(Received for publication, April 28, 1994; and in revised form, August 23, 1994)
From the
Thrombin production by the phospholipid-bound complex of blood
clotting factors Xa and Va (prothrombinase) was studied in a tubular
flow reactor. The inner wall of a glass capillary was coated with a
phospholipid bilayer of 25% phosphatidylserine and 75%
phosphatidylcholine. Prothrombinase was assembled on this bilayer by
perfusion with a mixture containing an excess of factor Va (2
nM) and a limiting amount of factor Xa (1-100
pM), either in the absence or presence of prothrombin. The
rate of assembly of prothrombinase in the presence of prothrombin
appeared to be limited by the transfer rate of factor Xa to the
phospholipid surface. A good agreement was found between the predicted
mass transfer coefficient for factor Xa and the observed pre-steady
state rate of thrombin production. The eventually obtained steady state
rates of thrombin production were proportional to the prothrombin
concentration and independent of the surface density of prothrombinase.
The observed rate of thrombin production was in excellent agreement
with the predicted mass transfer rate for prothrombin.
Transport-limited prothrombin conversion was observed for
prothrombinase densities exceeding 1 fmol/cm, which
corresponds to 0.05% occupation of available binding sites. The kinetic
parameters of the reaction were determined at low prothrombinase
densities (0.02-0.04 fmol/cm
). Even in this situation
the Michaelis-Menten equation had to be corrected for substrate
depletion near the catalytic surface. We hereto employed an accurate
approximation of the mass transfer coefficient. The kinetic parameter k
was 60 s
and the intrinsic K
had a surprisingly low value of 3
nM. Both parameters were not influenced by the wall shear
rate.
The process of blood coagulation encompasses a complex system of enzymatic reactions in flowing blood at the surface of damaged blood vessels. One of the final steps in this process is the conversion of prothrombin to thrombin(1, 2) . The activator of prothrombin is a multicomponent complex that consists of the serine protease factor Xa and the non-enzymatic cofactor factor Va. In the presence of calcium ions, the complex of factor Xa and factor Va assembles on a membrane containing negatively charged phospholipids, i.e. phosphatidylserine(3, 4) . These membranes become available when blood platelets are activated(5, 6) . Other cells like monocytes and endothelial cells have also been reported to support prothrombin activation by providing such a procoagulant surface(7, 8) .
Prothrombin activation at a macroscopic surface under flow conditions has been studied in our laboratory. We previously reported on prothrombin activation in a tubular flow reactor(9) . The findings of that study seemed to contrast with the findings on prothrombin activation at planar phospholipid bilayers(10, 11) . That is, during continuous flow through the tubular flow reactor, the rate of thrombin production appeared to be dependent on the prothrombin concentration in the bulk solution up to about 1 µM. Studies with an uniformly accessible surface, as offered by the rotating disc, however, showed that the concentration of prothrombin in the bulk solution that is required for the half-maximum rate of prothrombin conversion was about 7 nM(10) .
In the present paper we describe the results of a kinetic study on prothrombin activation in a tubular flow reactor. Our major goal was to identify the experimental settings, with respect to flow rate, substrate concentration, and prothrombinase surface density, that allow us to derive kinetic parameters from the thrombin production data. Hereto we explored the conditions that gave rates of thrombin production proportional to the true catalytic activity of prothrombinase and thus independent of the rate of transfer of prothrombin from the bulk solution to the catalytic surface. We, therefore, have carefully evaluated the transport limit in the flow reactor and used the theoretical considerations of mass transfer given by Brown(12) . Our findings support previous proposals that at a macroscopic phospholipid surface prothrombin activation by prothrombinase is extremely efficient because very low concentrations of prothrombin (a few nanomolar) are required to obtain half-maximal velocity of thrombin production(10, 11) . We also addressed the question whether the catalytic efficiency of prothrombinase was influenced by the wall shear rate, as has been reported for factor X activation in a flow reactor(13, 14) .
The factor Xa-containing samples were
diluted in Tris buffer containing 3 mM CaCl. An
aliquot (30 µl) of the dilution was added to a cuvette containing
108 µl of Tris buffer containing 1.4 nM factor Va, 14
µM phospholipid (25% phosphatidylserine, 75%
phosphatidylcholine), and 3 mM CaCl
. Thrombin
production was started by the addition of 12 µl of a 5 µM prothrombin solution and stopped after 4 min by the addition of
390 µl of 20 mM EDTA in Tris buffer. The amount of
thrombin formed was determined by adding 60 µl of S2238 (2.5
mM) and reading of the optical density at 405 nm. Reference
curves were constructed from known amounts of factor Xa.
with D the diffusion constant of the protein, V the mean fluid flow velocity and R the
radius of the capillary. At the entrance of the capillary, z = 0, C is equal to the perfusion concentration C
:
and at the capillary wall the diffusional flux must balance the conversion rate:
with V the maximal conversion rate per unit area
and K
the Michaelis constant.
A convenient and simple approximate solution of the boundary
value problem (a, b, and d) was derived by Leveque (19) using the
assumption that the boundary layer of substrate depletion is small
compared to the capillary diameter. With this so called Leveque
approximation, the total mass transfer, J (mol/s),
is given by:
where D is the diffusion constant of the substrate
(cm/s), Q the volumetric flow rate
(cm
/s), L the length of the capillary (cm), and C
the substrate concentration at the inlet
(mol/cm
). The total mass transfer can thus expressed as the
mass transfer coefficient
(cm
/s) times
the substrate concentration C
. It should be noted
that the mass transfer rate J
, expressed in
mol/cm
/s, depends on the distance z downstream
from the capillary inlet:
with C the solution of boundary value problem (a, b, and d) with C
= 1 and
the local mass transfer coefficient (cm/s) which
declines steeply in the downstream direction.
For low flow rates and relatively long capillaries of small diameter, the depletion layer, however, spans the entire capillary and the Leveque approximation becomes invalid. The depletion at the end of the capillary can be estimated by as:
For a capillary with R = 0.032 (cm), L = 12.7 (cm), a diffusion constant D =
10 (cm
/s) and a flow rate Q = 5
10
(cm
/s), this
amounts to C
/C
= 0.53
and will overestimate the transport rate by about 20%.
Errors of this size can be avoided by employing the semianalytical
approximate solution to the full problem (a, b, and d) as presented by
Brown (12, and Equation 15 and Table I therein). The equation, which
allows the calculation of the total mass transfer coefficient
for a given flow rate and diffusion coefficient, is
rather unwieldy. We, therefore, should like to confine ourselves to
refer the reader to the paper of Brown(12) . Table 1lists the numerical values calculated on basis of
Brown's semianalytical approximate solution for the total mass
transfer coefficient,
, for prothrombin and factor Xa
under the flow conditions used in our experiments. Also given are the
mass transfer coefficients calculated on the basis of the Leveque
approximation(19) . It is apparent that prothrombin transport
is overestimated by 10-30% in the latter approximation.
with as proportionality constant, allows a simple solution
of this problem. It is easily verified that
with C the substrate concentration at the
capillary wall (independent of z) and C
the solution of (a, b, and d) with C
= 1 satisfies a and
the boundary condition (b). Boundary condition (c) is equivalent to second-order equation in C
, which is independent of z:
Integration of this expression over the inner surface of the capillary results in
with J the total thrombin generation
(mol/s) and V
the maximal thrombin production rate
(mol/s) of the capillary. allows the following explicit
solution:
Non-linear regression of this equation to measured thrombin
production rates as a function of the perfusion concentration C of prothrombin then allows the determination of
the parameters V
and K
.
Fig. 1illustrates that the time to reach the steady state levels of thrombin production decreased with increasing amounts of factor Xa in the perfusion solution. Evidently, because a molar excess of factor Va over factor Xa was used, the rate of assembly of the prothrombinase complex at the macroscopic surface is limited by the transport of factor Xa and/or preformed factor Xa-factor Va complexes from the bulk solution to the surface. The initial part of the thrombin generation curve, therefore, reflects the rate of formation of prothrombinase activity at the phospholipid bilayer, and this rate will increase with increasing amounts of factor Xa in the perfusion mixture.
Figure 1:
Rate of thrombin production as a
function of factor Xa concentration. Phospholipid coated capillaries
(internal diameter of 0.32 mm) were perfused at 30 µl/min (wall
shear rate 20 s) with Tris buffer containing
prothrombin (0.2 µM), factor Va (2 nM), 3 mM CaCl
, and varying concentrations of factor Xa: 5
pM (
), 10 pM (
), and 20 pM (
).
It is also evident
from Fig. 1that the rate of thrombin formation reached a steady
state value independent of the amount of factor Xa present. Under the
flow conditions used and with 0.2 µM prothrombin in the
perfusion solution, the maximum rate of thrombin formation was 1.5
pmol/min. Table 1predicts a maximal mass transfer of 2.2 pmol
prothrombin/min for a prothrombin concentration of 0.2 µM.
Therefore, an obvious explanation for the identical steady state levels
is that in each experiment the prothrombin converting capacity of the
capillary wall ultimately exceeded the transport limit. The alternative
explanation that in all experiments the prothrombinase concentration at
the wall surface reached a maximal value was refuted by determination
of the prothrombinase densities attained in these experiments. The
amount of phospholipid-bound factor Xa at the end of the perfusion
increased linearly with the factor Xa concentration in the perfusion
mixture. Thus, 0.34, 0.65, and 1.13 fmol factor Xa/cm was
bound after 28 min perfusion with a mixture containing 0.2 µM prothrombin, 2 nM factor Va, and 5, 10, or 20 pM factor Xa, respectively. Thus, despite widely different surface
densities of prothrombinase we observed identical rates of thrombin
production. It is important to note that when factor Xa was perfused in
the absence of factor Va, no factor Xa could be detected in the EDTA
effluent of the capillary. Thus, the amount of prothrombinase in the
phospholipid-coated capillary is represented by the amount of factor Xa
found in the EDTA effluent.
Since the rate of thrombin production seemed to be limited not by the prothrombinase binding capacity of the phospholipid-coated capillary, but by the rate of supply of substrate to the catalytic surface, we expected to find that the steady state rate of thrombin production should be proportional to the concentration of prothrombin in the perfusion solution (see ). Thus, phospholipid-coated capillaries were perfused with varying concentrations of prothrombin in the presence of fixed amounts of factor Xa (50 pM) and factor Va (2 nM). The steady state rates of thrombin production are presented in Table 2as a function of the prothrombin concentration. It is seen that the rate of thrombin production increased linearly with the prothrombin concentration. Table 2also gives the rate of transport of prothrombin to the catalytic surface as a function of the prothrombin concentration in the perfusion solution. These values were calculated as described under ``Experimental Procedures.'' The mass transfer coefficient was taken from Table 1. The close agreement between the experimental rate of thrombin production and the rate of prothrombin transport to the catalytic surface strongly indicates that the system employed behaves as predicted by the hydrodynamic theory.
Figure 2:
Rate of thrombin production as a function
of prothrombinase density. Capillaries were first perfused with Tris
buffer containing factor Xa, at the concentrations indicated, factor Va
(2 nM), and 3 mM CaCl. Steady state rates
of thrombin production (
) were determined by perfusion with 0.5
µM prothrombin and 2 nM factor Va. At the end of
the perfusion experiment, the amount of phospholipid-bound factor Xa
(
) was determined by rinsing the capillary with EDTA containing
Tris buffer. The wall shear rate was 20 s
. Further
experimental conditions were as described under ``Experimental
Procedures.''
Figure 3:
Rate of thrombin production as function of
prothrombin concentration, shear rate, and prothrombinase density.
Capillaries containing phospholipid bound prothrombinase, 0.014
fmol/cm (
) and 0.018 fmol/cm
(
)
were perfused with Tris buffer containing the indicated concentrations
of prothrombin, 2 nM factor Va and 3 mM CaCl
, at a wall shear rate of 20 s
(
) and 3000 s
(
). The solid
lines represent the result of a non-linear fit to the
Michaelis-Menten equation. The estimated kinetic parameters are listed
in Table 3.
We have characterized the kinetics of prothrombin activation
by the complex of factor Va and factor Xa assembled at the macroscopic
phospholipid surface of a tubular flow reactor. The present study
confirms an earlier observation from our group (10, 11) that on macroscopic lipid bilayers the
Michaelis constant K is much lower than reported
for prothrombin activation at the surface of unilamellar vesicles with
a radius of 20-30 nm. It was noticed that the K
decreased with increasing phospholipid area per molecule of the
factor Xa-factor Va complex. Indeed the K
was
depressed by a factor of 4 when large unilamellar vesicles (radius of
60-80 nm) were used. Moreover, it was shown that on planar
phospholipid bilayers, with an area of 1
µm
/prothrombinase complex, the K
was 6-7 nM and thus 25-fold smaller than the K
(170 nM) on small unilamellar vesicles.
This marked difference was attributed to an efficient collection of
prothrombin from solution by a macroscopic surface, whereas the K
on small unilamellar vesicles represents an
apparent value reflecting the limited transport rate of prothrombin
from solution to the vesicle(10, 11) . The
phospholipid bilayers in these studies were present on a slide opposite
a stirring bar or on a rotating disc. The latter system especially has
the advantage that it presents a uniform accessible surface, with a
uniform mass transfer coefficient over the entire surface. The
capillary flow system used in the present study has the advantage of an
extremely stable and well defined fluid flow. Another advantage is the
high surface/volume ratio (0.05 cm
/µl) of the
capillary. Its potential disadvantage is the steep decline of the mass
transfer rate in the downstream direction (). This could
complicate the analysis of the intermediate enzyme kinetics. This
potential disadvantage is, however, largely circumvented by the
transport limited assembly of prothrombinase (see below), which implies
a local prothrombinase activity proportional to the mass transfer
coefficient and therefore the applicability of and (see ``Experimental Procedures'') for the
analysis of the experimental data.
Our initial experiments, in which
the phospholipid-coated capillary was perfused with a solution of
factor Xa, factor Va, and prothrombin showed that at a shear rate of 20
s the rate of thrombin production became independent
of the prothrombinase density at the surface when that density was
greater than 1 fmol/cm
and the prothrombin concentration of
the perfusion solution was 0.5 µM. From reported binding
data(20, 21, 22) , we could calculate a
maximum binding capacity of our phospholipid bilayer between 2 and 10
pmol prothrombinase/cm
. Thus, a fractional occupation of
about 0.05% is sufficient to enter a regime of thrombin production that
is controlled by the mass transfer rate of prothrombin. From the data
shown in Fig. 1, we also could estimate the rate at which
prothrombinase was assembled during the initial phase of the thrombin
production curve. If we assume a turnover of 60 s
(Table 3), then the initial slopes of thrombin production
indicate rates of prothrombinase formation of 0.05, 0.10, and 0.18
fmol/min with, respectively, 5, 10, and 20 pM factor Xa in the
perfusion mixtures. As a matter of fact these rates of prothrombinase
formation are in good agreement with the mass transfer of 0.01 fmol
factor Xa/min/pM factor Xa in the perfusate (Table 1).
The validity of our theoretical considerations regarding the thrombin production data in the flow reactor also became evident from the excellent agreement between the observed steady state rates of thrombin production under diffusion-controlled conditions (high prothrombinase density and low wall shear rate) and the calculated mass transfer of prothrombin as shown in Table 2. We like to note that the theory was developed for reactions at a homogeneously catalytic surface like that of platinum. The fact that catalysis at a macroscopic surface with a very low enzyme density follows the same theory is compatible with the notion that the lateral diffusion of reactants at the surface exceeds the flux of reactants toward the apparent non-homogeneously catalytic surface(11) .
When capillaries
with prothrombinase density below 0.6 fmol/cm were perfused
with 0.5 µM prothrombin at a wall shear rate of 20
s
, the steady state rate of thrombin production
became a linear function of the surface density of prothrombinase (Fig. 2). Yet, we have to emphasize that this observation does
not exclude prothrombin depletion at the catalytic surface. As outlined
under ``Experimental Procedures,'' thrombin production causes
a partial dependence of the rate of thrombin production on the mass
transfer rate of prothrombin to the catalytic surface. This is clearly
demonstrated by the results presented in Table 3. The apparent K
for prothrombin is much higher at a low shear
rate and high prothrombinase density than the K
determined under conditions where much less depletion of
prothrombin near the catalytic surface was expected. The apparent
difference in K
disappeared when the prothrombin
concentration was corrected for depletion of prothrombin near the
catalytic surface (). The results as summarized in Table 3show that both the V
and K
did not vary with the shear rate. In this
respect prothrombinase bound to a macroscopic surface seems to behave
differently from the factor X converting complex tissue factor-factor
VIIa. Nemerson and co-workers (13, 14) reported that
the V
of factor X activation in a tubular flow
reactor increased with increasing wall shear rate.
Whereas the k value for prothrombin activation in the
tubular flow reactor is in accordance with those reported for
prothrombin activation by prothrombinase in a vesicle
system(3, 4) , we found a much lower K
. However, the value 3 nM is in close
agreement with the K
value (6 nM)
determined for phospholipid bilayers on a rotating disc(10) .
The Michaelis constant observed for macroscopic lipid bilayers,
which is about 1000-fold lower than the plasma concentration of
prothrombin, might have interesting implications for the regulation of
thrombin generation in vivo. This low value of K implies the existence of two control regimes:
(i) for low surface densities of prothrombinase, with V
below the mass transfer coefficient times the
plasma concentration of prothrombin, the thrombin generation is
completely determined by the amount of prothrombinase, and (ii) for
high surface concentrations of prothrombinase the thrombin generation
rate is completely determined by the transport limit and thus by the
plasma concentration of prothrombin. The two regimes might also have
consequences for the inhibitor-regulated thrombin production. Because
of the high catalytic efficiency of prothrombinase for prothrombin,
inhibition by pseudo-substrates, like antithrombin, is hardly to be
expected. However, when thrombin production is limited by the supply of
substrate, inhibition by pseudo-substrates has to be expected, but will
not result in a reduced rate of thrombin production. In this respect,
it is of interest to see that these transport phenomena have the very
same implications for other surface-bound reactions involved in blood
coagulation like the activation of factor X(23) .