GLOSSARY

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODELING EXPERIMENTAL METHODS RESULTS DISCUSSION APPENDIX I REFERENCES

Note: All vector quantities are in bold. Variables with asterisk in superscript are in dimensionless form.

u|y=m	value of the variable u at position y = m.
a, ae	cell radius, radius of receptor-ligand encounter complex
A, A-1, Atot	adhesion flux, adhesion release flux, total adherent cell density
b	half chamber height
C, Cb	cell concentration per unit volume at any point, at the inlet of the flow chamber
Cr	rolling cell concentration per unit area
CL	ligand concentration per unit substrate area
C1, C2, C3, C4	intermediate constants
D	relative diffusion coefficient
fmax	fraction of substrate area occupied by cells at maximum density
G	velocity gradient tensor
i, j	grid element number in x and y direction
kad, kf, kin, kon	adhesion rate constant, forward rate constant, intrinsic reaction rate constant, lumped on-rate
L	length of flow chamber
m, m1, n	grid number in y direction for R2a, for R2b, grid number in x direction.
Nu, Pe	Nusselt number, Peclet number
Ncol	total number of cell-substrate collisions per unit time over the entire flow chamber area
NR	receptor number in contact region
P	capture probability
Q	flow rate
r	radial distance
R, RP, RS, R-1	total tethering flux, primary tethering flux, secondary tethering flux, rolling-release flux
t, t1/2	time, average time
te	receptor-ligand encounter duration
ur, uf, umax	rolling velocity, free stream velocity, maximum free stream velocity
vset, v	actual settling velocity, free settling velocity
V	relative velocity between cell surface and substrate
vx, vy, vz, vr, v, v	flow velocity in Cartesian and spherical coordinates
w	flow chamber width
x0, y0	coordinate at which cells enter flow chamber
Zcc	secondary collision frequency per unit substrate area

Greek symbols

	cell microvillus length
µ, c, m	fluid viscosity, cell density, medium density
w,w	wall shear rate, wall shear stress
	angular velocity of the cell
fr, ra, rf, ar, cc	primary capture frequency, firm-arrest frequency, rolling-release frequency, adhesion-release frequency, cell-cell capture probability

	INTRODUCTION

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODELING EXPERIMENTAL METHODS RESULTS DISCUSSION APPENDIX I REFERENCES

The parallel-plate flow chamber is used to study the biophysics of receptor-ligand interactions under physiologically relevant hydrodynamic flow conditions. This system has been applied to quantify the kinetics of leukocyte/bead binding to reconstituted ligand-coated substrates, activated endothelial cells, platelets, and other leukocytes (Diacovo et al., 1996; King and Hammer, 2001; Lawrence et al., 1987; Walcheck et al., 1996). It has also been used to quantify the progress and mechanism of processes like cancer metastasis and bacterial infection (Felding-Habermann et al., 1996; Mohamed et al., 1999).

Typically, studies carried out in the flow chamber have quantified the interactions between the cells and substrate in terms of the number of rolling cells and the number of adherent cells per unit area (Lawrence et al., 1987; Puri et al., 1997). Besides the biological adhesivity of cells, these measures are also functions of the physical features that affect the rate of cell-substrate interactions including the cell and media density, cell radius, inlet cell concentration, dimensions of the flow chamber, and the applied shear rate/stress (Munn et al., 1994; Patil et al., 2001; Rinker et al., 2001). For example, changing the cell concentration between runs dramatically alters the number of rolling cells by influencing the number of cell-substrate and cell-cell collisions. Also, in experiments that compare the rolling behavior of different cell types (e.g., neutrophils versus lymphocytes) the cell's physical properties, especially density and size, may play an important role in controlling the rate of cell-substrate collision. An understanding of the parameters controlling the number of rolling cells is important because it directly influences the number of adherent cells. To further complicate the issue, once the first few cells are rolling on the ligand-coated substrate, binding interactions between the rolling cells and cells in the free stream may also influence the rate of cell recruitment (Alon et al., 1996; Walcheck et al., 1996). Because of these issues, data from different laboratories or even different treatments cannot be readily compared.

The focus of the current paper is on neutrophil binding to substrates bearing adhesion molecules from the selectin and immunoglobulin supergene family. Specifically, we examine the multi-step process of cell adhesion to E-selectin and intercellular adhesion molecule-1 (ICAM-1) (Gopalan et al., 1997; Simon et al., 2000), because this type of cell recruitment provides a model system to study inflammatory diseases. In this adhesion process, endothelial E-selectin supports neutrophil capture and rolling (Abbassi et al., 1993), whereas following cellular activation firm arrest is mediated by the ₂-integrins (Simon et al., 2000). The ₂-integrins bind ICAM-1 and other ligands on the substrate.

To delineate between the role of fluid flow in controlling the flux of cells to the substrate and its role in modulating molecular bonding, we propose to analyze flow-chamber data using a kinetic model. The current work is similar to previous work by Munn et al. (1994) in that we assume that the flux of cells to the substrate is dependent on cell settling and convection velocities. However, whereas the focus of the previous paper was only on predicting the flux of cells to the substrate, we also predict the effect of this cell-substrate collision on the time-dependent evolution of cell rolling and adhesion densities in the flow chamber. Further, we account for lubrication effects near the plate surface (Brenner, 1961) and incorporate the role of cell microvilli in controlling adhesion rates.

Here, a series of first-order partial differential equations are set up to quantify the steady- and unsteady-state flux corresponding to the cells in the free stream, the rolling cells, and the firmly adherent cells. Frequency parameters are also introduced to measure the different interactions in the cell adhesion system: the capture of cells from the free stream onto the substrate to initiate rolling is quantified using the primary capture frequency, the transition from cell rolling to arrest is measured by the firm-arrest frequency, and the fraction of collisions between cells in the flow stream and surface-bound cells that result in the initiation of cell rolling is quantified using the cell-cell capture probability. Other parameters are also introduced to account for the release of cells from rolling and firm arrest. These model parameters are independent of each other. They are functions of the biological properties that influence the adhesivity of cells, including the number of receptors and ligands, their affinity, topography, activation levels and their response to applied forces. We apply the model to distinguish between cell-substrate binding events (primary capture) and cell-cell binding (secondary capture). We also estimate the on-rates of selectin bonds based on previously published data (Puri et al., 1997). Although the current paper studies selectin, integrin, and immunoglobulin-mediated neutrophil adhesion in a model of inflammation, the proposed analysis methodology may be applied to other experimental systems also.

	MATHEMATICAL MODELING

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODELING EXPERIMENTAL METHODS RESULTS DISCUSSION APPENDIX I REFERENCES

Trajectory of cells

The parallel-plate flow chamber has a defined flow profile that simulates the features of hydrodynamic forces found in the vasculature. A typical device consists of two parallel plates separated by a gasket. The thickness of the gasket controls the height of the flow chamber (Fig. 1). The bottom plate of the chamber is composed of either glass or plastic. It bears a ligand-expressing substrate, typically adsorbed extra-cellular matrix proteins, adhesion molecule-bearing cells, or isolated ligand molecules. A syringe pump precisely controls the flow rate of the cell suspension over the substrate. Cells entering the flow chamber interact with this ligand-bearing substrate as depicted in Fig. 1 A. In this apparatus, the wall shear rate (_w) is related to the flow rate of the syringe pump (Q), the chamber height (2b, equal to the thickness of the gasket) and flow section width (w) by the relation _w = 3Q/2b²w. The wall shear stress _w for a Newtonian fluid with viscosity µ is then defined as _w = µ_w.

View larger version (35K): [in this window] [in a new window]   FIGURE 1   Regions of the parallel-plate flow chamber. (A) The flow chamber of length L and height 2b is divided into n uniform divisions in the x direction and (m + m1 + 3) divisions in the y direction. Cells entering the chamber both convect with a velocity uf and settle at vset. Upon reaching the substrate, the cells may undergo rolling or firm-adhesion. The flow chamber is divided into four regions, labeled Region 1 (R1) to Region 4 (R4). (B) Mass flux in R1. The height of this region equals the cell diameter 2a. Cells enter by convection and leave either via convection or settling. (C) Mass flux in R2. R2 is divided into two sections: R2a consisting of m coarse divisions and R2b with m1 finer divisions. Cells enter this region either via convection or settling. In most cases, they leave the grid element via the same mechanisms. The exceptions to this are the bottom rows of R2b (3a +  > y  a + ) where secondary capture may also contribute to cell efflux. (D) Mass flux in R3. This region is of height , the cell microvillus length. In addition to settling and convection, the tethering of cells at a rate R and the release of rolling cells at a rate R1 also control the mass flux here. (E) Mass flux in R4. Adhesion flux (A) and adhesion-release rate (A1) are also important in this region. The cell rolling velocity is denoted ur.

Cells entering the flow chamber experience two types of forces: the force of gravity, which causes them to settle with a velocity v_set along the y axis, and the fluid convective force in the x direction, which causes them to translate axially with a velocity, u_f (Munn et al., 1994). When the cell is far from the flow chamber substrate, we assume that the cell is freely settling with a velocity v expressed by Stokes' equation,

(1)

where a is the cell radius, _c is the cell density, and _m is the media density. When close to the plate surface, however, the settling speed is less than the free settling velocity because of lubrication effects (Brenner, 1961). This reduction in settling velocity occurs because of balance between two opposing forces: the forces of gravity that drive the particle to the surface and the lubrication layer between the particle and the substrate that resists this motion. Overall, the settling velocity, v_set, at any point in the flow chamber is expressed as a function of y, the distance from the substrate to the cell's center. The exact solution of this problem is well approximated by (Davis and Giddings, 1985)

(2)

Here, (y  a) denotes the separation distance between the cell and the substrate. The convection velocity of the cells in the free stream varies quadratically with height from the substrate as described below, where u_max is the maximum convection velocity at the center of the flow chamber (Fig. 1),

(3a)

Close to the plate surface (at approximately y < 4a), the convection velocity is reduced by hydrodynamic wall effects. Exact results for u_f (Goldman et al., 1967) are well approximated by the far-field asymptotic formula,

(3b)

The equations of settling (Eq. 2) and convection (either Eq. 3a or 3b depending on the y-coordinate) can be combined to yield an equation that describes the (x, y) trajectory of a cell that enters the flow chamber at an initial coordinate (x₀, y₀). For example, above y = 4a, Eqs. 2 and 3a combine to give the trajectory equation (Eq.4). A similar equation can be derived by combining Eq. 2 and 3b for y < 4a.

(4)

Number of cell-substrate collisions

The cells are modeled to be spheres of radius a with microvilli protrusions of length . Thus, the tips of the microvilli contact the substrate when the cell is at a height y = a + . In the case of leukocytes, several families of adhesion molecules, including the selectins and their major ligands (Erlandsen et al., 1993; Moore et al., 1995), are found localized at the tips of microvilli. Thus, cell recruitment may occur after the cells settle to a height, y = a + .

To determine the total number of cell-substrate collisions per unit time (N_col) over the entire flow chamber area at a given shear rate, we determined a parameter called the critical y coordinate, y_crit. This is the largest y coordinate at the inlet of the flow chamber where the cell must enter if it is to just touch the chamber substrate before exiting the device at (x, y) = (L, a + ). All cells that enter the flow chamber below y_crit thus collide with the substrate and contribute to N_col. When cells at a concentration C_b enter a flow chamber with width, w, we thus define

(5)

The efficiency of adhesion in the flow chamber

All the cells observed in the flow-chamber are either in the free stream, rolling on the plate surface, or firmly adherent on the substrate. For the purpose of solving the concentration of cells in various regions of the flow chamber, we divide the apparatus using a two-dimensional (2D) mesh in the x and y direction (Fig. 1). There are n equal-sized divisions in the x direction. In the vertical y direction, there are m + m₁ + 3 divisions. Greater resolution is provided at the bottom of the flow chamber to more carefully resolve between the rolling cells and those in the free stream near the substrate. Each of the elements of the mesh is said to be located in one of 4 "regions" depicted from R1 to R4 (Fig. 1) depending on the nature of cell accumulation and the mass balance equation. A description of each Region follows.

Region 1 or R1 (2b >y 2b  2a, Fig. 1 B) is the topmost row of the flow chamber. In each mesh element of this region, cells enter by convection from the previous element and they leave either by convection to the right or by settling below. Together, these rates determine the cell concentration C (cells/volume) at any point (x, y) and time t:

(6)

The initial (IC) and boundary (BC) condition required for the solution are
Writing the above equations in dimensionless form using C* = C/C_b, t* = vt/(2b), x* = x/L and v^*_set = v_set/v, we get

(7)

The analytical solution of the above first-order partial differential equation is

(8)

where ^*₁ = 2a(2b  a)u_maxt*/(vLb)  x*.

Region 2 or R2 (2b  2a > y  a + , Fig. 1 C) constitutes the bulk of the flow chamber between the upper and lower surface. This region is subdivided into two parts: Region 2a (R2a) has m coarse divisions that cover the flow chamber volume four cell radii above the substrate (2b  2a > y  4a + ), and Region 2b (R2b) has m₁ finer divisions that cover the sections within 4 cell radii of the substrate (4a +  > y  a + ). In our simulations, m was set to 11 and m₁ to 25. Cells enter R2 (either R2a or R2b) either by settling from the mesh element above or via convection from the previous element. They leave the element in a similar fashion. A mass balance equation for any element in this region is

(9a)

Note that additional dimensionless parameters, y* = y/2b and u^*_f = u_f/u_max, have been introduced here.

We note here that some cells at the bottom section of R2b that are less than 1.5 cell diameter from the substrate (i.e., at 3a +  >y  a + ) may contact already adherent cells and thus tether via "secondary capture mechanism" (Alon et al., 1996; Mitchell et al., 2000). The mass balance in this bottom region of R2b thus includes an additional efflux term to account for secondary capture (Eq. 9b). We describe this secondary tethering mechanism and the mathematical form of R^*_S in more detail in the next section.

(9b)

Here, R^*_S = R_S/C_bv and y* = (3a/(m₁  1))/(2b). R_S is the tethering flux due to secondary capture in unit of cells/area/time. Unlike the equations for R1, Eq. 9 does not have an analytical solution. It is solved using finite difference as described later.

Region 3 or R3 (a +  > y  a, Fig. 1 D) is a region of height equal to the cell microvillus length, . It represents the lowest layer of the flow chamber with convective mass flow. In this region, cells come in contact with the substrate. They enter R3 either via convection from the previous element or by settling from above. In addition to exiting the element by convection to the next element, these cells may also bind or "tether" onto the ligand-coated substrate. Tethering marks the capture of cells from the flow stream onto the chamber substrate and this initiates cell rolling.

Two mechanisms may contribute to the tethering of cells: Adhesion molecules on the cell surface may bind ligands expressed by the flow chamber substrate. This process is termed "primary capture." Free-flowing cells may interact with previously recruited cells, and this may contribute to new tethering events through a "secondary capture" process. In this case, the previously recruited cells may either directly present ligands for cell capture, or they may alter the local hydrodynamic environment near the substrate, thus changing the rate of cell-substrate attachment.

In this model, to quantify these two modes of cell recruitment, we introduce the terms "primary capture frequency", _fr (unit of length¹), and "cell-cell capture probability", _cc (dimensionless unit). The primary capture frequency is analogous to a first-order reaction rate constant, and is defined as

(10)

In this equation, the rate of primary capture, denoted by the primary tethering flux R_P (unit of cells captured/area/time) is directly proportional to the local cell concentration in R3. Further, analogous to a first-order reaction with rate constant k, where the half-life of a reaction is given by ln(2)/k, it can be shown that the average distance the cell traverses in region R3 before primary capture equals ln(2)/_fr. The time taken to travel this distance (t_1/2) is thus ln(2)/(_fr u_f).

Cell-cell capture probability _cc is a measure of the fraction of collisions between cells in the free-stream and previously adherent cells that result in capture. This is analogous to cell-cell adhesion efficiency, which we have used elsewhere to quantify cellular binding kinetics in suspension (Neelamegham et al., 1997b). The cell tethering flux due to secondary capture, R_S (unit of cells captured/area/time), is a product of cell-cell capture probability and the frequency with which cells in the free stream collide with already bound cells, Z_cc (collisions/area/time):

(11)

To estimate Z_cc, we calculate the number of particles entering a "collision sphere" around a surface bound cell (Fig. 2). This collision sphere is an imaginary sphere (with radius = 2a) surrounding the surface-adherent cell. If the center of any other particle enters this collision sphere, cell-cell collision occurs. The term in the integral (Eq.11) denotes the total number of collisions taking place with any given adherent cell. Here, dA represents the projected area of an element of the collision sphere surface on the yz-plane (Fig. 2) and v_x (calculated from Eq. A5, Appendix) is the local free stream velocity along the x-direction at that point. The cell in the free stream may interact with either the rolling or firmly adherent cells on the substrate. This accounts for the factor C_r + A_tot. Here, A_tot is the density of adherent cells (cells/unit area) and C_r denotes the number of rolling cells per unit area. Methods to determine A_tot and C_r are described in the next section.

View larger version (32K): [in this window] [in a new window]   FIGURE 2   Estimating secondary tethering flux (RS). The schematic represents a dotted collision sphere of radius 2a surrounding a surface-bound cell (gray). If the center of any other cell enters this collision sphere, cell-cell collision occurs. The number of cell-cell collisions is estimated by calculating the mass flux of cells into the collision sphere (Eq. 11). dA depicts the projection of an area element located on the collision sphere surface at (2a, , ) onto the yz-plane.

The total tethering flux can be estimated by summing R_P and R_S when the density of rolling cells is low, i.e., during the initial phases of the experiment. However, it is evident that, at the later time points when a substantial fraction of the substrate area is occupied, the rate of cell attachment is lower, and it is never possible to completely pack the ligand-bearing substrate with rolling cells. To account for this feature, in the model, we introduce a parameter f_max, defined as the fraction of the substrate surface area (L · w) that is occupied when the substrate coverage reaches maximum. Assuming a linear relationship between tethering flux (R_P + R_S) and the substrate area available for cell recruitment, we thus determine an expression for the total tethering flux, R (Eq.12). Although we experimentally estimated f_max to be 0.025 in our system where neutrophils bound substrates bearing cotransfected cells, we note that this parameter may be higher in other systems, especially when cells bind to reconstituted ligand-bearing substrates,

(12)

Under some conditions where the off-rate of ligands is high or when the ligand density is sparse, some of the rolling cells may "release" from the surface and change back to the free stream. The flux of this transition is denoted by R₁ (number of cells release from rolling/area/time),

(13)

In this equation, the rate at which rolling cells are released to the free stream is quantified using a "rollingrelease frequency", _rf (units of length¹). This parameter is analogous to a first-order rate constant. From a physical standpoint, the average cell can be thought to roll a distance equal to ln(2)/_r before moving back into the free stream.

If we set R* = R/C_bv and R^*₁ = R₁/C_bv, the dimensionless form of the mass balance equation for R3 follows (Eq. 14). Solution of this equation yields an estimate of the cell concentration in R3.

(14)

Region 4 or R4 (a >y, Fig. 1 E) is the region of the flow chamber with the rolling cells. Although cells enter and leave this region by the fluxes described by R (Eq.10-12) and R₁ (Eq.13), the rate of firm adhesion also regulates the number of rolling cells. The rate of this arrest process, which is estimated by the adhesion flux parameter, A (number of adherent cells/area/time), is directly dependent on the number of rolling cells according to

(15)

Here, u_r is the cell rolling velocity. The frequency with which rolling cells change to firm-adhesion is quantified using the "firm-arrest frequency", _ra (length¹). Analogous to the frequency parameters described above, ln(2)/_ra is the average distance that the cell rolls before it switches to firm arrest on the substrate. The time taken for such a transition for an average cell equals ln(2)/(_ra u_r).

Similar to the release of rolling cells from the substrate, adherent cells may also be released back as rolling cells. To describe this phenomena, "adhesion-release frequency", _ar (units of time¹), is introduced. The rate of adhesion-release A₁ (number of cells release/area/time) is set to equal the product of _ar and the total adherent cells, A_tot (number of adherent cells/area),

(16)

We note here that it is possible that cells that are previously adherent may be released directly into the free stream, instead of rolling, in which case, similar modifications can be made to the model. The density of adherent cells at any time is based on the cumulative adhesion and release of cells over the time course of the experiment. At any time t, it is mathematically expressed as

(17)

The concentration of rolling cells, C_r, in Region 4 is determined from the mass balance equation of this region,

(18)

where A* = A/C_bv, A^*₁ = A^*₁/C_bv, C^*_r = C_r/aC_b.

Model solution and usage

A finite difference scheme was used to determine the cell concentration in each of the (m + m₁ + 3) × n grid elements of the flow chamber. For this, the concentration in R1 was determined analytically from Eq. 8. Then the differential equations for the other regions (Eqs. 9, 14, 18) were converted from the 2D form (i.e., in the x and y direction) into a set of first-order differential-algebraic equations. During this transformation process, the equation corresponding to any point, say the ith element in the jth row of the 2D grid was translated into the (j  1)*n + ith equation in the one-dimensional system of equations. The FORTRAN subroutine DDASPG in the IMSL library was then applied to solve the system of differential algebraic equations. The reference values for the parameters used in the simulations are given in Table 1. This corresponds to the case of neutrophil-like particles flowing and adhering on E-selectin and ICAM-1 bearing substrates. Whereas the first nine variables are determined from the physical parameters of the experimental system, the next two parameters (u_r and f_max) were determined directly from independent experiments that quantify cell-rolling velocity and maximum substrate occupancy. The final five variables are frequency and probability parameters that define the nature of the receptor-ligand interactions. These were obtained by fitting the experimental data.

To obtain estimates of _fr, _rf, _ra, _ar, and _cc for any experiment, the mathematical model was run for a range of frequency and probability parameters, and the output data was collected in terms of the number of rolling and adherent cell densities. Although a large number of combinations for the five parameters are possible, in most experimental situations, one or more of the frequency parameters can be set to zero. For example, in all our experiments, because we did not observe the release of either rolling or adherent cells back into the flow stream, _rf and _ar were set to zero. In some of the runs performed with DREG-56, which blocked secondary adhesion, we also set _cc to zero. Thus, by varying _fr and _ra in these simulations and upon comparing with the experimental data, we deduced the appropriate frequency- and probability-parameter values. Independent experiments and simulations were also performed in which we varied the inlet cell concentration in the flow chamber. Here, we confirmed that we could fit rolling and adhesion data over the range of inlet-cell concentrations with the identical frequency- and probability-parameter values at any given shear stress.

Estimating the intrinsic selectin on rate

Bonding between selectins and their ligands is facilitated by Van der Waal forces and electrostatic interactions, which eventually mediate the coordinated formation of a series of hydrogen bonds between the receptor and ligand (Graves et al., 1994). Formation of such interactions between a single receptor and ligand is termed a single bond. Currently, although it is thought that engagement of only a few neutrophil selectin-ligand bonds may be sufficient to initiate cell tethering and rolling over the range of physiologically relevant shear stresses, it is not established if a single selectin-ligand bond would be sufficient (Alon et al., 1995; Evans et al., 2001). In our analysis, we estimate the selectin on rate by defining a bond as the minimum set of hydrogen bonds that can mediate the tethering of cells. No assumption is made on whether this minimum set involves either single or multiple selectin molecules. We also assume that this tethering event only involves a single cell-surface microvillus because this is a likely scenario. In this regard, the selectins and their ligands are preferentially located on the neutrophil microvilli. Also, because our analysis is applied to analyze experiments with a high ligand density where rolling cells do not revert back into the free stream, we assumed that the formation of a single transient tether is sufficient to initiate stable cell rolling.

A previously published analysis method (Chang and Hammer, 1999) was adopted in conjunction with our estimates of primary capture frequency (_fr) to determine the selectin on rates from parallel-plate flow-chamber data. First, the capture or tethering of cells from the free stream in region R3 by the ligand-bearing substrate is described by a first-order rate expression (Eq.19). Here, k_ad (unit of time¹) is termed adhesion rate constant.

(19)

Thus, if t_1/2 denotes the time taken for the average cell in R3 to change from the free-stream to rolling, based on the above equation, k_ad = ln(2)/t_1/2. In our analysis, we have shown above that t_1/2 is related to the primary capture frequency according to t_1/2 = ln(2)/(u_f|_y=R3fr). Therefore,

(20)

As discussed elsewhere (Chang and Hammer, 1999), the k_ad estimated above is a linear function of the number of cell-surface receptors in the cell-substrate contact region N_R and ligand concentration C_L (sites/area). Thus, these authors defined a forward rate constant k_f (unit of area/time) that is independent of the receptor and ligand number,

(21)

The forward rate constant k_f depends not only on the intrinsic reactivity between the receptor and ligand, but also on the rate at which the transient receptor-ligand complex forms. As the cell flows in the free stream in contact with the substrate, both the fluid convective flow and the receptor/ligand surface diffusivity contribute to the formation of this receptor-ligand complex. The rate of complex formation is thus dependent on the receptor/ligand diffusivity (D), the size of the encounter complex (a_e), and the relative convective velocity (|V|) between the cell and the substrate. Because there is substantial slip near the flow-chamber substrate (Goldman et al., 1967), both the cell's free-stream velocity and rotation rate are accounted for while estimating the relative convective velocity, i.e.,  |V|  = u_f   · a, where the angular velocity  = _w(1   × (a/y)³)/2 (Goldman et al., 1967). Chang and Hammer solved the 2D convection-diffusion equation for cell interaction with the flow-chamber substrate. They introduced the Peclet number (Pe =  |V|a_e/D) to contrast the roles of diffusion and convection. In the context of selectin-mediated tethering and rolling where the radius of the selectin-ligand complex is 2.0 × 10⁷ cm (Springer, 1990) and D is ~10¹⁰ cm²/s (Chang and Hammer, 1999), Pe equals 10 at a wall shear rate (_w) of 28/s. Thus, typically, Pe 1 in the flow chamber, i.e., it is cell convection rather than receptor diffusivity which controls the rate of selectin-ligand encounter complex formation. Under these conditions, Chang and Hammer showed by the theory of first passage (Szabo et al., 1980) that the duration of each encounter complex, t_e, is 8a_e/(3|V|). The binding probability, P, which is defined as the probability that the selectin has bound its ligand before the dissolution of the encounter complex, is then

(22)

For a given value of the binding probability, P, and the duration of the encounter complex, t_e, we can then estimate the intrinsic on-rate k_in according to (Chang and Hammer, 1999)

(23)

In this analysis, the molecule on the cell surface is denoted as the "receptor" and the "ligand" is defined to be the surface-immobilized molecule. In the context of our flow-chamber experiments, it is not currently possible to estimate precisely a value for N_R (number of selectin ligands) because sufficient information on the nature of cell-substrate contact area at the instant when tethering occurs is not available. For this reason, we prefer to lump the intrinsic on- rate k_in and N_R into a lumped on rate denoted by k_on,

(24)

	EXPERIMENTAL METHODS

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODELING EXPERIMENTAL METHODS RESULTS DISCUSSION APPENDIX I REFERENCES

Cell culture and neutrophil isolation

Fresh human blood was collected by venipuncture into a sterile syringe containing 10 U/ml heparin (Elkins-Sinn, Cherry Hill, NJ). Neutrophils were isolated using a one-step Ficoll-Hypaque gradient (ICN Biomedicals, Aurora, OH) as described previously (Taylor et al., 1996). Isolated cells were kept in Ca²⁺ free HEPES buffer (NaCl 6.428 g/l; KCl 0.746 g/l; MgCl₂·6H₂O 0.427 g/l; Glucose 1.8 g/l; HEPES 7.149 g/l) with 0.1% human serum albumin (Bayer Corporation, Elkhart, IN) at 4°C before the experiment. All reagents were from Sigma Chemical Co. (St. Louis, MO) unless otherwise mentioned.

Parent mouse fibroblast L cells (abbreviated L cells), and L cells transfected to either express ICAM-1 (I cells), or both ICAM-1 and E-selectin (E/I cells) were kindly provided by C.W. Smith (Baylor College of Medicine, Houston, TX). Cells were cultured as described elsewhere (Gopalan et al., 1997; Simon et al., 2000). For the adhesion assays, the mouse cells were detached from tissue culture flasks by adding sterile PBS containing 5 mM EDTA, and then plated onto 35-mm tissue culture-treated petri dishes (Corning Class Works, Corning, NY) at 2-3 × 10⁶ cells/ml. Cells were grown for 2-3 days till confluence before the experiment.

Cell adhesion experiments

Neutrophil adhesion experiments were performed in a parallel-plate flow chamber (Glycotech, Rockville, MD) mounted on the stage of a phase-contrast optical microscope (CK40, Olympus, Japan) with a 10× objective. All runs were performed at 37°C. A syringe pump (Kd Scientific, New Hope, PA) was used to simulate a uniform laminar flow field in the flow chamber. Here, the petri dishes with the confluent E/I cell monolayers were used as the ligand-binding substrate. The monolayer was perfused with sterile PBS for 4-5 min before introduction of the isolated neutrophil suspension, resuspended in HEPES buffer with 1.5 mM CaCl₂, at a predetermined concentration. In some experiments, where antibodies were used to block secondary tethering, neutrophils were preincubated with 15 µg/ml L-selectin antibody DREG-56 (purified from ATCC hybridoma culture supernatant) for 10 min before the start of the experiment. Independent experiments performed in a neutrophil homotypic aggregation assay confirmed that DREG-56 blocks ~100% of the L-selectin interactions between neutrophils (data not shown). In our runs, the cell-rolling and adhesion data were recorded using a CCD camera (Model 77, MTI-Dage, Michigan City, IN) and time-lapse video recorder (TLC2100, GYYR, Anaheim, CA). During the first 9 min of each experiment, data was recorded at a fixed position of the flow chamber. Following this, the field of observation was moved to 5 other random locations and cell binding data was recorded at each position for 20s. Data analysis was performed after completion of the entire experiment by digitizing the images using a Scion LG3 board (Scion Corp., Frederick, MD) and using PC-based NIH-Image software (Scion).

In other experiments, performed with a chimeric E-selectin fusion protein (Glycotech, Rockville, MD), which consisted of the E-selectin extracellular domain fused to an IgG tail, we examined whether E-selectin was a prominent ligand for L-selectin on neutrophils. In these runs, we observed that the binding of the E-selectin fusion protein to isolated human neutrophils could not be blocked by an antibody against the lectin domain of L-selectin (DREG-56). However, this interaction could be blocked by anti-human E-selectin monoclonal antibody HAE-1f (Ancell, Bayport, MN). This suggests that E-selectin does not bind the neutrophil L-selectin lectin domain. This domain is thought to contribute to neutrophil tethering.

E-selectin density on cotransfected cells

The E-selectin site density on cotransfected cells was determined using the Quantum Simply Cellular microbead standards (Bangs Laboratories, Fishers, IN) in conjunction with mouse anti-human E-selectin antibody, CL2/6 (Biosource International, Camarillo, CA). These uniform microbeads have a calibrated number of goat-anti-mouse IgG sites on their surface. In these runs, both the calibrated microbeads and the cotransfected E/I cells were incubated with CL2/6 at saturating concentrations. After a brief wash in HEPES buffer, a secondary Alexa-488 conjugated F(ab')₂ goat anti-mouse IgG (H + L) antibody (Molecular Probes, Eugene, OR) was added for 10 min. The beads and cells were again washed rapidly and the samples were read using a flow cytometer. The number of E-selectin sites per E/I cell was then determined by quantifying the fluorescence intensity of the labeled cells, and translating this value to the number of bound antibodies using the microbead standards. Isotype matched controls were also performed to confirm the findings.

Data analysis

The number of rolling cells and adherent cells was determined at each time point. For these measurements, the density of adherent cells was determined at each time point by counting the number of cells that moved by less than 1 cell diameter in a given 20-s time period. The number of rolling cells was then estimated by subtracting the number of firmly adherent cells from the total number of cells.

In some runs, the rolling velocity of neutrophils was also determined by randomly choosing 40-50 rolling cells and following their motion for ~20 s. Rolling velocity was determined by dividing the distance traveled by these cells by the time taken. ANOVA analysis using the Student Newmann Keul's test was performed to assess statistical differences between experimental runs. p < 0.05 was considered significant.

	RESULTS

TOP ABSTRACT GLOSSARY INTRODUCTION MATHEMATICAL MODELING EXPERIMENTAL METHODS RESULTS DISCUSSION APPENDIX I REFERENCES

The mathematical model for cellular interactions in the parallel plate flow chamber was simulated over a range of conditions that are typical for biological experiments. A summary of the results is presented here. In addition, the model was fit to experiments that elaborate on the nature of selectin- and integrin-mediated adhesion. All reference parameters for the simulations are listed in Table 1, unless otherwise stated. These correspond to the case of isolated human neutrophils rolling on E/I cells.

The rate of cell-substrate collision depends on fluid convection and cell settling velocity

Figure 3 A presents the trajectory of cells in the flow chamber during typical biological studies. For these calculations, although particle convective velocity far above the substrate was assumed to follow a parabolic profile (Eq. 3a), it was estimated using Goldman's equation (Eq. 3b) near the substrate. The settling velocity near the substrate is also lower than free settling velocity due to the presence of lubrication layer (Eq. 2). For the flow chamber geometry considered, above a shear stress of 0.5 dyn/cm², cells that enter the flow chamber at a height above the ten-cell radius mark (i.e., y₀ > 10a) do not contact with the plate surface. Thus, typically, a majority of the cells introduced into the flow chamber do not contribute to cell rolling or adhesion.

View larger version (14K): [in this window] [in a new window]   FIGURE 3   Cell trajectory in flow chamber. (A) Trajectories of cells entering the flow chamber at a height of ten cell radii i.e., (x0, y0) = (0, 10a). Wall shear stress varies from 0.5 to 2 dyn/cm2. All other parameter values are given in Table 1. In typical experiments, a majority of cells introduced into the flow chamber do not collide with the substrate. (B) Total number of cell-substrate collisions decreases with increasing shear.

We examined whether the number of cells contacting the substrate per unit time, N_col, (Eq. 5) is a function of the applied shear stress (Fig. 3 B). Two competing features regulate this parameter: although increasing the shear stress increases the number of cells entering the chamber per unit time, the higher convective velocities simultaneously reduce the time available for cell settling onto the substrate. As seen in Fig. 3 B, N_col decreases by ~25% on increasing the applied shear stress from 0.5 to 20 dyn/cm². In general, at 2 dyn/cm², for the range of flow chamber sizes considered (L varied from 0.5 to 4 cm, and b from 0.0127 to 0.0254 cm), we observed that the percentage change in N_col with shear rate is independent of the flow chamber geometry. It is primarily regulated by the physical properties of the cells (density and size), the properties of the liquid (viscosity and density) and the applied shear rate.

Cell concentration near the plate surface may be higher than inlet cell concentration

Accurate estimation of the cell concentration close to the substrate is important because the density of rolling and adherent cells is a strong function of this parameter. For this reason, we compared the cell concentration: far from the substrate (in R2a at y  4a + ); near the substrate (in R2b, 4a +  > y  a + ); and in region R3 (a +  > y  a) where the cell microvilli are in contact with the substrate.

We observed that, away from the substrate in R2a, the steady-state cell concentration is independent of the nature of cell rolling and adhesion, quantitatively equal to the inlet cell concentration, and independent of the distance from the chamber entrance (data not shown). Further, the upper wall of the flow chamber does not affect the manner of cell settling near the ligand-coated substrate.

Near the substrate in R2b, however, we observed that the cell concentration was higher than the inlet concentration, C_b (Fig. 4, A and B). In this region, steady-state cell concentration was achieved rapidly, typically in less than 10s. (Fig. 4 A). The time taken was approximately equal to the time taken for the cells to convect from the entrance of the flow chamber to that region. The steady-state concentration at the bottom of R2 increased with proximity to the substrate (Fig. 4 B). This is apparently because of the effect of the lubrication layer near the substrate, which reduces the cell settling velocity and consequently increases the accumulation near the substrate. In support of this proposition, we observed that, upon neglecting the lubrication effects (i.e., assuming that v_set was equal to the free settling velocity v independent of position in the flow chamber), the predicted cell concentration was uniform and equal to C_b in all sections of R2 (data not shown). The lubrication layer also causes positional variations in the cell concentration at the bottom of R2 with distance from the flow chamber entrance. We observed, using our default simulation parameters (Table 1), that, although the concentration near the entrance equals the inlet concentration C_b, it increases with distance from the entrance (data not shown). This lubrication feature may thus contribute to the variation in cell concentration with position in the flow chamber.

View larger version (12K): [in this window] [in a new window]   FIGURE 4   Cell concentrations near the plate surface. (A) Temporal evolution of cell concentration in the last row of R2b at dimensionless position x* = 0.4. Steady state is achieved within seconds and the final cell concentration in this region is ~6 times inlet cell concentration. (B) Steady-state cell concentration in R2b increases dramatically with proximity to the flow chamber substrate. Data are presented for dimensionlessposition x* = 0.4. (C) Positional variation in cell concentration in R3 at 10 min. The concentration of cells may either decrease or increase with distance from the flow chamber entrance depending on the relative magnitude of the cell settling velocity and primary capture frequency. All simulation parameters, except fr in (C), are given in Table 1.

Finally, we considered the cell concentration in R3 (Fig. 4 C). The concentration of cells in this region is not only dependent on the rate of settling and convection, it is also influenced by the rate of cell tethering onto the flow chamber (Eq. 14). If the rate of cell tethering is lower (i.e., _fr is small) than the flux of cells into this region, then there is net accumulation of cells in this region. This results in an increase in cell concentration with distance from the flow