Surface-dependent Coagulation Enzymes: Flow Kinetics of Factor Xa Generation on Live Membranes

The initial surface reactions of the extrinsic coagulation pathway on live cell membranes were examined under flow conditions. Generation of fXa (activated coagulation factor X) was measured on spherical monolayers of epithelial cells with a total surface area of 41-47 cm$^{2}$ expressing TF(tissue factor) at $>25$ fmol/cm$^{2}$. Concentrations of reactants and product were monitored as a function of time with radiolabeled proteins and a chromogenic substrate at resolutions of 2-8 s. At physiological concentrations of fVIIa and fX, the reaction rate was $3.05 \pm 0.75$ fmol fXa/s/cm$^{2}$, independent of flux, and 10 times slower than that expected for collision-limited reactions. Rates were also independent of surface fVIIa concentrations within the range 0.6-25 fmol/cm$^{2}$. The transit time of fX activated on the reaction chamber was prolonged relative to transit times of nonreacting tracers or preformed fXa. Membrane reactions were modeled using a set of nonlinear kinetic equations and a lagged normal density curve to track the expected surface concentration of reactants for various hypothetical reaction mechanisms. The experimental results were theoretically predicted only when the models used a slow intermediate reaction step, consistent with surface diffusion. These results provide evidence that the transfer of substrate within the membrane is rate-limiting in the kinetic mechanisms leading to initiation of blood coagulation by the tissue factor pathway.

The initial surface reactions of the extrinsic coagulation pathway on live cell membranes were examined under flow conditions. Generation of activated coagulation factor X (fXa) was measured on spherical monolayers of epithelial cells with a total surface area of 41-47 cm 2 expressing tissue factor (TF) at >25 fmol/cm 2 . Concentrations of reactants and product were monitored as a function of time with radiolabeled proteins and a chromogenic substrate at resolutions of 2-8 s. At physiological concentrations of fVIIa and fX, the reaction rate was 3.05 ؎ 0.75 fmol fXa/s/cm 2 , independent of flux, and 10 times slower than that expected for collision-limited reactions. Rates were also independent of surface fVIIa concentrations within the range 0.6 -25 fmol/cm 2 . The transit time of fX activated on the reaction chamber was prolonged relative to transit times of nonreacting tracers or preformed fXa. Membrane reactions were modeled using a set of nonlinear kinetic equations and a lagged normal density curve to track the expected surface concentration of reactants for various hypothetical reaction mechanisms. The experimental results were theoretically predicted only when the models used a slow intermediate reaction step, consistent with surface diffusion. These results provide evidence that the transfer of substrate within the membrane is rate-limiting in the kinetic mechanisms leading to initiation of blood coagulation by the TF pathway.
Blood coagulation reactions mediate fibrin deposition in hemostasis and many pathological processes. Blood clots are directly implicated in the lethal complications of cardiovascular disease and contribute significantly to the pathogenesis of infectious, autoimmune, and neoplastic diseases (1)(2)(3)(4)(5)(6).
The blood coagulation process is initiated by an assembly of complexes comprised of an essential cofactor, TF 1 (tissue factor) and a protease component, fVIIa. The functional complex, TF⅐fVIIa, cleaves the natural substrates, fVII, fIX (factor IX), and fX at specific sites, generating fVIIa, fIXa, and fXa, respectively (4 -6). Factors VII, IX, and X circulate in the blood and extravascular fluids (7)(8)(9)(10), whereas TF is expressed on the membranes of many extravascular tissues (11). The anatomic distribution of cells expressing TF is consistent with its role as the initiator of hemostatic reactions. Cell surfaces in contact with blood do not appear to express functional TF constitutively. However, inflammatory stimuli induce expression of functional TF on endothelial cell membranes and blood monocytes (12)(13)(14).
Factors VII, IX, and X are vitamin K-dependent proteins, and their functional interaction with negatively charged procoagulant membranes has a calcium-dependent, electrostatic component (15)(16)(17)(18)(19). The interaction sites are located in highly homologous ␥-carboxyglutamic acid (Gla)-rich regions near the N terminus of all vitamin K-dependent coagulation proteins (4,19). The specific binding and functional kinetics of interaction between coagulation proteins and biological membranes have been studied extensively under equilibrium steady-state conditions (20 -27). Although equilibrium binding parameters vary significantly among vitamin K-dependent proteins, adsorption parameters are similar, suggesting nonspecific initial contact occurs (28,29). Anionic phospholipid membranes modify the apparent kinetic parameters of coagulation reactions relative to kinetics in solution. The membrane effect is manifested by a large decrease in the apparent K m of substrates to values far below their respective plasma concentrations. The mechanisms by which this effect manifests itself during TFmediated coagulation remain speculative. Achieving useful time resolutions has been one of the main obstacles to developing experimental systems to study the presteady-state transients of coagulation factor adsorption and activation on cell membranes. Blood clotting in vivo and in vitro can be completed faster than the sampling intervals of traditional batch systems used to measure membrane reactants.
Measurements of fVIIa binding and fXa generation on intact cell membranes under steady-state conditions indicate that TF⅐fVIIa functional activity is fully expressed before the binding interaction between fVIIa and TF reaches equilibrium (24,25). Furthermore, under steady-state conditions, the overall rate of coagulation substrate activation on membranes preequilibrated with enzyme was close to the theoretical collisional limit (27). These findings suggest intermediate noncovalent steps on the membrane linking the initial adsorption step to the assembly and catalysis of substrate in the A-E-S (activator-enzyme-substrate).
Coagulation zymogens and active proteases are subject to local microcirculation controls (3,(7)(8)(9)(10)14), because they are found in extravascular lymphatic, synovial, and alveolar fluids. The importance of flow control in coagulation reactions has been demonstrated in vivo. Tracer studies with radiolabeled fibrinogen and vasoactive agents indicate a direct correlation between changes in vascular permeability and fibrin deposition (3). Several studies using lipid-coated capillaries also indicate that flow rates influence the activity of coagulation proteases (30,31). In the present study, we use high resolution tracerdilution analyses (32)(33)(34)(35) along with numerical modeling to identify the surface and flow-dependent kinetics of fX activation via the TF pathway. We show that the generation of fXa from plasma fX proceeds via an intermediate step within the membrane. For reactions initiated with fVIIa and fX, the rate of this step and of the overall reaction is limited by the transfer of fX from the adsorption sites to the catalytic sites on the cell's surface.

EXPERIMENTAL PROCEDURES
Cell Culture and Reaction Chambers-Cells and cell cultures have been described and characterized in detail elsewhere (29,34). Briefly, Vero cells (American Tissue Type Collection) were grown to confluency on microcarrier beads (Cytodex 2, Amersham Pharmacia Biotech) of 150-m average diameter. Reaction chambers were assembled with a 2to 3-ml suspension of cell-covered microspheres loosely packed in a thermoregulated column fitted with flow adapters. In some experiments, reaction chambers were assembled using microcarriers without cells. In these control experiments, the naked microcarriers were incubated and processed in the same way as the cell-covered microcarriers. A schematic of the reaction flow chamber is shown in Fig. 1 below.
Cell viability and metabolic integrity, as demonstrated by amino acid uptake, were maintained for periods exceeding the kinetic measurements described here (34). The cells expressed surface-TF constitutively and interacted functionally with human fVIIa and fX with apparent steady-state kinetic parameters similar to other procoagulant cells, as reported previously (29,35). The TF activity of the monolayer suspension in the reaction chamber was equivalent to 6149 Ϯ 847 fmol/ml recombinant TF reconstituted into phospholipid vesicles. The chambers were perfused with medium (M-199, Life Technologies, Inc.), buffered with HEPES, and supplemented with 0.1 mM ovalbumin and 3 mM CaCl 2 . The overall geometrical parameters and TF content in the reaction chamber are summarized in Table I below.
Tracer Dilution Techniques-Concentrations of reactants and products in the bulk, flowing, and aqueous phases and on the cell surface during fXa-generating reactions were measured using double-and triple-tracer techniques. These techniques were adapted from previously described and validated methods in perfused organs (32-34, 36, 37). Coagulation reactions were carried out at physiological concentrations, under flowing conditions. Reactions were initiated by injecting 48 -500 ng of fVIIa and 3000 -9000 ng of fX into a 1.23-ml reaction chamber. The reactants were rapidly (1-2 s) delivered via a 100-l bolus and were quickly flow-dispersed in the 1.23-ml reaction chamber to attain maximal initial concentrations of ϳ0.8 -8 nM fVIIa and 45-135 nM fX. A control tracer of 14 C-labeled ovalbumin (833 ng) was also included in the bolus injection yielding an initial maximal concentration of ϳ14 nM. To measure membrane concentrations, 3 H-labeled fVIIa was used as an adsorbing tracer. Reactants and tracers were in a perfusing medium containing 0.1 mM unlabeled ovalbumin and 3 mM CaCl 2 .
After introducing reactants into the chamber via the inflow tubing, the effluent was collected in 72 samples of 46 Ϯ 2.8 l each at time resolutions ranging from 2 to 10 s per sample, depending on the perfusion rate. At a flow rate of 13 l/s, the typical flow velocity in the reaction chamber was estimated at Շ0.2 mm/s. This value is within the ranges measured in rabbit microvasculature in vivo (38).
Samples were collected in microtiter wells preloaded with Tris buffer solution, pH 8.3, containing 0.2 M EDTA and 0.8 M NaCl. A final 5-ml sample was collected to complete the recovery of tracer and to determine perfusion rates. Standard curves for 14 C and 3 H tracers were constructed from serial dilutions of the solutions used in the bolus injection. Concentrations of control and test radioactive tracers in standard dilutions and effluent samples were measured by scintillation counting. The concentration of product, fXa, was measured by amidolytic assays (13,14) with chromogenic substrate (methoxycarbonyl-Dcyclohexylglycil-arginine-p-nitroanilide acetate) before scintillation counting of radioactive tracers.
Functional Tests for TF Activity in Cells-Functional activity of TF was determined from fXa generation rates in purified systems using amidolytic assay and recombinant proteins as standard. Recombinant TF used to construct standard curves was relipidated into 30:70 phosphatidylserine/phosphatidylcholine vesicles as done before (17,31,39). The molar ratio of TF/lipid in the standard TF-phosphatidylserine/ phosphatidylcholine preparations was 1/4100. The TF activity measured in intact monolayers was 68.5 Ϯ 19% of that measured in monolayers lysed by freezing/thawing. Radioactive Tracers-The control tracer used to measure concentrations in bulk aqueous phase was 14 C-labeled ovalbumin (Sigma, St. Louis, MO) with a specific activity of 33 Ci/mg. The test tracer for adsorption measurements was fVIIa radiolabeled with tritium using the technique of Van Lenten and Ashwell (40), with modifications (41). Labeled preparations had specific activities of 1.8ϫ10 8 cpm/mg of fVIIa and a functional activity comparable to unlabeled factor in clotting tests and activating mixtures with purified components (29).
Reaction Scheme and Mathematical Model-The surface reactions leading to fXa generation were analyzed according to the following scheme, where E ϵ fVIIa⅐TF is the fVIIa and TF complex ("enzyme") that forms and dissociates with rate constants k ϩE and k ϪE , respectively. The substrate⅐enzyme complex denoted by E⅐fX associates and dissociates with a second-order rate constant, k ϩa , and a first-order rate constant, k Ϫa , respectively. The effective rate of product (fXa) formation from the complex and its irreversible release are denoted by the first-order rate constant, k ϩ . Denoting the surface concentrations of each species by (⌫ 1 , ⌫ 2 , ⌫ 3 , ⌫ 4 , ⌫ 5 , ⌫ 6 ) ϵ (fVIIa, fX, TF, E, E⅐fX, fXa), the full kinetic equations consistent with Eqs. 1 and 2 are where ⌫ i (t) ϵ d⌫ i (t)/dt, with i ϭ (1, 2, 3, 4, 5, 6). The constants k ϮE and k Ϯa correspond to effective rates of fVIIa-and fX-binding interactions with TF and E, respectively. These coefficients include the time delays of all intermediary processes on the membranes before the interactions. The time distributions of these unspecified processes are accounted for in our numerical predictions of the time course for the overall fX activating process. The possibility of inhibition or fX/fXa-destroying sinks is precluded from our data, because, within experimental error, all the absorbed fX is recovered as fXa.
In the above nonlinear differential equations, ␤ i and ␣ i are desorption and adsorption rates, respectively. Because the total area fraction of adsorbed species is negligible under our experimental conditions, species adsorption from bulk is simply proportional to the bulk concentration, C i (t), at the surface of each microsphere. As the fluid passes through the ensemble of microcarriers, certain flow lines are faster or slower than the mean flow velocity, resulting in a distribution of reactant velocities. A "lagged normal density curve" (LNDC) has been successfully used to approximate the dispersion resulting from the combined effects of random velocity distribution and molecular diffusion in the human circulatory system (36,37). We find good agreement between a fitted lagged density curve and the sequentially measured concentrations in the outflow of the reaction chamber (Fig. 3). Therefore, to simplify the modeling process, we assume that the dispersion and diffusion of all species are equal and use the LNDC to approximate the source, C i (t), surrounding each microcarrier. The parameters used in the lagged density curve will reflect chamber packing characteristics, bulk diffusion constants, and the imposed volume flow rate, J V . Additional details, analysis, and simplifications of Eqs. 3-8 are provided under "Appendix." Calculation of Reactant Concentrations in Membranes-The proportion of fX and fVIIa adsorbed from the flowing phase into the membrane was determined from the difference between the normalized concentrations of control, 14 C-labeled ovalbumin, and 3 H-labeled fVIIa. Concentrations of factor VIIa adsorbed at time t were estimated using where [ 14 C(t)] and [ 3 H(t)] are the fraction of the total nonadsorbed control and adsorbed test tracer, respectively, collected in the effluent. Q T (in femtomoles) is the total amount of fVIIa added, and S T is the total membrane surface area (Ϸ41-47 cm 2 ). Previous studies measuring adsorption of various coagulation factors, including fVIIa and fX, indicated that adsorption rates are proportional to their aqueous-phase concentration and not significantly different among vitamin K-dependent proteins (28,29). Based on these data, the normalized concentration of 3 H-labeled fVIIa was used to trace both fVIIa and fX adsorption. For substrate fX, the membrane concentration ⌫ 2 (t) was determined by where Q T is the total amount (fmol) of fX added, and [fXa(t)] is the fraction of that total released into the aqueous phase as fXa.
Miscellaneous-Tissue factor antigen expressed by the cells was determined in cell lysates using a commercial enzyme-linked immunosorbent assay kit with recombinant soluble TF as standard (American Diagnostics). Protein determinations were performed in the same cell lysates with a commercial reagent (Bio-Rad Laboratories), using bovine serum albumin as standard. Coagulation fVIIa was human recombinant, kindly donated by Dr. Ulla Hedner (Novo Nordisc, Denmark). Recombinant tissue factor (used as standard in enzyme-linked immunosorbent assay and functional TF determination in cells) was purchased from American Diagnostics. Human fX and fXa were purchased from Enzyme Laboratories. Data reduction, plotting, and statistical analyses were performed using StatView software (Brain Power, Inc.). Numerical solution of the nonlinear kinetic Eqs. 3-8 was performed using adaptive Runge-Kutta method implemented through Matlab.

Rate of fXa Generation at Different fVIIa Concentrations-
The generation of fXa from fX on live procoagulant cell membranes was examined in reaction chambers filled with spherical cell monolayers. The geometrical and flow characteristics of these reaction chambers are summarized in Fig. 1 and Table I, respectively. The distribution of concentrations of reactants in the flowing bulk aqueous phase was followed using control tracer [ 14 C]ovalbumin. Reactions were initiated with fX and fVIIa, and the product, fXa, was measured by amidolytic assay. Reactions were followed until 70 -90% of the nonreacting control tracer was recovered in the effluent. The amounts of 14 C and 3 H tracer collected and the amount of 3 H adsorbed to the cell are shown as functions of time in Fig. 2A.
The time evolution of reactants inside the reaction chamber can be fairly well approximated by the LNDC, as shown in Fig.  3. This agreement indicates that dispersion of reactants in the chamber due to the random flow distribution and diffusion is qualitatively similar to that encountered in human circulation (36 -38, 43, 46 -47). Fig. 3 also illustrates the time/concentration distribution of fXa generated in the 1.23-ml reaction chamber in a typical reaction initiated with 500 ng of fVIIa and 9000 ng of fX. No fXa was detected in the control experiments where microcarriers were used in the absence of cells.
Under the conditions of these experiments, aqueous-phase concentrations ranged from 0.3 to 10 Ϯ 2 nM and from 4 to 137 Ϯ 29 nM for fVIIa and fX, respectively. A time trace measuring the total amount of fXa collected and fX adsorbed on cell membranes is shown in Fig. 2B. Factor Xa profiles were weakly sigmoidal with a linear middle segment. Table II shows that the average fXa production rate, calculated from the linear segment, did not change when average concentrations of fVIIa in aqueous phase were decreased by 10-fold, from 5 to 0.5 nM (membrane concentrations ranged from 0.7 to 25.0 fmol/cm 2 ). No fXa was generated in the absence of fVIIa, and reaction rates did not differ significantly when fVII was substituted for fVIIa. The observation that maximal constant catalytic activity is reached at very low concentrations of TF⅐fVIIa complexes allows for simplifying substitutions in the model equations (Eqs. 3-8) for ⌫ 1 and ⌫ 4 ("Appendix").
Comparison between the Reaction Rate and the Theoretical Collisional Rate-The independence of average reaction rates and enzyme concentrations suggests that substrate transfer to the catalytic sites is rate-limiting. Two possibilities were investigated: (i) the rate-limiting step may occur during the adsorption of reactants from bulk to the cell membranes, and (ii) the rate-limiting step occurs after the adsorption step. To differentiate these two possibilities, the rate of fXa generation was  b Tissue factor antigen was measured by enzyme-linked immunosorbent assay in detergent-lysed monolayers. Activity equivalent was estimated from TF activity in suspensions of intact and lysed (by one freeze/thaw cycle) cell monolayers relative to the activity of standard dilutions of human recombinant TF reconstituted in phospholipid vesicles (30:70, phosphatidylserine/phosphatidylcholine). The TF activity in suspensions of intact cell monolayers was 68.5 Ϯ 19% of that in suspensions of lysed monolayers. compared with the theoretical collisional rate between reactants and microspheres, given the aqueous phase concentrations of fX used and the flow rates, J V , imposed. Because average adsorption rates were shown to approach or to exceed the collisional limit (29), activation rates below this limit support the second possibility.
Theoretical steady-state collisional rates were calculated from the aqueous-phase concentrations of fX and the radius of the spherical microcarriers using Smoluchowski's relationship for steady-state diffusion (42,57), where k coll is the rate of collisions between reactant molecules and a unit area of membrane (collisions/cm 2 /s), D 1,2 is the diffusion constant for fVIIa, fX in water (ϳ5 ϫ 10 Ϫ7 cm 2 /s), R Ӎ 7.5 ϫ 10 Ϫ3 cm is the microcarrier radius, and C 1,2 is the fVIIa, fX concentration (molecules/cm 3 ). Fig. 3 contrasts the number of fXa molecules released by the monolayer and the aqueous phase concentration, C 2 (t), of fX as a function of time. Because the collisional rate follows Eq. 11, collision-limited rates are expected to be directly proportional to C 2 (t). However, the rate of fX activation on the monolayer was not correlated with fX-membrane collisions. The rate of fXa production (molecules/cm 2 /s) reached maximal values after the peak in C 2 (t) and collisional rates. Furthermore, high fXa rates were sustained during the rapid decrease in collisions between fX and the membrane, following the concentration peak. Averaged over 13 experiments, the activation rate was 3.05 Ϯ 0.72 fmol/cm 2 /s, corresponding to 1.8 Ϯ 0.43ϫ10 9 (molecules/cm 2 /s), below the theoretical maximum of 2.4 Ϯ 0.57 ϫ 10 10 (collisions/cm 2 /s).
Apparent second-order rate coefficients, calculated from initial reaction rates and aqueous-phase concentrations of fVIIa and fX, did not have a constant value but increased continuously during the observation time. Using aqueous-phase fX and membrane-phase fVIIa to calculate apparent second order rate coefficients also resulted in increasing coefficient values, consistent with the observation that the average reaction rates are essentially independent of fVIIa concentration (Table II). These results indicate that the initial adsorption from bulk to membrane is not rate-limiting in the overall fXa generation reaction. The results also imply that fX is activated via a membrane-bound intermediate rather than directly from the bulk aqueous phase.
Mean Transit Times of fX through the Reaction Chamber-The existence of a slow membrane step was further investigated by comparing average transit times of the fXa generated in the chamber to the transit times of preformed fXa. The presence of a rate-limiting step between membrane adsorption  a Membrane density and adsorption rate of fVIIa were derived from the flux and adsorption rate coefficients previously measured (29). Values are the initial and final concentrations measured during the linear interval of the progression curve. Tissue factor density in these experiments was estimated at Ͼ25 fmol/cm 2 from cell protein and specific activity assays.
b Average rate was calculated from the concentration of fXa measured by chromogenic assay in samples collected during the linear segment of the reaction progression curve.
c Average fluxes of fVIIa and fX in each experiment were determined from the concentration of 14 C-control tracer in the effluent. The total amounts of fVIIa and fX initially added to the 1.23-ml reaction chamber were 48 -500 ng and 2800 -9000 ng, respectively. and catalytic cleavage is expected to delay the transit of the substrate that is adsorbed and catalyzed as compared with bulk aqueous-phase reactants. A mean transit time, T D , was determined from the concentration of fXa and control tracer in 72 consecutive samples of the effluent according to the expression, where [*C(t n )] is the fraction (of the total amount added or theoretical maximum) of either control tracer or fXa collected in aliquot n at time t n Ӎ n ϫ (2-8) s, depending on the particular experiment. Results shown in Table III indicate that the mean transit time of fX, T D (fX), activated in the reaction chamber is increased relative to the T D of aqueous-phase control tracer. In contrast, T D (fXa) for fXa formed before being introduced in the reaction chamber is indistinguishable from that of the control tracer. Furthermore, the increase in T D (fX) is inversely correlated with flow rate. These results are consistent with a slow membrane step following the fast, flow-dependent adsorption step. Kinetic Modeling of Surface Reactions-The hypothesis that the reaction pathway proceeds with fast equilibration of enzyme activity followed by a rate-limiting step involving reactant surface diffusion was also tested by comparing experimental measurements to the solutions of the kinetic equations (Eqs. 3-8). Eqs. 3-8 were solved numerically using initial estimates for intrinsic rate constants based upon results of previous steady-state kinetic studies (15,20,21,(25)(26)(27). Heuristic arguments for initial guesses for all the rate parameters are provided under "Appendix." A continuous function for aqueous-phase concentrations C i (t) is derived from a least-squares fit to a LNDC ("Appendix") shown in Fig. 3. The remaining parameters in the model were then adjusted until the best visual fit of ⌫ 6 (t) to fXa collected was achieved. Because ⌫ 2 (t) was indirectly measured and subject to larger experimental errors, we used only varied rate parameters to get an order-ofmagnitude agreement between the measured fX (Fig. 3) and ⌫ 2 (t) (Fig. 4B), using measured fXa and ⌫ 6 (t) to more precisely fit the parameters. The solutions and the associated best-fit parameters are shown in Fig. 4. We found that the magnitudes of ⌫ i (t) match the measurements only when the amount of TF assumed in the simulations was 0.32 fmol/cm 2 , much smaller than the actual amount expressed on the cell membranes. This finding is consistent with our hypothesis that enzyme complexes form domains, which is further developed under "Discussion." The model also shows that, within reasonable ranges, the shape and magnitude of the product curve, ⌫ 6 (t), are sensitive to k ϩa , k ϩ and ␣ i , ␤ i , but less sensitive to the other parameters. If the association step, k ϩa , was fast, the theoretical model would predict a premature overproduction of fXa, as shown in Fig. 5A. The sensitivity to a slow intermediate membrane step associated with k ϩa is shown in Fig. 5C, and the corresponding predicted values for effective enzyme, TF, and fX on the membranes are shown in Fig. 5D. Note also that for parameters differing from those used in Fig. 4, the magnitudes of ⌫ 2 (t) change dramatically and are no longer close to the measured fX (Fig. 3). For the parameters used to fit the measurements in Fig. 2, the numerical solution for E(⌫ 4 ) plateaus to a value ϳ⌫ 4 * after t ϳ 40 s and remains nearly constant for the duration of the 300-s interval under consideration. This quasi-steady state exists even for the cases where k ϩa is too large (Fig. 5B) or too small (Fig. 5D). We show under the "Appendix" that this quasisteady-state behavior allows us to define an approximate effective rate constant (s Ϫ1 ), which approximately describes the rate of fXa production on the cell membranes via, ⌫ 6 Ϸ k eff ⌫ 2 ͑t͒ ϩ adsorption/desorption terms (Eq. 14) For large k ϩ (the fast chemical step), k ϩa becomes the limiting rate, because k eff Ϸ k ϩa . The experimental data are consistent with model predictions both qualitatively and quantitatively, when k ϩa is in the range expected for lateral diffusion of proteins on membranes. An estimate for a mechanistically relevant diffusion length can be derived from where D 2 surf is the surface diffusion constant of fX in the cell  [3][4][5][6][7][8] was solved numerically, and the associated parameters were adjusted to obtain the best visual fit between computed and experimentally measured fXa concentration curves. A, bulk fX concentration (0.1 C 2 (t), black) and product fXa (⌫ 6 , line fitted to data points). The approximate parameters achieving the best fit are: k ϩ ϭ 15, k ϩa ϭ 0.06, k Ϫa ϭ 6, k ϩE ϭ 0.06, k ϪE ϭ 0.0005, ␣ 1,2 ϭ 0.8, ␣ 6 ϭ 0.1, ␤ 1 ϭ ␤ 2 ϭ 0.001, and ␤ 6 ϭ 0.12. The amount of TF present in accessible enzyme complexes was assumed to be 0.32 fmol/cm 2 . B, corresponding surface concentrations ⌫ 2 /100, ⌫ 3 , ⌫ 4 , and ⌫ 5 in fmol/cm 2 . membranes. Using a typical value D 2 surf ϳ 10 Ϫ10 cm 2 /s (45), and k eff ϳ 0.02 s Ϫ1 obtained from the mathematical model, we find that ᐉ D ϳ 0.7 m. However, ᐉ D can be shorter if obstructions in the membrane hinder surface diffusion and reduce D 2 surf . Factor Xa Generation Rate as a Function of Surface Density of Reactants-The observations described in the previous sections indicate that reaction rates are not directly related to the aqueous-phase concentration of substrate. To further investigate the rate-limiting step, we analyzed reaction rates as a function of flow rates and membrane concentrations of substrate.
The instantaneous fraction of membrane fX converted to fXa was not directly proportional to the fX concentration on the membrane. Instead, at all flow rates tested, it increased linearly with time. Interestingly, the increase in the proportion of adsorbed fX encountering catalytic sites per unit time was essentially independent of flow. This observation suggests that adsorbed fX is not immediately available to the catalytic sites and is consistent with a rate-limiting surface diffusion process. That the fraction of membrane fX available for catalysis increases with time suggests diffusive transfer of fX from initial adsorption sites to the catalytic sites. The value of the slope of the function ⌫ 6 (t)/⌫ 2 (t) was 0.001-0.0005 s Ϫ1 and largely independent of flow rate (Table IV).
Table IV also lists the product yield and average reaction rates measured at different flow rates. The yield was strongly correlated with flow rate (correlation coefficient, 0.86), whereas the average reaction rate was independent of flow rate. Again, these results are as expected for a kinetic mechanism that includes a flow-dependent adsorption step followed by a ratelimiting intermediate step on the membrane. DISCUSSION In this paper, we have analyzed the surface reaction kinetics of blood coagulation initiated by interaction among coagulation factors on biological membranes. Reactions were initiated on live epithelial cells expressing TF with physiological concentrations of fVII⅐fVIIa and fX. The aqueous and membrane concentration of reactants as well as product formation were measured at time resolutions relevant to plasma clotting. Although adsorption of vitamin K-dependent proteins on procoagulant membrane surfaces has been shown to be fast and correlated to aqueous-phase flux (29), the rate of fXa generation was independent of both enzyme density on the membrane and flow rate, J V . Moreover, using tracer dilution analyses we found that the transit time of the fX participating in the reaction was prolonged relative to transit times of nonreacting control tracers. Rate coefficients calculated from reaction rates and either the aqueous or membrane concentration of reactants changed with time. Flow velocities influenced the total amount of reactants adsorbed to the membrane and the total yield but not the intrinsic rate of product formation. Taken together, the experimental results provide evidence for a kinetic mechanism limited by a slow transfer of substrate between initial membrane adsorption sites and reaction sites. The hypothesis of a slow membrane step was further tested by numerically solving a set of nonlinear kinetic equations describing the evolution of all membrane reaction species. The experimental results were reproduced when the rate-limiting step followed the adsorption of substrate to the membrane and preceded the chemical catalysis. Testing the alternative hypothesis of either slow adsorp-FIG. 5. Model predictions for alternative reaction mechanisms. A, the predicted fXa generation as a function of time if the surface diffusion is much faster (k ϩa ϭ 1.0) than that assumed in the simulations depicted in Fig. 4, with all other parameters identical to those used in Fig. 4. B, the corresponding surface concentrations. Note that membrane fX, ⌫ 2 , is much smaller than that in Fig. 4 and estimated from measurements (not shown). C, the predicted fXa production if the surface diffusion was slower than optimal. With k ϩa ϭ 0.01, factor Xa is generated in lower quantities and at later times. D, the predicted ⌫ 2 , however, is much greater than that observed. a The fraction of membrane-bound fX converted to fXa increased linearly with time giving the slopes (roughly 0.001 s Ϫ1 ) indicated. The density of fX on the membrane was calculated from the difference between fX adsorbed and fX released as fXa. The concentration of the fXa released was measured directly in the effluent samples by amidolytic assay.
b The yield of fXa is expressed as the percentage of the total fX added to the reaction chamber. Average reaction rates were determined from the steepest linear segment of the progression curves as in Fig. 2B. tion or slow catalytic steps resulted in product yield and profiles markedly different from those observed experimentally.
Maximal surface catalytic activity was observed when the membrane concentration of fVIIa was at least two orders of magnitude lower than the surface concentration of TF, estimated either by immunoassay or functional tests. The experimentally measured fXa levels best matched solutions of the kinetic equations (Eqs. 3-8) when the intrinsic rate constants shown in Fig. 4 were used along with a maximal enzyme concentration of ⌫ 4 Ӎ 0.32 fmol/cm 2 . This observation suggests that only a fraction of the available membrane E ϭ fVIIa⅐TF is involved in catalysis as would be expected if catalytic sites were in large molar excess over the substrate. However, the substrate concentration reached, Ͼ100 fmol/cm 2 , is much higher than TF or enzyme concentrations.
These results can be explained using a model in which the enzyme concentration, ⌫ 4 , is in relative local excess, and only a fraction of the surface enzyme effectively participates in catalysis. If enzyme complexes fVIIa⅐TF form domains, the fast catalytic cleavage reaction would be expected to occur only near the perimeter of these enzyme domains, where the substrate initially encounters the enzyme after diffusing a typical distance, ᐉ D , following adsorption. The interiors of these domains are rarely accessed by the rapidly converted substrate and therefore do not participate in the overall reaction. This process is shown schematically in Fig. 6.
Surface segregation of molecules is a common phenomenon. It has been shown that even small molecules can form domains in lipid monolayers (44,48,49) and bilayer vesicles (52). A large body of experimental research also provides evidence that protein and lipid domains exist on live cell membranes (50,51). For example, adhesion molecules localize at sites of cell-cell contact, and receptors are often found concentrated at the tips of filopodia and lamelopodia of moving cells. Neurotransmitter receptors in postsynaptic terminals have also been shown to form dynamic aggregates (53,54). Direct visualization reveals localization of certain proteins in areas of high or low curvature in artificial vesicles (55). We have evidence of the existence of domain formation on the membrane of the epithelial cell line used in these studies. Using gold immunochemistry on cells fixed after a short exposure to fVIIa, the enzyme was localized primarily on the ruffled border of the cell membrane. Furthermore, analysis of nearest-neighbor distances indicated a nonrandom distribution of the enzyme (29). Although the mechanisms of domain formation are unknown, possibilities may involve electrostatic or dipole-induced phase transitions (48) and membrane elasticity-induced protein-protein attractions (56).
The flow characteristics and time resolution achieved with this experimental system are relevant to reactions on biological membranes after exposure of TF to flowing plasma coagulation proteins. The experimental approaches and mathematical model used here to identify the early kinetic mechanisms of fXa generation will also be useful for studying novel pharmacokinetic mechanisms occurring on biomembranes. APPENDIX In this section, we give details of the mathematical model (Eqs. [3][4][5][6][7][8] and the associated approximations used in its analysis. Because no measurable amount of fXa is generated by fVIIa and fX in bulk solution, we have assumed that chemical reactions can only occur when molecules are adsorbed on the surface of each cell-covered sphere.
The kinetic equations (Eqs. [3][4][5][6][7][8] are solved numerically using finite difference approximations. All surface densities, ⌫ i , are in units of fmol/cm 2 , whereas all bulk concentrations, C i , are measured in units of pmol/cm 3 . With this convention, the rates take on the following units: We assume that the adsorption rates of species from the bulk onto the microsphere surfaces are proportional to the local bulk concentration. For the sake of completeness, and to motivate more quantitative modeling, we write the governing equations for surface adsorption of reactants. In the bulk phase, the concentration of species i follows the convection-diffusion equation, where C ϭ C 1 , C 2 , C 6 , and D i are the bulk solution concentrations (number per volume) and associated diffusion constants of fVIIa, fX, and fXa, respectively. Although a closure relation is required to specify the detailed velocity field, V, around each sphere, we will assume that the identical microspheres each are subject to an equivalent, averaged, effective flow velocity, V. The boundary conditions within continuum theory at the sphere surface are found by balancing the diffusive flux with the desorption and adsorption rates on the surface of each microsphere (at r ϭ R), where the area fraction available for adsorption ⌽ Ӎ 1 at low coverage. The above equations constitute the exact continuum equations for the species in solution. The complete set of equations consists of the convection-diffusion equation (Eq. 16), the boundary conditions (Eq. 17), and the surface reaction equations (Eqs. [3][4][5][6][7][8]. Significant simplifications and decoupling of some of Eqs. 3-8 can be realized by assuming that the bulk concentrations at the spheres' surfaces C i (R,t) can be approximated by the LNDC (37). As the fluid passes through the ensemble of cellcovered microspheres, certain flow lines are faster or slower than the mean flow velocity. The reactants in the aqueous phase are randomly advected through the microsphere chamber at a distribution of velocities. The LNDC is a convolution of random advection velocities with molecular diffusion and has been used to approximate advection-diffusion in blood flow (37). Although the source concentration, C i (t), may depend on the position of the microsphere within the reaction chamber, we assume that the dispersion and diffusion of all species are equal and use the lagged density curves to approximate the source, C i (t), surrounding each microsphere. The parameters used in the LNDC depend upon average microsphere packing, the bulk diffusion constants, and the imposed constant volume flow rate, J V . The concentration of the ith species in a random flow environment is assumed to obey where m i is the total number of femtomoles of species i added via the bolus injection, and J V is the constant flow rate measured in l/s. The intrinsic delay time, T i , is inversely related to the mean ͉V͉, whereas i and i describe the width and effects of molecular dispersion, respectively. The amount of spreading embodied in i is proportional to the bulk diffusion constant, D i . The solution to the initial value problem (Eq. 18) is The concentration, C i , above is given in nanomolar units. These solutions determine the sources, ␣ i C i (R,t), for the surface kinetic equations (Eqs. [3][4][5][6][7][8]. We take the entire reaction chamber and the inlet and outlet tubes to constitute a single, effective flow system. The zero used in Eq. 19 corresponds to the time when nonbinding species are first detected (for the experiment in Fig. 3, ϳ36 s after adding reactants). Upon fitting (by adjusting i , i , T i until a local minimum in the least-squares is found) the parameters in Eq. 19 to the concentration, we find (for this experiment at J V Ӎ 13.4 l/s) that i Ϸ 21 s, Ϸ 91 s, and T Ϸ 19 s for i ϭ 1, 2. The fitted LNDC is shown in Fig. 3.
The initial rate parameters used to solve Eqs. 3-8 were estimated as follows. The absorption rates of fVIIa and fX, from previous studies, were found to be similar and close to the collisional limit (29). Here, we first assumed that these adsorption rates were diffusion-limited. Such high absorption rates were needed to obtain the right magnitudes of fXa formation, regardless of the other rate parameters. Enough reactant must simply reach the membranes within the time limit imposed by the flow rate, J V . The maximum rate of particles reaching and being absorbed into the membrane of an isolated microsphere's surface, in the absence of flow, is given by This upper limit assumes that every molecule coming into contact with the sphere is absorbed. With a diffusion constant of D i ϳ 7.5 ϫ 10 Ϫ7 cm 2 /s, the absorption rate under zero flow conditions is approximately ␣ i Ͻ ϳ1000 D i /R Ϸ 0.1 cm/s (the factor 1000 converts pmol/cm 3 to fmol/cm 3 ). Now consider the effects of advection due to the imposed volume flow, J V . Purcell (57) gives an expression for the flux to a spherical surface under flow, Although at first glance the V 1/3 dependence is weak, quantitative changes in ␣ due to flow can dramatically influence the yield of the surface reactions. For the density of microspheres and flow rates used, we estimated the typical velocity in the reaction chamber to be V Ϸ J V /A eff Ϸ 0.02 cm/s, where J V Ӎ 13.4 l/s, and the reaction chamber effective cross-sectional area A eff Ͻ (0.7 cm) 2 due to partial obstruction. Therefore, we used ␣ 1,2 ϳ 0.6 cm/s as an initial guess for the absorption rates in the reaction scheme (Eqs. [3][4][5][6][7][8]. For the association rates k ϩa and k ϩE , we used the surface diffusion of fVIIa and fX to set upper limits. For a surface diffusion constant of D i surf ϳ 10 Ϫ10 cm/s, we found that k ϩa ϳ k ϩE Ͻ 0.1 cm 2 /(fmol⅐s). Moreover, from previous aqueous-phase equilibrium binding studies, k ϪE /k ϩE ϳ 10 Ϫ10 M (24 -26). To estimate the corresponding ratio for two-dimensional reactions, we made a qualitative estimate by assuming the energetics of E formation are not significantly different from those in bulk. Thus, 10 Ϫ10 M corresponds to a typical particle-particle distance of 2.5 m. Translating this to a surface density, we estimated very roughly that, for the surface enzyme formation reaction, k ϪE /k ϩE ϳ 10 Ϫ2 fmol/cm 2 . Finally, from previous steady-state measurements on cell membranes at saturated concentrations of fX, k ϩ Ͼ ϳ14/s (25). Within these limits, we explored the parameter space to obtain a reasonable fit to the data. The data and the fit of ⌫ 6 (t) calculated from Eqs. 3-8 are shown in Fig. 4A. The concentration of product collected, Q 6 (t), is calculated from Q 6 ͑t͒ Ϸ ␤ 6 ⌫ 6 ͑t͒S T /J v (Eq. 22) where S T Ӎ 47 cm 2 is the total membrane area in the reaction chamber. The membrane concentration ⌫ 2 is shown in Fig. 4B.
The results were consistent with the data in Table II and showed that, after a short initial transient, the concentration of membrane enzyme (⌫ 4 ) reaches a plateau. This behavior permitted simplification of Eqs. 3-8 and approximate analytic solutions for the surface concentrations ⌫ 2,6 (t). For large k ϩ ϩ k Ϫa , the concentration ⌫ 5 (t) (E⅐fX) is always small. From a typical simulation (Figs. 4 and 5) we observed a short transient in ⌫ 3 (TF). At times beyond this transient, ⌫ 3 Ϸ 0, and the enzyme concentration, ⌫ 4 , reaches a nearly steady value, From the equation for ⌫ 1 , we see that, over long time intervals, fVIIa approximately follows the adsorption and desorption processes, It is evident from Figs. 4 and 5 that ⌫ 4 (E) also reaches a quasi-steady state shortly after TF. Therefore, setting ⌫ 4 Ϸ 0, The remaining time-dependent surface quantities at these quasi-steady-state times obey ͩ ⌫ 2 ͑t͒ ⌫ 6 ͑t͒ ͪϭͩ Ϫk eff 0 k eff Ϫ ␤ 6 ͪ ͩ ⌫ 2 ͑t͒ ⌫ 6 ͑t͒ ͪϩͩ ␣ 2 C 2 ͑t͒ ␣ 6 C 6 ͑t͒ ͪ (Eq. 26) where k eff given by Eq. 13 is the effective rate of conversion from fX to fXa during quasi-steady-state times when the enzyme concentration is ⌫ 4 Ϸ ⌫ 4 *. Assuming that ␣ 6 C 6 is negligible, Eqs. 26 admit analytic solutions, and, considering the approximate nature of our model, a further simplification can be made: Using Eq. 22, the second equation in Eq. 26 becomes Therefore, an independent measurement of ⌫ 2 and the collected product Q 6 can be used to estimate the unknowns ␤ 6 and k eff . Although in our analyses we have numerically solved the full kinetic equations (Eqs. 3-8), a simplified set of equations (Eqs. 24 and 26) provide an analytic model to the reaction kinetics for times beyond the initial short transient (t Ͼ ϳ40 s for the run shown in Fig. 4).