Surface-dependent Coagulation Enzymes
FLOW KINETICS OF FACTOR Xa GENERATION ON LIVE CELL
MEMBRANES*
Maria P.
McGee
§ and
Tom
Chou¶
From the
Department of Medicine, Wake-Forest
University School of Medicine, Winston-Salem, North Carolina 27157 and
the ¶ Department of Biomathematics, UCLA, Los Angeles, California
90095-1766
Received for publication, April 17, 2000, and in revised form, November 14, 2000
 |
ABSTRACT |
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 cm2 expressing tissue factor
(TF) at >25 fmol/cm2. 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/cm2, 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/cm2. 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.
 |
INTRODUCTION |
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 (1-6).
The blood coagulation process is initiated by an assembly of complexes
comprised of an essential cofactor,
TF1 (tissue factor) and a
protease component, fVIIa. The functional complex, TF·fVIIa,
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
(7-10), 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
(12-14).
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
(15-19). 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 Km of substrates to values far below their
respective plasma concentrations. The mechanisms by which this effect
manifests itself during TF-mediated 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 TF·fVIIa 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 pre-equilibrated 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, 7-10, 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 tracer-dilution analyses (32-35) 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.
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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-µm average diameter. Reaction chambers were assembled
with a 2- to 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 ± 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 CaCl2. 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-µl bolus and
were quickly flow-dispersed in the 1.23-ml reaction chamber to attain
maximal initial concentrations of ~0.8-8 nM fVIIa and 45-135 nM fX. A control tracer of 14C-labeled
ovalbumin (833 ng) was also included in the bolus injection yielding an
initial maximal concentration of ~14 nM. To measure membrane concentrations, 3H-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
CaCl2.
After introducing reactants into the chamber via the inflow tubing, the
effluent was collected in 72 samples of 46 ± 2.8 µl each at
time resolutions ranging from 2 to 10 s per sample, depending on
the perfusion rate. At a flow rate of 13 µl/s, the typical flow
velocity in the reaction chamber was estimated at
0.2 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 14C and 3H 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-D-cyclohexylglycil-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 ± 19% of that
measured in monolayers lysed by freezing/thawing.
Radioactive Tracers--
The control tracer used to measure
concentrations in bulk aqueous phase was 14C-labeled
ovalbumin (Sigma, St. Louis, MO) with a specific activity of 33 µCi/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 1.8×108 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,
|
(Eq. 1)
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|
(Eq. 2)
|
where E
fVIIa·TF is the fVIIa and TF complex
("enzyme") that forms and dissociates with rate constants
k+E and k
E, respectively. The substrate·enzyme complex denoted by
E·fX associates and dissociates with a second-order rate
constant, k+a, and a first-order
rate constant, k
a, respectively.
The effective rate of product (fXa) formation from the complex and its
irreversible release are denoted by the first-order rate
constant, k+. Denoting the surface
concentrations of each species by (
1,
2,
3,
4,
5,
6)
(fVIIa, fX, TF, E, E·fX, fXa), the full
kinetic equations consistent with Eqs. 1 and 2 are
|
(Eq. 3)
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(Eq. 4)
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(Eq. 5)
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(Eq. 6)
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(Eq. 7)
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|
(Eq. 8)
|
where
i(t)
d
i(t)/dt, with
i = (1, 2, 3, 4, 5, 6). The constants
k±E and k±a 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,
Ci(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,
Ci(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, JV. 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, 14C-labeled ovalbumin, and
3H-labeled fVIIa. Concentrations of factor VIIa adsorbed at
time t were estimated using
|
(Eq. 9)
|
where [14C(t)] and
[3H(t)] are the fraction of the total
nonadsorbed control and adsorbed test tracer, respectively, collected in the effluent. QT (in femtomoles) is the total
amount of fVIIa added, and ST is the
total membrane surface area (
41-47 cm2). 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 3H-labeled fVIIa was used
to trace both fVIIa and fX adsorption. For substrate fX, the membrane
concentration
2(t) was determined by
|
(Eq. 10)
|
where QT 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.
 |
RESULTS |
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 [14C]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 14C and 3H tracer collected and the amount
of 3H adsorbed to the cell are shown as functions of time
in Fig. 2A.

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Fig. 1.
Schematic of the reaction chamber and
cell-covered microspheres. The reactive surface in the reactor is
the surface of viable Vero cells grown to confluency on microcarriers
with the indicated dimensions. The microcarriers (~5-20 × 104) were packed in a thermoregulated column fitted with
flow adapters and perfused at constant flow rates of 5-25 µl/s.
Reactants and control tracer were added via the inflow (lower
diagram) and collected via the outflow (upper diagram)
tubing in 72-140 consecutive samples of 46 ± 8.9 µl each. For
most experiments, reactants were added as a rapid bolus and reactions
followed for 150-300 s by collecting samples at a resolution of 2-10
s per sample.
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Fig. 2.
Adsorption of reactants and factor Xa
generation under flow. The reaction chamber was maintained at
37 °C and perfused at 13.4 µl/s with HEPES buffered medium, pH
7.2, containing 0.15 N NaCl, 3 mM
CaCl2, and 0.1 mM nonlabeled ovalbumin. Maximal
initial concentrations of reactant were 8 nM
[3H]fVIIa and 130 nM fXa. The TF density on
the monolayer surface was estimated at >25 fmol/cm2 from
both functional and immunological assays. A, total amounts
of reactant (either fVIIa or fX, ) adsorbed to the monolayer were
determined from the difference between the normalized concentrations of
control tracer, 14C ( ), and test tracer, 3H
( ), collected in effluent samples. Tracer amounts were normalized as
the fraction of the total added to the reaction chamber. B,
the amount of fX on the membrane ( ) was determined from the
difference between fX adsorbed ( ) and fXa ( ) released. The
average rate of fXa generation for this experiment calculated from the
slope of the middle linear segment of the progression curve (100-150
s) was 2.1 ± 0.02 fmol of fXa/s/cm2. The mean from 13 similar experiments was 3.05 ± 0.72 fmol/s/cm2.
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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.

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Fig. 3.
The lagged normal density curve and
distributions of substrate and product concentrations. The
open circles correspond to concentrations of fX (in
nanomolar) in aqueous phase determined from the concentration of
nonreacting, nonadsorbing control tracer. The zero of the
time axis was chosen to correspond to initial detection of
14C. The qualitative fit to the lagged normal density curve
(LNDC) yields the parameters 2 21 s,
2 91 s, and T2 19 s. The filled circles correspond to the fXa
concentration released into each aliquot in the reacting
system (×10 in the figure to facilitate comparison with fX
values).
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Under the conditions of these experiments, aqueous-phase concentrations
ranged from 0.3 to 10 ± 2 nM and from 4 to 137 ± 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/cm2). 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 TF·fVIIa 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 compared with the theoretical
collisional rate between reactants and microspheres, given the aqueous
phase concentrations of fX used and the flow rates,
JV, 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),
|
(Eq. 11)
|
where kcoll is the rate of
collisions between reactant molecules and a unit area of membrane
(collisions/cm2/s), D1,2 is the
diffusion constant for fVIIa, fX in water (~5 × 10
7 cm2/s), R
7.5 × 10
3 cm is the microcarrier radius, and
C1,2 is the fVIIa, fX concentration (molecules/cm3).
Fig. 3 contrasts the number of fXa molecules released by the monolayer
and the aqueous phase concentration,
C2(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
C2(t). However, the rate of fX
activation on the monolayer was not correlated with fX-membrane
collisions. The rate of fXa production (molecules/cm2/s)
reached maximal values after the peak in
C2(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 ± 0.72 fmol/cm2/s, corresponding to 1.8 ± 0.43×109 (molecules/cm2/s), below the
theoretical maximum of 2.4 ± 0.57 × 1010
(collisions/cm2/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 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,
TD, was determined from the concentration of fXa
and control tracer in 72 consecutive samples of the effluent according
to the expression,
|
(Eq. 12)
|
where [*C(tn)] is the fraction (of
the total amount added or theoretical maximum) of either control tracer
or fXa collected in aliquot n at time tn
n × (2-8) s, depending on the
particular experiment. Results shown in Table
III indicate that the mean transit time
of fX, TD(fX), activated in the reaction chamber
is increased relative to the TD of aqueous-phase
control tracer. In contrast, TD(fXa) for fXa
formed before being introduced in the reaction chamber is
indistinguishable from that of the control tracer. Furthermore, the
increase in TD(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, 25-27).
Heuristic arguments for initial guesses for all the rate parameters are
provided under "Appendix." A continuous function for aqueous-phase
concentrations Ci(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-of-magnitude 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/cm2, 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."

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Fig. 4.
Kinetic modeling of surface reactions.
The set of nonlinear equations (Eqs. 3-8) 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 C2(t), black) and product
fXa ( 6, line fitted to data points). The
approximate parameters achieving the best fit are:
k+ = 15, k+a = 0.06, k a = 6, k+E = 0.06, k E = 0.0005, 1,2 = 0.8, 6 = 0.1, 1 = 2 = 0.001, and 6 = 0.12. The amount of TF present in accessible
enzyme complexes was assumed to be 0.32 fmol/cm2.
B, corresponding surface concentrations
2/100, 3, 4, and
5 in fmol/cm2.
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The model also shows that, within reasonable ranges, the shape and
magnitude of the product curve,
6(t), are
sensitive to k+a,
k+ and
i,
i, but less sensitive to the other parameters. If the association step,
k+a, 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+a 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).

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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+a = 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+a = 0.01, factor Xa is generated
in lower quantities and at later times. D, the predicted
2, however, is much greater than that observed.
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For the parameters used to fit the measurements in Fig. 2, the
numerical solution for E(
4) plateaus to a value
~
after t ~ 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+a is too large (Fig.
5B) or too small (Fig. 5D). We show under the
"Appendix" that this quasi-steady-state behavior allows us to
define an approximate effective rate constant (s
1),
|
(Eq. 13)
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which approximately describes the rate of fXa production on the
cell membranes via,
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(Eq. 14)
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For large k+ (the fast chemical step),
k+a becomes the limiting rate,
because keff
k+a. The experimental data are consistent with model predictions both qualitatively and
quantitatively, when k+a is in the
range expected for lateral diffusion of proteins on membranes. An
estimate for a mechanistically relevant diffusion length can be derived
from
|
(Eq. 15)
|
where D
is the surface
diffusion constant of fX in the cell membranes. Using a typical value
D
~ 10
10
cm2/s (45), and keff ~ 0.02 s
1 obtained from the mathematical model, we find that
D ~ 0.7 µm. However,
D can be shorter
if obstructions in the membrane hinder surface diffusion and
reduce
D
.
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
1 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 rate-limiting
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 fVII·fVIIa 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,
JV. 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 adsorption 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
0.32 fmol/cm2. This
observation suggests that only a fraction of the available membrane
E = fVIIa·TF is involved in catalysis as would be expected if
catalytic sites were in large molar excess over the substrate. However,
the substrate concentration reached, >100 fmol/cm2, 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 fVIIa·TF 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,
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.

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|
Fig. 6.
Diagram representing protein distribution on
the cell surface. If the formation of E·fX occurs upon nearly
each encounter of E and fX on the membrane surface, only E molecules
near the perimeters of the domains will participate in catalysis of fX
fXa.
|
|
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.
 |
FOOTNOTES |
*
This work was supported by National Science Foundation
Grants MCB-9601411, DMS-9804370, and NIH-HL57936.The costs of publication of this
article were defrayed in part by the
payment of page charges. The article
must therefore be hereby marked
"advertisement" in accordance with 18 U.S.C. Section
1734 solely to indicate this fact.
§
To whom correspondence should be addressed: Dept. of Medicine,
Wake-Forest University School of Medicine, Medical Center Blvd., Winston-Salem, NC 27157. Tel.: 336-716-6716; Fax: 336-716-9821; E-mail:
mmcgee@wfubmc.edu.
Published, JBC Papers in Press, December 6, 2000, DOI 10.1074/jbc.M003275200
 |
ABBREVIATIONS |
The abbreviations used are:
TF, tissue factor;
LNDC, lagged normal density curve.
 |
APPENDIX |
In this section, we give details of the mathematical model (Eqs.
3-8) 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. 3-8) are solved numerically using finite
difference approximations. All surface densities,
i, are in
units of fmol/cm2, whereas all bulk concentrations,
Ci, are measured in units of pmol/cm3.
With this convention, the rates take on the following units: [
] = [k
a] = [k
E] = [k+] = s
1,
[k+E] = [k+a] = cm2/(fmol·s), and [
] = 10
3
cm/s.
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,
|
(Eq. 16)
|
where C = C1,
C2, C6, and
Di 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 = R),
|
(Eq. 17)
|
where the area fraction available for adsorption
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. 3-8).
Significant simplifications and decoupling of some of Eqs. 3-8 can be
realized by assuming that the bulk concentrations at the spheres'
surfaces Ci(R,t) can be
approximated by the LNDC (37). As the fluid passes through the ensemble
of cell-covered 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, Ci(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,
Ci(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, JV.
The concentration of the ith species in a random
flow environment is assumed to obey
|
(Eq. 18)
|
where mi is the total number of femtomoles of
species i added via the bolus injection, and
JV is the constant flow rate measured
in µl/s. The intrinsic delay time,
Ti, is inversely related to the mean
|V|, 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, Di. The solution to the initial
value problem (Eq. 18) is
|
(Eq. 19)
|
The concentration, Ci, above is given in
nanomolar units. These solutions determine the sources,
iCi(R,t), for the
surface kinetic equations (Eqs. 3-8). 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, ~36 s after adding reactants). Upon fitting (by adjusting
i,
i, Ti until a local minimum
in the least-squares is found) the parameters in Eq. 19 to the
concentration, we find (for this experiment at
JV
13.4 µl/s) that
i
21 s,
91 s, and T
19 s
for i = 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, JV. 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
|
(Eq. 20)
|
This upper limit assumes that every molecule coming into contact
with the sphere is absorbed. With a diffusion constant of Di ~ 7.5 × 10
7
cm2/s, the absorption rate under zero flow conditions is
approximately
i < ~1000
Di/R
0.1 cm/s (the factor 1000 converts pmol/cm3 to fmol/cm3). Now consider
the effects of advection due to the imposed volume flow,
JV. Purcell (57) gives an expression for the
flux to a spherical surface under flow,
|
(Eq. 21)
|
Although at first glance the
V1/3 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
JV/Aeff
0.02 cm/s,
where JV
13.4 µl/s, and the reaction
chamber effective cross-sectional area
Aeff <
(0.7 cm)2 due to partial
obstruction. Therefore, we used
1,2 ~ 0.6 cm/s as an
initial guess for the absorption rates in the reaction scheme (Eqs.
3-8).
For the association rates k+a and
k+E, we used the surface diffusion of fVIIa and
fX to set upper limits. For a surface diffusion constant of
D
~ 10
10
cm/s, we found that k+a ~ k+E < 0.1 cm2/(fmol·s). Moreover,
from previous aqueous-phase equilibrium binding studies,
k
E/k+E ~ 10
10 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
10 M corresponds to a typical
particle-particle distance of 2.5 µm. Translating this to a surface
density, we estimated very roughly that, for the surface enzyme
formation reaction,
k
E/k+E ~ 10
2 fmol/cm2. Finally, from previous
steady-state measurements on cell membranes at saturated concentrations
of fX, k+ > ~14/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, Q6(t), is calculated
from
|
(Eq. 22)
|
where ST
47 cm2 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+ + k
a, the
concentration
5(t) (E·fX) 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
0, and the enzyme concentration,
4, reaches a nearly steady value,
|
(Eq. 23)
|
From the equation for
1, we see that, over
long time intervals, fVIIa approximately follows the adsorption and
desorption processes,
|
(Eq. 24)
|
It is evident from Figs. 4 and 5 that
4 (E) also
reaches a quasi-steady state shortly after TF. Therefore, setting
4
0,
|
(Eq. 25)
|
The remaining time-dependent surface quantities at
these quasi-steady-state times obey
|
(Eq. 26)
|
where keff 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
4*.
Assuming that
6C6 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
|
(Eq. 27)
|
Therefore, an independent measurement of
2 and the
collected product Q6 can be used to estimate the
unknowns
6 and keff. 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 > ~40 s for the run shown in Fig.
4).
 |
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Copyright © 2001 by The American Society for Biochemistry and Molecular Biology, Inc.
Copyright © 2001 by the American Society for Biochemistry and Molecular Biology.