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J. Biol. Chem., Vol. 277, Issue 21, 18322-18333, May 24, 2002

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A Model for the Stoichiometric Regulation of Blood Coagulation^*

Matthew F. Hockin^§, Kenneth C. Jones^¶, Stephen J. Everse, and Kenneth G. Mann

From the  Department of Biochemistry, College of Medicine, University of Vermont, Burlington, Vermont 05405

Received for publication, February 5, 2002, and in revised form, March 12, 2002

	ABSTRACT

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

We have developed a model of the extrinsic blood coagulation system that includes the stoichiometric anticoagulants. The model accounts for the formation, expression, and propagation of the vitamin K-dependent procoagulant complexes and extends our previous model by including: (a) the tissue factor pathway inhibitor (TFPI)-mediated inactivation of tissue factor (TF)·VIIa and its product complexes; (b) the antithrombin-III (AT-III)-mediated inactivation of IIa, mIIa, factor VIIa, factor IXa, and factor Xa; (c) the initial activation of factor V and factor VIII by thrombin generated by factor Xa-membrane; (d) factor VIIIa dissociation/activity loss; (e) the binding competition and kinetic activation steps that exist between TF and factors VII and VIIa; and (f) the activation of factor VII by IIa, factor Xa, and factor IXa. These additions to our earlier model generate a model consisting of 34 differential equations with 42 rate constants that together describe the 27 independent equilibrium expressions, which describe the fates of 34 species. Simulations are initiated by "exposing" picomolar concentrations of TF to an electronic milieu consisting of factors II, IX, X, VII, VIIa, V, and VIIII, and the anticoagulants TFPI and AT-III at concentrations found in normal plasma or associated with coagulation pathology. The reaction followed in terms of thrombin generation, proceeds through phases that can be operationally defined as initiation, propagation, and termination. The generation of thrombin displays a nonlinear dependence upon TF, AT-III, and TFPI and the combination of these latter inhibitors displays kinetic thresholds. At subthreshold TF, thrombin production/expression is suppressed by the combination of TFPI and AT-III; for concentrations above the TF threshold, the bolus of thrombin produced is quantitatively equivalent. A comparison of the model with empirical laboratory data illustrates that most experimentally observable parameters are captured, and the pathology that results in enhanced or deficient thrombin generation is accurately described.

	INTRODUCTION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

The blood coagulation system is composed of a set of pro- and anticoagulant systems that maintain the balance of blood fluidity. Defects in this balance can result in either thrombotic or bleeding tendencies. Qualitative or quantitative alterations in this hemostatic balance can have devastating effects, producing hemorrhagic diseases (hemophilia A, hemophilia B, hemophilia C, para-hemophilia, hypoprothrombinemia) (1-6) or thrombotic diseases (antithrombin III (AT-III)¹ deficiency, protein C deficiency, protein S deficiency, and factor V^Leiden) (7-10). The initial protein components involved in the essential coagulation cascade and its stoichiometric regulation includes 10 species. However, during thrombin generation by the tissue factor (TF)-initiated reaction, multiple transitory species are produced, bringing the total to 27, many of which play multiple roles at different stages of the process. An example of these multiple roles is presented in the activation of prothrombin. The product, thrombin, participates in feed back processes leading to its own generation by activating factor V, factor VIII, and factor VII; using a different binding partner, it initiates the anticoagulation cascade through protein C activation (11). It is in the balance of this complex interplay that the response to each injury or stimulus is determined.

The rational design of therapeutic antihemorrhagic and antithrombotic strategies has primarily relied upon intuitive approaches to judge how qualitative or quantitative alterations in a natural product or a therapeutic agent might behave in this complex reaction milieu. These estimates are usually made on the basis of oversimplifications, which ignore the dynamic interplay between coagulation factor reactions; consequently, judgments are made based upon the presumption of isolated defective functions. In even the simplest cases, such as the single gene deficiencies (hemophilia A or B) in which a fundamental knowledge of the defect is available, less than satisfactory algorithms exist for determining the dosage required of the missing factor. Even less obvious is the case in which an otherwise "normal" individuals present with complex disorders such as venous or coronary thrombosis, or an unaccounted for bleeding diathesis (12, 13). A more complete understanding of the interplay between the pro- and anticoagulant factors involved in hemostasis should permit the kinetics of "composite" deficiencies to be better understood. Toward this end, we have extended our numerical model (14) for the thrombin generation reaction by including the stoichiometric inhibitor systems and updating the mechanism to the current level of knowledge.

Over the past 25 years, a reasonably comprehensive understanding of the inventory of the proteins and the associated biophysical and enzymologic processes involved in blood clotting has been developed through the efforts of numerous investigators (15). A large body of rigorously obtained data describes association states, membrane binding thermodynamics, enzyme complex assembly kinetics, and reaction kinetics for the inventory of processes (16). This in turn has led to the description of an increasingly overwhelming network of pathways and interactions (17-19). In the face of this complexity, intuition frequently leads to paradox. For example, increasing the concentration of a procoagulant factor such as factor VII "should" lead to an increase in thrombin generation and decreased clotting time. In empirical studies and in laboratory experiments, the opposite is true (20). Computational modeling provides the opportunity to integrate and quantify reaction details, which, in turn, aids in the design of the more expensive empirical "wet" experiments. Rapid advances in mathematical algorithm development and computational power have enabled modeling of ever more complex systems (21-26). These technologies, when coupled with the expanding base of empirically defined mechanism, allow the development of rational models that predict the product generation profiles to be expected for diverse experimental conditions. The construction of models also provide the ability to look into reactions at a "microscopic" level, at which the concentrations of substrates and products being investigated are below detection limits accessible through direct analysis. Such evaluations can provide estimations of the initiating events in a process and lead to the design of laboratory experiments that are designed to explore unforeseen computational results.

Our laboratory provided one of the first computational models successful in predicting empirical outcomes in the biochemistry of the TF-initiated procoagulant pathway (14). Subsequent empirical studies have evaluated the influence of combinations of the stoichiometric and dynamic anticoagulant systems on the thrombin generation process (27-32). However, our attempts to develop a model that would produce the unique kinetic forms observed in empirical experiments were limited by insufficient mechanistic and kinetic data. The elucidation of robust kinetic schemes for TFPI (33) and AT-III inhibition (34-37), factor VII/VIIa competition (20), factor VIIIa dissociation (38-40), and factor VII (41), prothrombin (42), and factor V activation (43), which adequately account for empirical observations, has enabled undertaking the integration of the stoichiometric anticoagulation system into a more comprehensive model of the coagulation system. The present model includes 34 species, 42 rate constants, and accounts for the peculiar inhibitory "thresholds" produced by combinations of TFPI and AT-III.

	MATERIALS AND METHODS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

Computation-- A software package was written to enable rapid transformation of chemical equilibrium expressions to the necessary time-dependent partial differential equations required for this model and their solution. The software package Speed Rx utilizes an Internet-based interface with a generally applicable fourth order Runge-Kutta solver that provides solutions to a family of time-dependent differential equations. After the user inputs the chemical equations, initial concentrations, and rate constants for all relevant species and steps, the software generates a series of time-dependent concentration profiles for any/all reactants over any time frame of interest. A set of simulations can be developed representing titration of individual species or varying individual rate constants to represent qualitative or quantitative alterations in the reactant. The results of each simulation are stored in a relational data base architecture utilizing SQL standards. For this work all computations were carried out on a Pentium III computer running LINUX (RedHat version 7.1); however, the software may be installed on any computer capable of running UNIX.

The validity of our implementation of the fourth order Runge-Kutta solver was tested through use of the freely available backward differentiational formula in the livermore solver for ordinary differential equations (LSODE) that utilizes third order polynomials (www.llnl.gov/casc/odepak). LSODE utilizes dynamic step sizing and a matrix of partial differential equations to approach the solution. Utilizing the Livermore algorithm, we were able to demonstrate exact correspondence to the solutions obtained with our Runge-Kutta solver. This correspondence was found between both simple chemical systems and numerous pre-publication versions of the coagulation model.

The Coagulation Model-- The present effort is based upon our prior publications describing a procoagulant model of blood coagulation (14). Extensive revisions were made to incorporate additional steps to our procoagulant system. In all cases the processes and rate constants described are representative of reaction paths and rates experimentally observed under the condition of saturating phospholipid concentrations. In the instances for which experimental data are not available for the processes modeled, rate constants and mechanisms were incorporated by analogy with similar processes within the coagulation cascade. The entire model, in tabular form, is presented as Table I. The rate constants are presented in Table II, and the initial ("normal") concentrations for all reactants are presented in Table III.

Literature values for rate constants were utilized as starting points. Additional fitting was required either because these values were evaluated under nonphysiologic conditions or because they made use of extensively modified proteins. For example, the reported equilibrium constant for factor VIIa-TF-membrane was estimated utilizing a truncated TF protein and evaluated by surface plasmon resonance utilizing chemical cross-linking to a substrate (44). In this case, we maintained the ratio of the reported reverse and forward rate constants (K_d) to fit published kinetic functional data (45-47).

For description purposes, it is convenient to define the coagulation reactions leading to thrombin generation in three phases: initiation, propagation, and termination. This oversimplification provides a convenient vehicle for communication.

Initiation-- The extrinsic factor Xase (factor VIIa-TF-membrane) forms through the assembly of membrane-associated TF and factor VII/VIIa and was modeled as a equilibrium process (Table I, nos. 1 and 2). Rate constants for this process were estimated by analogy with the assembly processes for the intrinsic factor Xase (factor Xa-factor VIIIa-membrane) and prothrombinase (factor Xa-factor Va-membrane) (Table II). The extrinsic factor Xase functions by the activation of factor X (Table I, nos. 6 and 7) and factor IX (Table I, no. 8) and, in a self-propagating loop, the activation of factor VII (Table I, no. 3). Factor VII activation also occurs by thrombin and factor Xa (41) and is included (Table I, nos. 4 and 5) in this version of the model.

Our prior model assumed that the initial activation of factor V by factor Xa (14). Empirical analyses of early events during the TF-initiated generation of thrombin (47) convincingly demonstrated that early factor V activation occurs exclusively by thrombin. Further, anisotropy data² for factor Xa binding show that factor V at biologically relevant concentrations (20 nM) does not bind factor Xa and thus cannot participate in prothrombinase formation prior to its proteolytic activation. An evaluation of the activation of prothrombin by factor Xa-phospholipid in the absence of factor Va at 37 °C at enzyme concentrations relevant to the initiation phase of the reaction gave K_m 0.3 ± 0.05 µM and k_cat 2.3 × 10³ s¹. Accordingly, in the model the initial factor Va is produced by thrombin activation (Table I, no. 16) with the latter activated by factor Xa-PCPS (Table I, no. 9) using the appropriate constants (Table II, no. 16).

The activation of factor VIII is as previously described, whereas a chemical mechanism for factor VIIIa activity loss by spontaneous A₂ domain dissociation (38-40) (Table I, nos. 13 and 15) is incorporated. The rate constants for these dissociation processes and their governing binding constants at physiological pH are utilized (Table II, nos. 23-25) (40).

Propagation-- The production of factor Xa by both the intrinsic and extrinsic factor Xases is as previously described, as is the formation and activity of the prothrombinase complex (14) (Table I, nos. 6-8 and 17). The thrombin self-propagation by activation of components of the complexes catalyzes its formation is implemented (Table I, nos. 5, 10, and 16). The thrombin precursor, meizothrombin, is less efficient at these processes and was therefore not included as a separate entity² (48).

Termination-- The stoichiometric inhibitors AT-III and TFPI are included, enabling the analyses of their isolated and combined effects on thrombin generation. The contribution of AT-III is straightforward, using existing literature for the second-order rate constants for AT-III inhibition of IIa, factor Xa, factor VIIa, and factor IXa (Table I, nos. 23-27). TFPI inhibition proved challenging. This inhibitor has been the subject of numerous investigations. The most satisfactory explanation for TFPI behavior is provided by Baugh et al. (33), and we have adapted their scheme (Table I, nos. 20-22) and rate constants (Table II, nos. 33-37). TFPI acts through two pathways, one of which involves the inhibition of the enzyme product complex TF·VIIa·Xa (Table I, no. 21). The other pathway involves a three-step mechanism: 1) inhibition of the product factor Xa by TFPI (Table I, no. 20); 2) binding of the Xa·TFPI complex to the TF·VIIa complex through the substrate interaction domain of factor Xa (Table I, no. 22); and 3) the inhibition of the bound TF-factor VIIa through a first order "rearrangement," wherein the Kunitz domain of TFPI interacts with the factor VIIa active site (33), producing a product indistinguishable from that of the second order addition to the enzyme-product complex.

Implementation of the second order, enzyme-product-TFPI pathway (Table I, no. 21) is straightforward using the published rate constants. The alternative mechanism creates problems resulting from the unimolecular inhibition process as the final step. Our initial attempts at modeling this system led to computational instability that was relieved by condensing the final two steps of this mechanism into a single second order collisional process dictated by the limiting forward or reverse rates of the two steps. This eliminates the problems associated with interpreting a collisional rate constant in the context of a first order process. Confirmation of the validity of this approach was obtained through construction of a model involving only the extrinsic factor Xase components and TFPI. This TFPI model generated data comparable with empirical TFPI inhibition rate data.

Simulations-- Unless otherwise indicated, all simulations were performed utilizing the mean plasma concentrations for all proteins (Table III): prothrombin (1.4 µM), factor X (160 nM), factor IX (90 nM), factor V (20 nM), factor VII (10 nM), factor VIIa (100 pM), factor VIII (700 pM), TFPI (2.5 nM), and AT-III (3.4 µM). TF concentrations were varied between 1 and 25 pM to simulate estimates of a physiologically relevant challenge (46). Total thrombin (IIa and mIIa) with units corresponding to thrombin-seconds is obtained by integrating the thrombin concentration over an experimental time interval. This value represents the quantitative exposure of the experimental system to thrombin activity. The model was tested by simulation of experimental conditions that have shown unique thrombin profiles (27, 30). Comparisons between simulations and experimental data were used in assessing the fidelity of the model to the empirical system.

Refinement-- Analysis of the model TFPI mechanism included tests using the isolated inhibitory loop in simulations involving only TF, factor VIIa, TFPI, and factor X, in which the amount of factor Xa formed over time was examined. In a typical simulation, the TF concentration was varied from 1 to 1024 pM, while the factor VIIa (2 nM), TFPI (2.5 nM), and factor X (170 nM) were held constant. The factor Xa profiles were contrasted with published data and the model parameters adjusted until correspondence was obtained. Further simulations in which the initial conditions included preformed TFPI-factor Xa complexes were conducted to verify the response. The effect of TFPI was explored in terms of thrombin response, lag phase, and maximal rate of thrombin generation at multiple TF concentrations. Independent titrations of TFPI (0-150 nM), TF (0.01-1000 pM), and factor VIIa (0.1-20 nM), under the set of otherwise normal concentrations (Table III), were conducted as well as simulations examining the thrombin generation profile for conditions mimicking severe hemophilia A (zero factor VIII) in the presence or absence of TFPI. The results of each set of simulations were compared with experimental data (33).

Model Validation-- The aggregate effects of TFPI and AT-III on the procoagulant model were assessed by including both inhibitors in the procoagulant model. Initial simulations were conducted to evaluate the thrombin response profile, and the integrated thrombin levels generated after stimulus with TF over a range spanning 0.01-1000 pM. Subsequent analyses included the examination of the thrombin response in hemophilia A at various TF stimuli (1, 5, and 25 pM TF) at various factor VIII concentrations (100, 10, and 1% factor VIII). Further analyses of the hemophilia A conditions were conducted quantifying the thrombin response to factor VIIa titration in severe hemophilia (<1% factor VIII). Simultaneous variations in the level of AT-III within the clinically normal range (50 or 150%) and in the level of II (150 or 50%) were analyzed at 5 and 25 pM TF and the results contrasted with published reports.

	RESULTS

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

The Procoagulants in the Absence of Inhibition-- Simulations (Fig. 1, open symbols) were performed in the context of only the procoagulation model (Table I, nos. 1-19). Increasing TF concentrations (1, 5, and 25 pM) result in reduction of the duration of the initiation phase, which is arbitrarily defined as the time from introduction of TF necessary to generate ~20 nM thrombin. The concentration of thrombin is represented as the activity measured using the synthetic substrate S-2238. The "bump" observed prior to the stable final value (1.4 × 10⁶ M) is a consequence of the 20% greater activity displayed toward the chromogenic substrate by meizothrombin (14, 48). Over the range of TF illustrated in the absence of TFPI (open symbols), the maximum rate of thrombin production varied ~5-fold. In the present model the initial activation of prothrombin occurs by factor Xa-membrane and the initial activation of factor V occurs by thrombin generated from the former reaction. The elimination of the factor Xa-PCPS activation of prothrombin (Table II, k₁₆ = 0) under the set of equations detailed in Table I (no. 9), generated simulations with no thrombin production. The factor VIIIa decay term based upon the empirically measured A₂ dissociation rate (Table I, nos. 13-15) (40) increases the sensitivity of the reaction to reductions in factor VIII concentration. The most notable characteristic of the procoagulant-alone data, also observed in empirical studies (30, 49), is its biphasic behavior, a lag or initiation phase followed by a propagation phase. The results obtained here are similar to those reported by van't Veer for similar procoagulant mixtures in the absence of inhibitors.³ However, in contrast to the results in Fig. 1, the published studies of van't Veer et al. were conducted in the absence of factor VII and with reaction initiation conducted by the addition of preformed factor VIIa-TF-membrane complex to the reaction system. Modeling experiments conducted under the explicit conditions described by van't Veer et al. are nearly identical to those published (data not shown).

View larger version (21K): [in this window] [in a new window]   Fig. 1.   Total thrombin generation (thrombin + meizothrombin) as a function of TF concentration with (closed symbols) and without (open symbols) TFPI. The concentrations of TF illustrated are 25 pM (circles), 5 pM (squares), and 1 pM (diamonds). The filled symbols represent experiments conducted with 2.5 nM TFPI present.

At fixed TF concentration, increasing the concentration of factor VIIa shortens the duration of the initiation phase in a saturable manner. As observed in empirical experiments, the dependence of the duration of the lag phase appears in the VII/VIIa ratio, rather than the absolute factor VIIa concentration. This dependence is a function of the competitive binding equilibria between factor VII and factor VIIa and holds under conditions in which the binding isotherm (Table I, nos. 1 and 2) is at least partially saturated (i.e. >1 nM total VII/VIIa), given by the ratio of the rate constants k₁/k₂. These results illustrate the sensitivity of TF-induced coagulation to levels of the TF·VIIa complex and thus the rate of factor Xa production at any point in the cascade as observed in the empirical experiments by van't Veer et al. (30).

When TFPI at 2.5 nM is added to the titrations observed in Fig. 1 (closed symbols), a significant extension of the initiation phase of the reaction is observed with only a relatively small effect on the propagation rate. This observation is in agreement with empirical experiments published by van't Veer et al., which illustrated that the major effect of TFPI is in prolonging the initiation phase of the reaction.

Effect of the Combination of AT-III and TFPI on TF-initiated Thrombin Formation-- The addition of AT-III to the procoagulant reaction requires rate equations for IIa, mIIa, factor Xa, factor IXa, and TF-factor VIIa complexation with this inhibitor (Table I, nos. 23-27). When compared with the procoagulant-alone system, simulations including AT-III exhibit bell-shaped curves for thrombin generation at all TF concentrations tested. When challenged with 25 pM TF in the presence of 3.4 µM AT-III (Fig. 2, diamonds), thrombin production is slightly delayed, is at a maximum near 150 s, subsequently decreases, and is nearly consumed by 400 s. Reactions with TFPI, in the absence of AT-III, 25 pM TF (triangles) yield maximal rates of thrombin production at ~200 s and quantitative activation by 300 s. As observed in empirical experiments, AT-III does little to alter the duration of the initiation phase or the maximum rate of thrombin formation, whereas TFPI (Table I, nos. 20-22) results in extension of the initiation phase of thrombin generation (Fig. 2, triangles). The addition of both TFPI and AT-III to the reaction system results in equivalent total thrombin generation at 25 pM TF (squares), but the reaction is significantly delayed.

View larger version (20K): [in this window] [in a new window]   Fig. 2.   Active thrombin present as a function of time for a reaction initiated with 25 pM tissue factor. The reactions represented are no inhibitors (circles), AT-III only (diamonds), TFPI only (triangles), and both inhibitors present (squares).

Fig. 3 illustrates a TF titration (1-25 pM) of the procoagulant system complemented with 2.5 nM TFPI and 3.4 µM AT-III. Active thrombin is plotted versus time. The data illustrate that, between the TF concentrations of 5 pM (filled squares) and 1 pM (filled diamonds), there is virtual attenuation of the thrombin formation response, i.e. a threshold in this reaction. This synergistic effect of the two inhibitors acting in concert is similar to the empirically observed synergy observed in the empirical chemistry experiments reported by van't Veer et al., when these two inhibitors were combined with all procoagulants in TF-initiated reactions.

View larger version (25K): [in this window] [in a new window]   Fig. 3.   Total thrombin as a function of time is represented for varying initiating TF concentrations: 25 pM (filled circles), 20 pM (open triangles), 15 pM (open circles), 10 pM (filled triangles), 5 pM (filled squares), and 1 pM (filled diamonds).

A quantitative interpretation of the data of Fig. 3 in terms of the total amount of thrombin produced over a 700 s time interval is illustrated by an exponential plot of thrombin peak area (IIa·seconds) versus TF concentration in Fig. 4. For the concentration range from 3 to 10 pM, and extending to 25 pM TF (data not shown), the total thrombin and active thrombin are unchanged. At TF concentrations below 3 pM, an exponential concentration dependence is observed with almost a thousandfold decrease in active thrombin (filled symbols) between 3 and 1 pM. Thus, the theoretical model mirrors the empirically observed effect of the two inhibitors, TFPI and AT-III, acting in concert.

View larger version (14K): [in this window] [in a new window]   Fig. 4.   Peak area of active thrombin (thrombin·seconds) is plotted versus TF concentration. Total thrombin is represented by open squares; active thrombin is represented by filled squares.

Initiation Phase of the Procoagulant Response-- Following complexation of factor VIIa with TF-PCPS (Table I, no. 2), the initiation phase begins with the activation of factor IX and factor X to their respective enzyme products (Table I, nos. 6-8). As noted, the duration of this initiation phase is largely a consequence of factor VIIa and TF and regulation by factor VII and TFPI (Table I, nos. 1, 2, 21, and 22). The factor Xa generated initially by the factor VIIa-TF complex (Table I, nos. 6 and 7) activates a small amount of prothrombin to thrombin (Table I, no. 9). That thrombin begins the process of catalyst building by activating factor V and factor VIII (Table I, nos. 10 and 16).

Although factor Xa-PCPS has the capacity to activate factor V (50), empirical data (28) show conclusively that thrombin is the essential early activator in empirical chemical experiments. Thus, crucial to the initiation phase is the activation of some prothrombin to thrombin by factor Xa-PCPS exclusive of factor Va. This initial catalyst generates the thrombin, which initially activates some factor V and factor VIII to their respective cofactor (factor Va, factor VIIIa) products.

Fig. 5A illustrates a simulation of a reaction initiated with 5 pM TF during the first 30 s. Displayed, on an exponential scale, are the concentrations of active thrombin (squares), factor Xa (triangles), factor Va (circles), and factor VIIIa (diamonds) in the reaction as a function of time. The data are plotted on an exponential vertical axis, which reflects the diminishingly small concentrations of products in the early part of the reaction.

View larger version (22K): [in this window] [in a new window]   Fig. 5.   A, concentration of various metabolites as a function of time for the first 30 s of a reaction initiated by 5 pM TF. Represented are active thrombin (squares), active factor VIIIa (diamonds), active factor Va (circles), and active factor Xa (triangles). B, active thrombin (squares) and active factor VIIIa (diamonds) as a function of time in the first 30 s for reactions with factor Va present (filled symbols) or absent (open symbols).

The point at which one observes the initial contribution of factor Va to the generation of thrombin is shown graphically in Fig. 5B, which illustrates a comparison of thrombin (squares) and factor VIIIa (diamonds) formation over the initial 30 s in the presence (filled symbols) and absence (open symbols) of factor V. This figure illustrates that, during the first 12 s of the reaction, thrombin is produced by factor Xa, independent of a factor Va contribution. Subsequently, after 12 s, the feedback activation of factor V permits formation of prothrombinase, which provides increased thrombin levels (filled squares). This may be discerned by the deviation between the factor V-replete (closed symbols) and factor V-deficient (open symbols) reactions at approximately the 15 s time points. It should also be noted that factor VIII activation (diamonds) is dominated by thrombin generated by factor Xa-PCPS during this interval, as the presence of factor V has no influence on factor VIIIa formation (filled and open diamonds).

As the active cofactors are generated, the concentrations of prothrombinase and intrinsic factor Xase are rapidly increased. Fig. 6 illustrates the first 100 s of the reaction initiated by 5 pM TF. By 100 s, the factor Va concentration (filled circles) is ~ 50 pM whereas factor VIIIa concentrations (filled diamonds) are ~ 1 pM. It should be noted here that the factor Xa concentration (filled triangles) is the limiting component for prothrombinase catalyst (open circles) formation, which is ~0.8 pM at 100 s. In contrast, the intrinsic factor Xase (open diamonds) at 100 s (~0.3 fM) is governed by near equivalent concentrations of factor VIIIa (filled diamonds) (~1.0 pM) and factor IXa (open squares) (~1.0 pM). Thus, the K_d for the intrinsic factor Xase plays a major role in regulating the total catalyst concentration to ~0.3 fM.

View larger version (24K): [in this window] [in a new window]   Fig. 6.   Metabolite concentrations over the first 100 s of the reaction initiated with 5 pM TF. Represented are active thrombin (filled squares), active factor VIIIa (filled diamonds), active factor IXa (open squares), intrinsic factor Xase complex (open diamonds), factor Va (filled circles), active factor Xa (filled triangles), and prothrombinase (open circles).

At the 100-s interval (Fig. 7), it can be seen that the original factor VIIa concentration (filled diamonds) (~0.2 nM) is only slightly increased by thrombin feedback activation (Table I, no. 5) and the concentration of extrinsic factor Xase (~4.0 fM) (open diamonds) is declining because of the action of TFPI and AT-III. The small inflection should be noted in total active thrombin concentrations (filled squares), which occurs between ~10 and 20 s. This discontinuity is associated with the beginning of the transition from factor Xa-membrane to prothrombinase activation of prothrombin. By 100 s, thrombin concentrations are approaching 0.5 nM.

View larger version (16K): [in this window] [in a new window]   Fig. 7.   The concentrations of active thrombin (closed squares), active factor VIIa (filled diamonds), and extrinsic factor Xase (open diamonds) as a function of time for the first 100 s for a reaction initiated with 5 pM TF.

Propagation Phase of the Reaction-- An expanded view of the reaction which includes the propagation phase is presented in Fig. 8 (A and B). Active thrombin generation (Fig. 8A, filled squares) continues briskly until 700 s and then begins to slow as AT-III consumes thrombin and the catalysts that produce it. By 700 s, thrombin production and consumption are equivalent. If fibrinogen were present, clotting would have occurred at ~400 s in this reaction (~20 nM IIa), based upon evaluations of the similar reaction conducted in whole blood (31, 32, 51). Factor VIIa generation (filled diamonds) continues to increase, leading to some largely irrelevant increases in the extent of the extrinsic factor Xase (open diamonds).

View larger version (23K): [in this window] [in a new window]   Fig. 8.   A, the concentration of active thrombin (filled squares), active factor VIIa (filled diamonds), and extrinsic factor Xase (open diamonds) are plotted as a function of time over the entire course of the reaction (1200 s) initiated with 5 pM TF. B, metabolites are plotted as a function of time over the entire (1200 s) course for the reaction initiated with 5 pM TF. Represented are active factor Xa (filled triangles), active factor Va (filled circles), prothrombinase (open circles), active factor IXa (open squares), active factor VIIIa (filled diamonds), and intrinsic factor Xase complex (open diamonds). C, the concentration of factor Xa produced by the intrinsic factor Xase (filled triangles) and the extrinsic factor Xase (open triangles) is presented as a function of time. The inset to C illustrates the relative percentage of factor Xa produced by each catalyst.

In Fig. 8B it can be seen that, by 300 s, all of the factor V and factor VIII have been activated to factor Va (20 nM) (filled circles) and factor VIIIa (~0.7 nM) (filled diamonds). Factor VIIIa declines in concentration noticeably beyond 600 s, because of the dissociation of the factor VIIIa A2 domain (Table I, no. 13). The factor VIIIa dissociation process and the inhibition of factor IXa by AT-III lead to a progressive decline in the intrinsic factor Xase (open diamonds) beyond 600 s. Especially interesting is that, over most of the time course, the prothrombinase concentration (open circles) is equivalent with the factor Xa concentration (filled triangles) generation curve illustrating the prominence of factor Xa as limiting component in the expression of prothrombinase, an observation initially made in the studies of Lawson et al. (49) and extended in subsequent studies of the TF induction of coagulation in whole blood (31).

The ultimate dominance of the intrinsic factor Xase (filled triangles) over the extrinsic factor Xase (open triangles) in factor Xa generation is illustrated in Fig. 8C, which shows the concentrations of these two complexes over the time course of the reaction, whereas the inset to Fig. 8C displays the relative percentage of factor Xa delivered by the two catalytic complexes. Initially the extrinsic factor Xase (open triangles) is the major contributor to factor Xa generation because it is the catalyst at the highest concentration. However, by ~300 s, the concentration of the extrinsic factor Xase is superseded by the concentration of the intrinsic factor Xase (closed triangles) whose catalytic properties are ~50 times more efficient than those of the factor VIIa-TF complex. As a consequence, even by 400 s, factor Xa generation is dominated by the intrinsic factor Xase. These observations should be compared with the thrombin curve in Fig. 3, which illustrates that the maximum rate of accumulation of active thrombin occurs at ~650 s, whereas the peak of intrinsic factor Xase generation of factor Xa occurs at ~600 s (Fig. 8C). Fig. 8A also illustrates the dominant role that thrombin plays in the formation of the extrinsic factor Xase; both catalysts peak between 600 and 700 s. The flattening of the two factor Xase catalyst propagation curves is a consequence of factor VIIIa dissociation and factor IXa and extrinsic factor Xase inhibition by antithrombin III.

Termination Phase of the Reaction-- Termination of the thrombin generating reaction is essential to eliminate ever-expanding thrombin generation and clot formation. Each event of catalyst formation is accompanied by a catalyst depletion mechanism. The clearest illustration of catalyst termination is the reduction in the concentration of thrombin under all model conditions. Thrombin inhibition is the ultimate result of complex formation with the stoichiometric inhibitor, antithrombin III. However, equally important are reductions of activities of the extrinsic and intrinsic factor Xases which contribute to further thrombin formation. AT-III, TFPI, and factor VIIIa dissociation are the principle contributors to catalyst elimination in plasma coagulation. The role of TFPI is largely evident in the initiation phase of the reaction (Figs. 1 and 2). When integrated with AT-III, the combination of inhibitors ensures that thrombin formation only takes place when sufficient initiating concentrations of TF are presented (Figs. 4 and 5). TFPI binds with several species, including factor Xa (Table I, no. 20) and the TF-factor Xa-factor VIIa product complex (Table I, nos. 21 and 22). The limited concentration of TFPI plays a significant role by delaying initiation by inhibiting the factor Xa produced. The major role of AT-III in termination is related to nearly quantitative, general serpin inhibition.

The decay of factor VIIIa activity caused by dissociation of the A₂ domain also contributes to the termination phase of the reaction. In our earlier published model (14), we employed an abstract mathematical construction to accommodate the known reduction in factor VIIIa effectiveness to explain the slowing of factor X activation observed under empirical conditions. In this updated model, we employ the rate constants for the established mechanism for factor VIIIa dissociation. This approach yields results that approximate the empirical observations, which display attenuation of factor Xa activation rates.

The factor VIIIa A₂ domain dissociation and reaction termination is essential to the regulation of the concentration of the procoagulant during the termination phase of the reaction. Factors V and VIII are completely converted to their active forms during the propagation phase, and their depletion through active protein C (APC) (for factor Va) and subunit dissociation (for factor VIIIa) is enhanced when those cofactors are dissociated from their active enzyme complexes. Therefore, it is necessary to keep factor Xa and factor IXa concentrations from expanding too rapidly (to allow cofactor dissociation). Factor VIIIa A₂ dissociation is key to the decreased activity of the intrinsic factor Xase activity. Just as the propagation phase is controlled by the expanding concentration and function of the intrinsic Xase activity, the termination phase is controlled by a reversal of this process.

Fig. 9 represents the states and accumulation of the various serpin-AT-III complexes and the factor VIIIa dissociation products associated with the reaction termination during the course of the process. Factor VIIIa-A₂ domain dissociation and accumulation (filled diamonds) is a major contributor to the demise of the efficacy and concentration of the intrinsic factor Xase. This dominance is illustrated by the relative contribution of AT-III to factor IXa inhibition. The product of this complex, factor IXa-AT-III (open triangles), is observed to be at much lower concentration than the concentration of the factor VIIIa-A₂ dissociation product (filled diamonds). A relatively modest contribution of AT-III combining with factor VIIa-TF (filled circles) illustrates the larger role of TFPI in inactivating this complex.

View larger version (24K): [in this window] [in a new window]   Fig. 9.   The concentrations of the inactivation products of the reaction are plotted over the entire 1200-s course for the reaction. Represented are the factor VIIIa-A2 domain dissociation product (filled diamonds), factor Xa-AT-III complex (filled triangles), factor IXa-AT-III complex (open triangles), factor VIIa-TF-AT-III complex (filled circles), and the complex of all thrombin species with AT-III (filled squares).

Table IV illustrates anticipated residual levels of thrombin, factor Xa, factor VIIa-TF, and factor IXa, which would exist at 1200 s in a closed system. The explicit conclusion of the model in this regard (that some enzyme remains) has not been verified by empirical observation, nor has it been tested in empirical experiments. The suggested residual levels of enzymes available at the conclusion of the reaction may be significant, even though they represent only tiny fractions of the amounts of zymogen consumed in the overall reaction. At the low concentrations predicted, they would escape detection in typical empirical experiments designed to evaluate the efficacy of AT-III interaction with any of its serpin companions.

The present "plasma" model does not provide for regulation of factor Va in the decay of prothrombinase because blood (31), plasma, and this model do not include significant levels of thrombomodulin, an essential element of the dynamic protein C system which serves to deplete factor Va. For this reason, the present mathematical model does not offer a comprehensive picture of termination. This will be one focus of continuing work in this laboratory.

Validation-- Few empirical experiments have been published that display the complexity of the reaction system described by the numerical model. The empirical model that most closely approximates the model's conditions is represented by a previous report from this laboratory, which examined the influence of alterations of blood clotting proteins within the normal range of plasma concentration (i.e. 50-150% of the mean plasma value) on the generation of thrombin (27). These studies utilized an in vitro reaction system conducted with purified components at saturating PCPS concentrations; however, in this empirical model, TF and VIIa were preincubated prior to the reaction system, i.e. preformation of the factor VIIa-TF complex. In the empirical experiments, prothrombin and AT-III had the most influence on the total amount of thrombin formed.

The numerical model was therefore computed with a factor VIIa-TF-preincubation term, which eliminates the formation time for the extrinsic factor Xase. The empirical representation of the numerical and empirical experiments exploring the influence of prothrombin concentration are presented in Fig. 10 (A and B), which shows the relative amounts of active thrombin produced as a function of time when prothrombin concentration is varied from 0.7 to 2.1 µM (i.e. 50-150% of the mean plasma value). The comparison of the empirical (Fig. 10B) and numerical (Fig. 10A) representations display great similarity in the relative amounts of thrombin produced for each experimental condition with similar peak values of thrombin observed. The major nonconformity of the numerical analysis with the empirical experiment is in the duration of the initiation phase observed in the empirical experiment, which is noticeably shorter than that observed in the numerical analysis. A likely cause of this discrepancy is illustrated in Fig. 10C. The simulation in Fig. 10C is identical to the simulation in Fig. 10A, except that it assumes the presence of 1% factor Va contamination in the factor V used in the empirical system experiment. With the assumption of 1% factor Va contamination in the experimental system, the numerical and empirical experiments are nearly identical (Fig. 10, compare A and C). The presence of a 1% contamination Va in human factor V preparations is highly likely, based upon previous experiments and experience with natural preparations of this difficult molecule (52-55).

View larger version (33K): [in this window] [in a new window]   Fig. 10.   A, concentration of active thrombin as a function of time produced when the reaction is initiated with varying concentrations of prothrombin. Experimental conditions include 2.1 µM prothrombin (150%, filled diamonds), 1.75 µM prothrombin (125%, filled triangles), 1.4 µM prothrombin (100%, filled squares), 1.05 µM prothrombin (75%, filled circles), and 0.7 µM prothrombin (50%, asterisk). B, empirical data taken from the report of Butenas et al. (27). Active thrombin present as a function of time for 30-150% concentrations of prothrombin. See A for legend identification. C, a representation of the theoretical thrombin produced as a function of time under the experimental conditions of A and B with initial conditions representing a combination of 99% factor V and 1% factor Va.

In a similar experiment, AT-III was varied over the same relative range. The results obtained were again similar to those observed in empirical experiments (data not shown).

	DISCUSSION

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES

The theoretical model for the formation of thrombin and other products by the TF-induced coagulation pathway in the presence of stoichiometric inhibitors presented here provides a reasonable description of the reactions as described by empirical data obtained under similar reaction conditions. The complex reaction pathway of thrombin formation/inhibition observed in empirical analyses is approximated by the theoretical curves generated by our computer model. Most unique qualities of the thrombin-producing reactions, including the observation of reaction thresholds governed by combinations of TFPI and AT-III, are represented in simulations generated by this model. In addition, the quantitative influence of alterations of coagulation protein concentrations within the normal range that influence empirical experimental outcomes are represented in a quantitatively respectable fashion by thrombin generation, as predicted by the theoretical model.

Our model system for the TF pathway with stoichiometric inhibitors involves 27 chemical expressions and 42 rate constants for a reaction that involves 10 initial reactants and 24 intermediates and final products. The goal of developing this model was not to replace empirical chemistry experiments to describe this complex process, but to provide a useful framework for the design and execution of experimental protocols involving this complex array of reagents and reactants. The evaluation of complex reaction arrays using intuition can be extraordinarily misleading in the anticipation of the influence of qualitative or quantitative alterations in individual constituents or reactions on a reaction system outcome. Also of central importance is the utility of numerical models in predicting presently inaccessible quantitative parameters whose required existence is anticipated and assured by the ultimate presence of catalysts, cofactors, serine proteases, and their inhibitor complexes, which must exist to give rise to the responses observed. The computer model has the capacity to anticipate the presence of minute concentrations of reactants and enzymes, which must be present from estimation of the measurable products of their activation. The diminishingly small concentrations that must exist are frequently beyond the realm of the quantitative limits of current analytical devices and technologies. A prime example of this is the anticipated concentration of the factor VIIIa-factor IXa complex illustrated in Fig. 6. To produce the amounts of factor Xa and its complex with factor Va required to generate the thrombin concentrations that have been empirically measured, factor VIIIa-factor IXa complex concentrations between 10¹⁸ and 10¹⁴ M must have been formed.

Comparisons of reaction profiles predicted to occur following preincubation of factor VIIa and TF-PCPS to those predicted without mixing the initial factor VIIa and TF solutions are extraordinary when viewed from a kinetic perspective. At concentrations deemed biologically relevant, the association rate constant (Table II, no. 4) between factor VIIa and TF clearly plays a significant role in the onset of this reaction. Empirical studies have illustrated the essential step of the activation of factor V by thrombin (Table I, no. 16). Although the rate of prothrombin activation by factor Xa-membrane (Table I, no. 9) is ~0.0001 of that for prothrombinase, the model predicts that this tiny level of direct prothrombin activation is sufficient to provide the necessary thrombin initially required to catalyze the initial activations of factor V and factor VIII. The influence of factor VIIIa inactivation by A₂ fragment dissociation (Table I, nos. 13 and 15) and the significance of the regulatory influence of this process are also clearly manifest. Similarly, the relatively slow association between the factor VIIa-TF complex and AT-III might be presumed to be irrelevant, but, as illustrated by the model, this inhibitor has significant influence on the ultimate propagation of the reaction.

Our laboratory developed the first accurate conceptual model involving the formation and expression of a procoagulant complex (prothrombinase) with the original "clot speed" model, which described prothrombinase expression toward prothrombin as a substrate (21). This initial venture at modeling a membrane-bound procoagulant complex successfully provided a quantitative description, based upon theoretical considerations, which described many seemingly paradoxical observations in the generation of thrombin. The clot speed model predicted situations under which enzyme, substrate, and membrane could be observed to be inhibitory, and all of these predictions, when subsequently tested in "wet" chemistry experiments, were found to occur. Subsequently, our laboratory developed a TF procoagulant model that successfully modeled the TF pathway to thrombin at physiologically relevant reagent concentrations. The present model incorporates the stoichiometric inhibitors TFPI and AT-III and provides a reasonably quantitative description of the generation of thrombin and other products and the regulation of this reaction under conditions incorporating normal plasma concentrations of protein with saturating concentrations of membrane. Future refinements will be required that deal with the expression of selective, independent binding sites for the formation of procoagulant complexes on peripheral blood cells (56) and the incorporation of the dynamic protein C system (11) in the overall reaction. However, even in the absence of models incorporating membrane presentation and activated protein C formation and its inhibitory function, the present model has utility in the design of agents intended to alter the qualities and sensitivity of the procoagulant reactions of the TF pathway governed by stoichiometric regulators and is relevant to plasma coagulation. The development of interventions that will both accelerate and decelerate the TF-induced procoagulant response may be useful in the management of thrombophilia and hemophilia.

	ACKNOWLEDGEMENTS

We thank Sriram Krishnaswamy for helpful discussions; Ty Adams for reaction modeling; Hang Zu for prothrombin activation kinetic data; and Thomas Smith, Dan Braucher, and Kihachiro Umezaki for assistance in programming.

	FOOTNOTES

^* This work was supported by National Institutes of Health NHLBI Grants PO-1 HL 46703 and HL 34575.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.

^§ Present address: Howard Hughes Medical Inst., Dept. of Human Genetics, University of Utah, Salt Lake City, UT 84112-5331.

^¶ Present address: Green Mountain Inst. for Environmental Democracy, Montpelier, VT 05602.

To whom correspondence should be addressed.

Published, JBC Papers in Press, March 13, 2002, DOI 10.1074/jbc.M201173200

² T. Orfeo, K. Cawthern, M. Nesheim, and K. G. Mann, manuscript in preparation.

³ In the van't Veer model, the reaction was initiated by preformed 1.25 nM TF-VIIa complex. When simulation is carried out in identical fashion using preformed complex, similar results to those published by van't Veer are obtained.

	ABBREVIATIONS

The abbreviations used are: AT-III, antithrombin III; TFPI, tissue factor pathway inhibitor; TF, tissue factor; PCPS, phospholipid vesicles composed of 25% phosphatidylserine and 75% phosphatidylcholine.

	REFERENCES

TOP ABSTRACT INTRODUCTION MATERIALS AND METHODS RESULTS DISCUSSION REFERENCES