(Received for publication, May 31, 1995)
From the
A mathematical model has been developed to simulate the generation of thrombin by the tissue factor pathway. The model gives reasonable predictions of published experimental results without the adjustment of any parameter values. The model also accounts explicitly for the effects of serine protease inhibitors on thrombin generation. Simulations to define the optimum affinity profile of an inhibitor in this system indicate that for an inhibitor simultaneously potent against VIIa, IXa, and Xa, inhibition of thrombin generation decreases dramatically as the affinity for thrombin increases. Additional simulations show that the reason for this behavior is the sequestration of the inhibitor by small amounts of thrombin generated early in the reaction. This model is also useful for predicting the potency of compounds that inhibit thrombosis in rats. We believe that this is the first mathematical model of blood coagulation that considers the effects of exogenous inhibitors. Such a model, or extensions thereof, should be useful for evaluating targets for therapeutic intervention in the processes of blood coagulation.
The clotting of blood is an exquisitely complex process. The simultaneous requirements for the free flow of blood under normal conditions and rapid clotting to prevent blood loss in the case of injury require a delicate balance between clot formation and clot lysis. Taken together, these processes involve more than 30 proteins, at least 10 of which are serine proteases. Inhibitors of serine proteases occur as natural anticoagulants, and natural and synthetic inhibitors have been widely studied for therapeutic applications in the prevention of thrombosis(1) .
When choosing candidate serine protease inhibitors for therapeutic use, the choice of a target enzyme may be less than obvious. Many of the enzymes have multiple activities, there is positive and negative feedback regulation, and there are alternative pathways for activation and inactivation. In addition, serine protease inhibitors (especially synthetic compounds) generally have a spectrum of activities due to the high degrees of homology among the blood coagulation factors. This raises the question of whether the best antithrombotic compound would be specific for a single enzyme or show affinities for several serine proteases(2) .
Mathematical modeling can help us understand such complex systems, but published models of blood coagulation suffer from the following limitations: consideration of only a small part of the coagulation cascade(3, 4, 5, 6, 7) , empirical description of interactions for which molecular mechanisms were known(3, 8, 9) , determination of some (4, 9, 10) or all (3, 11) of the parameter values from the experimental data to which the model predictions were then compared (curve fitting), and the absence of comparisons of model predictions to experimental data (6, 12, 13, 14) . In the best of these studies(4, 5, 7) , most or all of the parameter values were independently determined, model predictions were compared to experimental results that had not been used to estimate parameter values, and the model was used to understand and interpret unexpected experimental observations.
An experimental system to
study the tissue factor pathway to thrombin (15) and a
mathematical model of this system (9) were recently described.
The human tissue factor (TF)()-VIIa complex was used as the
initiator of coagulation; human cofactors V and VIII, serine protease
zymogens IX and X, and prothrombin were used at their normal plasma
concentrations.
In this report, we describe a substantial revision of the mathematical model (9) of this experimental system(15) . The revised model gives reasonable predictions of the reported experimental results with no adjustment of parameter values. We then extended this revised model to include the effects of serine protease inhibitors with affinity for any or all of the factors VIIa, IXa, Xa, and thrombin. The extended model was used to predict the affinity profile of the optimum inhibitor in this experimental system and to predict the potency of compounds that inhibit thrombosis in rats. In common with the best studies described above, all the model parameter values are independently determined, model predictions are compared to experimental data, and the model is used to explain unexpected results. In addition, our model is as comprehensive in its coverage of the clotting reactions as any other published model, and it is the first to include the effects of exogenous inhibitors.
Figure 1: Model reaction scheme. Colons (:) indicate complex formations; asterisks (*) indicate irreversibly inactivated species. Complexes containing the inhibitor (I) are assumed to be reversibly inactivated. Subscripts (Xa) and (IIa) indicate that VIII has been activated by Xa or (m)IIa, respectively. Note that I indicates an inhibitor, not Factor I (fibrinogen).
Whenever possible, parameter values determined at 37 °C using human proteins were used in the model. Kinetic constants measured in the presence of phospholipid vesicles were used instead of those measured in the presence of cell surfaces, consistent with the experimental system being modeled. The parameter values used in the model are shown in Table 1.
Figure 2:
Comparison of model predictions to
experimental results (see Fig. 6 of (15) ). The labeled lines
are the model predictions; the line for IXa is indistinguishable from
the base line at this scale. The symbols show the experimental data for
Va (), VIIIa (
), IXa (
), Xa (
), and thrombin
(
). Meizothrombin is more active toward the chromogenic
substrate than is thrombin(31) , so accumulation of this
activation intermediate can lead to total activities greater than those
expected from complete activation to
thrombin.
A concentration
of 10 nM inhibitor was assumed, and the K values for each enzyme were allowed to vary over ranges similar
to those observed for compounds in our library of inhibitors. These
ranges were 0.1-1000 nM for VIIa, 1-1000 nM for Xa, and 0.01-1000 nM for (m)IIa. We currently
do not have a good assay for the inhibition of IXa, so K
values were assumed to range from 1-1000 nM.
Simulations were run to determine the time required for the generation
of 80% of the final thrombin activity. We used the ratio of the
inhibited time to the uninhibited time as a measure of inhibitor
potency; the higher this ratio, the more potent the inhibitor. The K
ranges were covered in half-log steps, with
simulations run at all of the 4,851 K
combinations.
The results shown in Fig. 3lead to
several conclusions. First, the predicted thrombin generation times are
relatively insensitive to changes in the K values
until these values are on the order of 1 nM or less (except as
noted below). Second, inhibitors with affinity for only one of the
serine proteases are not very potent compared to inhibitors with a
spectrum of affinities. Third, compounds that simultaneously have high
affinity for VIIa, Xa, and thrombin are predicted to be very potent,
regardless of the IXa K
(Fig. 3, bottom
row, back corners). Fourth, and most surprising, for
compounds that simultaneously have high affinity for VIIa and IXa,
inhibition of thrombin generation decreases as the affinity for
thrombin increases from 1000 to 5 nM (Fig. 3, left
column, back right planes). This effect is especially
dramatic for compounds that also have high affinity for Xa (Fig. 3, bottom left, back right plane).
Figure 3: Predicted inhibitor potency as a function of affinity for the various serine proteases. The potency shown on the vertical axes is the ratio of the times required to generate 80% of the final thrombin activity in the inhibited and uninhibited systems; the larger the ratio, the more potent the inhibitor. The ratios for inhibitors highly specific for a single serine protease, shown by the labeled circles, are 1.8 (VIIa), 1.6 (IXa), 1.3 (Xa), and 1.1 (thrombin). The ratio for the best inhibitor, shown by the unlabeled circle, is 5.1.
To explain this unexpected result, we used the model to determine where the inhibitor was bound. In Fig. 4, the amount of inhibitor bound to each of the serine proteases is shown for two inhibitors, each very potent against VIIa, IXa, and Xa; one has high affinity for thrombin, and the other has low affinity for thrombin. For the case of high affinity for thrombin (Fig. 4A), the inhibitor binds to thrombin in preference to IXa and Xa as soon as a small amount of thrombin is formed. With a reduction in the amount of inhibitor bound to IXa and Xa, the activation of prothrombin occurs very rapidly. For the case of low affinity for thrombin (Fig. 4B), the inhibitor remains bound to IXa and Xa even after a substantial amount of thrombin is formed, and the activation of prothrombin occurs only gradually.
Figure 4: Distribution of inhibitor. A, an inhibitor with high affinity for thrombin. B, an inhibitor with low affinity for thrombin. The lines show the generation of thrombin activity (right vertical axes). Areas are proportional to the amount of inhibitor bound to each species (left vertical axes): free inhibitor (hatched area), bound to thrombin or meizothrombin (graphed area), bound to Xa (shaded area), bound to IXa (cross-hatched area), and bound to VIIa (white area). In each case, only 1% of the total inhibitor is shown; the rest is free or bound to thrombin or Xa.
Figure 5: Potency of inhibitors of rat arterial thrombosis. A, correlation with the inhibitor concentration required to double the APTT. B, correlation with model predictions. Compounds that are more potent in the rat than would be predicted by the APTT are labeled with identifying codes. Note that on the horizontal axes, potency increases from right to left for APTT and from left to right for the model predictions.
Model predictions of the thrombin generation time ratio for 25
compounds tested in the rat were compared to the OT values (Fig. 5B). The ranges of K
values
for the compounds tested in rats were 0.65-1330 nM for
VIIa, 0.84-920 nM for Xa, and 0.024-206 nM for thrombin. Because we did not have IXa K
values for these compounds, the simulations were done assuming
that the IXa K
was equal to the VIIa K
. Similar results were obtained when the IXa K
was assumed to be equal to the Xa K
; predictions were somewhat worse when all the
inhibitors were assumed to be active (K
= 1
nM) or inactive (K
= 1000
nM) against IXa (data not shown). All of the compounds
predicted to be potent by the model (ratio > 1.2) were potent in the
rat (OT
< 10 µg/kg/min); all of the compounds
impotent in the rat (OT
> 10 µg/kg/min) were
predicted by the model to be impotent (ratio < 1.2). In addition,
the model correctly predicted the potency of the APTT outliers.