/H_p  1) is the time-averaged probability of survival (and 1  / that of spontaneous break-up) of preformed doublets in the interval [t₁, t₂]. / = 0 indicates that there would be no doublets remaining at t = t₂ as a result of spontaneous break-ups; and / = 1 means that all doublets formed at low shear rate survived this mixing phase. Likewise, the probability distribution of bonds in the doublets that survived at the end of the mixing period can be obtained by taking the time average of doublets that were formed at different times but all linked by the same number of bonds n at t = t₂ and then renormalizing by the time-averaged survival probability:

(13)

Break-up of doublets at high shear rate

To predict the fraction of doublet break-ups at high shear rate, Eq. 5 was again solved for time t > t₂, using Eq. 13 as an initial condition. Here, the dependence of the reverse rate coefficient on force and the bond number, as given by Eq. 3, was taken into consideration. The periodic nature of the force (each half-rotation through  having an identical period) enables the solution to be expressed as (see Appendix A)

(14a)

where m_min = min(m_r, m_l). The relationship between time t and the polar angle ₁ of rotation is given by

(14b)

where r_e (= 1.98) is the equivalent axis ratio of the doublet (Goldsmith and Mason, 1967; Wakiya, 1971). Note that p_n[t(0)] = p_n(t₂), which is given by Eq. 13. Thus, only the conditioned probabilities p_n|m (assuming that there were m bonds initially) in [0, ] need to be solved. And here the Runge-Kutta numerical scheme was employed, for an analytical solution was no longer possible, as the force acting on the rotating doublet varied continually with its orientation (Eq. 1). The remaining problem is reduced to one of matrix multiplication.

B_i in Eq. 14a is a renormalization factor. In the experiments of individual break-ups of doublets, each doublet was continuously observed from the time it was first subjected to a high shear rate until it broke up or left the field of view. If the doublet was still intact at the end of a half-rotation, one knows that the probability of having no bond should be zero at that point in time. The probability of having nonzero bonds can thus be renormalized by B_i = 1  p₀[t(i)]. Not only does this enable us to use experimental data to reduce the degree of uncertainty of our prediction, but it also allows the predicted probability p₀ to be expressed in exactly the same way as the experimental data, i.e., as the fraction of doublets broken up per rotation (the fraction of the total number of doublets observed in that rotation that broke up; Tees et al., 1993). By contrast, in the population studies, B_i is unity because the doublet number density was only measured at the end of applying shear for a given period of time (Kwong et al., 1996).

The computations reported here were all carried out using the mean angle factor C_f (C_f = sin²₁ sin 2₁). Using C_f appeared justifiable, because within the measured variation of C_f in the population of doublets observed, neither the predicted probabilities nor the fitted kinetic parameters varied significantly (data not shown). These results are corroborated by Monte Carlo simulations.

Data analysis

The theoretical model was fitted to the experimental data by using a numerical routine that employs the Levenberg-Marquart method to evaluate the parameters that minimize the error (²) between the data and the predictions (Press et al., 1989). The chi square statistic, or weighted sum of square of errors, was defined by

(15)

where y_i, y(x_i), and _i are the measurement, prediction, and standard deviation at x_i, respectively, and N is the number of data points. The reduced chi square statistic, _v² = ²/, where  is the number of degrees of freedom (= N  N_f, where N_f is the number of fitting parameters), can be used to measure both the appropriateness of the proposed model and the quality of the data (Bevington and Robinson, 1992). In the previous experiments, the standard deviations were measured only in the population studies but not in the individual break-up studies. Therefore, the predicted standard deviations were used in Eq. 15 in the curve fit of the individual doublet break-up data, as simply setting _i = 1 would yield misleadingly small ² values as a result of the very small values of the measurements themselves (y_i 1). The predicted standard deviation, _i, is that of the Bernoulli trials (e.g., Hines and Montgomery, 1990),

(16)

as the doublets at any given time can only be observed in one of two states: break-up (with a probability p₀) or intact (with a probability 1  p₀). The predicted standard errors, defined by _i = _i/, where N_i is the number of observations (e.g., number of doublets employed in a simulation) comprising the ith data point, indicate the expected fluctuations of predictions.

	COMPARISON WITH EXPERIMENT

Top Abstract Glossary Introduction Analysis Comparison with experiment Discussion Conclusions Appendix A Appendix B References

Doublet formation and bond evolution at low shear rate

The formation of the first bond and the evolution of additional bonds in a doublet at low shear rate are the first two time phases of doublet formation and breakage. Not only do their solutions introduce an initial condition (Eq. 13) for solving Eq. 5 to predict doublet break-up at high shear rate, but they also provide new predictions that can be compared with data. In the case of the population study with doublets of A/S latexes (Kwong et al., 1996), the experimental results are available for such a comparison. The relevant doublet formation parameters are the instantaneous two-body collision capture frequency per unit volume and its weighted time average, H_p and _p, as well as the two-body collision capture efficiency and its weighted time average,  and . In the experiments of Kwong et al. (1996), particle suspensions containing 0.9 nM protein G and A/S latex spheres of radius R = 2.38 µm and singlet density N_S = 8 × 10³ µl¹ were sheared at a low rate G  8 s¹ for a duration t₂  t₁ = 30 min. It follows from Eq. 7b that the two-body collision frequency per unit volume is H_c = 36.8 s¹ · µl¹. The doublet density measured at the end of the low shear mixing phase was 768 ± 194 µl¹ (SD, N = 117). This resulted in _p = 0.427 ± 0.108 s¹ · µl¹ and  = 1.16 ± 0.29% by definition (Eqs. 11 and 12). To calculate from the convolution integral (far right-hand side of Eq. 12), the bond kinetic parameters are required. The values, A_cm_rm_lk_f^L = 3.46 × 10² s¹ and k_r⁰ = 8.05 × 10³ s¹, were taken from Table 1 (second column from the right), where A_cm_rm_lk_f^L denotes the per-cell forward rate constant at low shear rate. As will be explained below, these parameters were derived from curve fitting of data from not only the low but also the high shear rate phases, and as such are not totally freely adjustable parameters for the purpose of calculating _p or alone. The predicted doublet formation parameters are H_p = 0.500 ± 0.128 µl¹ · s¹ (Eq. 9),  = 1.36 ± 0.35%, _p = 0.440 ± 0.103 s¹ · µl¹ (Eq. 11), and  = 1.20 ± 0.31%. The latter two values are in excellent agreement with the measured data. Note that the time-averaged fraction of spontaneous break-up of preformed doublets, 1  / = 0.118, is small but still significant at low shear rate, even when the influence of applied force has been neglected. This is consistent with the stochastic nature of bond association and dissociation for small bond numbers. In principle, doublet break-up can occur under no applied load, and such doublet break-up was indeed observed by Tees et al. (1993).

The ability to treat the low shear rate mixing phase and to connect it with the high shear rate phase provides analytical tools for a new experiment to measure the dependence of the doublet formation efficiency, , on the time, t₂, during which the particle suspension is subject to a low shear rate. The basic idea is that doublets subjected to a longer low shear rate mixing phase are more likely to develop a higher number of bonds. Indeed, evidence for this has recently been obtained using doublets of A/S latexes bearing covalently coupled platelet _IIb3 integrin, cross-linked by divalent human fibrinogen. It was found that when the low shear rate mixing phase was only 5 min, 13% of the doublets could be broken up when subjected to high shear rates. However, when sheared at low rate for more than 20 min, none of the doublets could be broken up at the same high shear rates (Goldsmith and McIntosh, unpublished results).

Probability distribution of initial bonds

To predict or simulate doublet break-up in the high shear rate phase requires an initial condition, namely, the probability distribution of bonds in a doublet at t = t₂. In the previous studies using Monte Carlo simulations, this was done by assuming a Poisson distribution for the initial bonds (Tees et al., 1993). This introduced an additional curve-fitting parameterthe average number of bonds, nwhich was needed to construct the Poisson distribution. In the present study, the two phases before the application of high shear rate (preceding section) were also considered, and the resulting probability distribution at the end of the low shear rate phase, p_n(t₂) given by Eq. 13, was then used as the required initial condition for solving the time course of probability of doublet break-ups at high shear rate. Not only does such a treatment remove an unnecessary assumption, but it also enables evaluation of the forward rate constant, A_cm_rm_lk_f^L, a parameter that has more intrinsic physical meaning than the average number of initial bonds n. In addition, the analysis of the low shear rate period allows one to test the validity of the Poisson approximation for initial bond distribution.

To make this test, Eq. 5 was solved under two different initial conditions, and the results were expressed as the fraction of doublets breaking up at various time points after being sheared at high rate as well as the two initial bond distributions themselves. To isolate the effect of the distribution, the influence of the fitting parameters must be eliminated. Hence, the initial condition for the first solution was taken to be the probability distribution of bonds at the end of the low shear rate phase, p_n(t₂), given by Eq. 13. The average number of initial bonds n at t₂ = 30 min was computed as the mathematical expectation from this p_n(t₂). It was this very same n that was used to construct a Poisson distribution that was assumed to be the initial condition for the second solution. Also kept identical in the two solutions were the other two bond kinetic parameters, i.e., the zero-force reverse rate constant, k_r⁰, and the bond interaction parameter, a, both evaluated by curve fitting the data with the first solution. It was found that both initial conditions, one calculated from Eq. 13, and the other its Poisson approximation, predicted virtually the same time course of break-up that was in equally good agreement with the experimental data. Hence, only the curve computed from Eq. 13 is shown in Fig. 3 a. Such a visual impression was confirmed by the similar quantitative measures for the goodness of fit of the two solutions, which are ² = 8.99 and 8.59, respectively. This conclusion still held true when an additional fitting parameter, the per-cell forward rate constant at high shear rate, A_cm_rm_lk_f^H = 5.23 × 10³ s¹ (Table 1, first column from the right), was included in the analysis, which resulted in two very similar ² values (= 6.82 and 6.58, respectively). This is not surprising, as the Poisson distribution does an excellent job in approximating the probability of initial bonds (Fig. 3 b), and its standard deviation (_n = 2.75) is almost identical to that predicted from p_n(t₂) given by Eq. 13 (_n = 2.74). It is worth mentioning that even when the parameters used to solve Eq. 5 under the Poisson initial condition were allowed to vary freely instead of being required to match those under the initial conditions of Eq. 13, the two approaches still yielded equally good agreement with the data (not shown) and predicted very similar parameters (Table 1, third and fourth or fifth and sixth columns from the right). Thus the Poisson distribution is a very good approximate initial condition for solving the break-ups of doublets at high shear rate.

View larger version (22K): [in this window] [in a new window]   FIGURE 3   Test of Poisson distribution as an approximate initial condition for solving Eq. 5. (a) Comparisons among the data (points, from Kwong et al., 1996, by permission) and the predicted break-up (solid curves) of doublets of A/S spheres cross-linked by 0.9 nM divalent protein G. The best fit predicted fractions of doublet break-up as functions of time, p0(t), were calculated from Eq. 14a with FN,max = 85 and 185 pN and Cf = 0.950 (derived from the corresponding individual break-up experiments; Kwong et al., 1996), using as initial condition pn(t2) or Poisson distribution for three-parameter fittingthe latter is not shown, as the curves are almost identical. The parameter values that resulted in the best fit are listed in Table 1 (second column from the right). (b) Comparison of the Poisson distribution (solid bars) to the probability distribution of bonds, pn(t2), calculated from Eq. 13 (open bars), linking the doublets at the end of the low shear rate phases (t = t2). The mean bond number, n, of the Poisson distributions was required to match that calculated as mathematical expectation from the probability pn(t2) predicted by Eq. 13 (both n = 2.42). Both probability distributions have been renormalized, so that p0 = 0.

Doublet break-up at high shear rate

The data for break-up of A/S latex doublets at high shear rate have already been compared with the predictions (Fig. 3 a). It should be pointed out that, in the present probabilistic model, the calculations of the bond kinetic parameters utilize information from all three experimental time periods. The benefit of this approach is twofold: it keeps the number of freely adjustable fitting parameters to a minimum and, at the same time, increases the reliability of the computed values of these intrinsic properties. The per-cell forward rate constant, A_cm_rm_lk_f^L, and the zero-force reverse rate constant, k_r⁰, were calculated not only by fitting the doublet formation data (i.e., , via Eqs. 9-12) at the low shear rate period, but also by adjusting the initial