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Modeling of human factor Va inactivation by activated protein C

Background Because understanding of the inventory, connectivity and dynamics of the components characterizing the process of coagulation is relatively mature, it has become an attractive target for physiochemical modeling. Such models can potentially improve the design of therapeutics. The prothrombinase complex (composed of the protease factor (F)Xa and its cofactor FVa) plays a central role in this network as the main producer of thrombin, which catalyses both the activation of platelets and the conversion of fibrinogen to fibrin, the main substances of a clot. A key negative feedback loop that prevents clot propagation beyond the site of injury is the thrombin-dependent generation of activated protein C (APC), an enzyme that inactivates FVa, thus neutralizing the prothrombinase complex. APC inactivation of FVa is complex, involving the production of partially active intermediates and “protection” of FVa from APC by both FXa and prothrombin. An empirically validated mathematical model of this process would be useful in advancing the predictive capacity of comprehensive models of coagulation. Results A model of human APC inactivation of prothrombinase was constructed in a stepwise fashion by analyzing time courses of FVa inactivation in empirical reaction systems with increasing number of interacting components and generating corresponding model constructs of each reaction system. Reaction mechanisms, rate constants and equilibrium constants informing these model constructs were initially derived from various research groups reporting on APC inactivation of FVa in isolation, or in the presence of FXa or prothrombin. Model predictions were assessed against empirical data measuring the appearance and disappearance of multiple FVa degradation intermediates as well as prothrombinase activity changes, with plasma proteins derived from multiple preparations. Our work integrates previously published findings and through the cooperative analysis of in vitro experiments and mathematical constructs we are able to produce a final validated model that includes 24 chemical reactions and interactions with 14 unique rate constants which describe the flux in concentrations of 24 species. Conclusion This study highlights the complexity of the inactivation process and provides a module of equations describing the Protein C pathway that can be integrated into existing comprehensive mathematical models describing tissue factor initiated coagulation.


Background
One of the critical events in blood coagulation is the conversion of large amounts of the zymogen prothrombin to the enzyme thrombin by the enzymatic complex prothrombinase. The prothrombinase complex is formed by the non-covalent interaction between the enzyme factor Xa (FXa) and the non-enzymatic cofactor factor Va (FVa) on a phospholipid surface in the presence of calcium [1][2][3]. The procofactor, factor V, is activated by thrombin to generate a calciumassociated two chain molecule, composed of a heavy chain (FVa HC ) and light chain (FVa LC ) ( Figure 1) [4,5]. The prothrombinase complex increases thrombin generation over FXa alone by 5 orders of magnitude [2]. The explosive burst in prothrombin conversion is essential to rapidly form a stable fibrin clot in response to a vascular injury. The fibrin clot is formed when thrombin cleaves fibrinogen to form an insoluble polymer network [6,7]. This fibrin network becomes cross-linked and stabilized by factor XIIIa which also traps platelets and red blood cells sealing the wound [7][8][9].
It is critical that once a procoagulant response to vascular injury has been initiated that an appropriate anticoagulant response is concurrently mounted. As such, the components of the prothrombinase complex are targets of multiple regulatory mechanisms to terminate the production of thrombin. First, FXa availability is regulated by formation of inhibition complexes with antithrombin (AT), α 1 -antitrypsin, and tissue factor pathway inhibitor (TFPI) [10,11]. The cofactor, FVa, is a target for degradation by activated protein C (APC). APC, a serine protease derived from its plasma precursor protein C (PC) in a thrombin dependent process [12], plays a critical role in inactivating the non-enzymatic cofactor components of both the prothrombinase and the intrinsic tenase complexes, factors Va and VIIIa, respectively [13][14][15]. Additional key components of the PC pathway include: thrombomodulin and endothelial protein C receptor (EPCR) which contribute to APC formation [16]; protein S which functions as a cofactor enhancing APC efficacy [17]; and protein C inhibitor, a suppressor of APC formation [18,19].
An increased risk of thrombotic disease has been associated with partial deficiencies or loss of function mutations in the PC pathway, including deficiencies in PC, its cofactor protein S, or proteins involved in the activation of PC [20]. The most prevalent defect in the PC pathway is a result of a genetic mutation in FV that renders the cofactor resistant to APC inactivation [21]. This resistance to APC was first characterized by Dahlbäck and coworkers [21] and derives from a mutation at one of the APC inactivation sites (Arg 506 ! Gln 506 ) on FV/FVa rendering it resistant to APC cleavage [22].
APC catalyzed inactivation of FVa cofactor activity is a complex, membrane dependent process involving cleavage at multiple sites in the FVa HC , the generation of transient species with partial cofactor activity, and the ultimate disassociation of a fragment of the FVa HC from the molecule rendering it catalytically inactive. APC interacts with FVa or partially proteolyzed FVa species through their light chains to form enzyme • substrate complexes [23][24][25]. Additional studies have identified other regions of interaction, including the proteolytic target residues and the surrounding regions, in the FVa heavy chain involved in the enzyme • substrate complex formation [26][27][28]. Three arginine residues are targeted in human APC • FVa complexes: Arg 306 , Arg 506 , and Arg 679 [15,29]. A number of studies have defined the activities of the partially proteolyzed FVa species [30,31]. Cleavage at either Arg 306 or Arg 506 results in cofactor species, FVa i 306 and FVa i 506 , respectively, with reduced but similar cofactor activity in the prothrombinase complex ( Figure 1) [30,31]. Cleavage at Arg 306 has been shown to be essential for full loss of cofactor activity [32]. The significance of the cleavage at Arg 679 remains undetermined. The final inactive cofactor, FVa i , is a two chain molecule composed of the FVa LC and FVa 1-306 fragment ( Figure 1) [32]. The inactive cofactor binds APC with the same affinity as the intact cofactor [23,33], suggesting a potential role of product inhibition in regulating APC efficacy [33].
The complexity of this process led Hockin and coworkers to construct an ordinary differential equation based description of the network of reactions characterizing the process of bovine APC inactivation of bovine FVa using rate constants gleaned from studies with bovine, human and recombinant human proteins [33]. Empirical studies from a number of laboratories detailing APC inactivation of FVa have shown the cleavage at Arg 506 preceding that at Arg 306 , with these data initially (often) being interpreted as indicating a sequential or ordered kinetic mechanism [15]. However, based on their mathematical modeling, Hockin et al. [33], concluded that the reaction pathway was better described as a random order cleavage process for Arg 306 and Arg 505 (Arg 506 in human FVa) in which each of the cleavages is independent of the other, but occurs at different rates ( Figure 2), and where full loss of cofactor activity requires disassociation of the fragment of the FVa HC downstream from the Arg 306 cleavage [32,33].
Numerous studies examining how components like FXa, prothrombin and protein S modulate APC inactivation of FVa have been reported [14,[34][35][36][37][38][39][40]. These studies routinely employ approaches like progress curve analysis based on curve fitting to compare initial rates, or natural and recombinant cleavage site mutants to detail mechanistic features. To date, however, an integrated approach, one combining the use of physicochemical model construction based on prior research and model verification via construction of corresponding reaction systems using purified proteins, has not been extended to understand APC regulation of the prothrombinase complex stability and function. In this study we build on the prior studies modeling APC inactivation of FVa [33] by incorporating the presence of additional components of the prothrombinase complex (FXa and prothrombin) to construct an empirically validated mathematical model. To test model constructs, a step-wise approach to increasing the number of components was used. Experiments were replicated multiple times with different preparations of proteins in order to generate robust data sets that included estimates of measurement error. At each level of complexity, we empirically monitored multiple analytes, including starting reactants, intermediates and final products to provide multiple points of comparison to evaluate each model's performance.

APC Inactivation of Factor Va Empirical Analyses
The inactivation of FVa (20 nM) was monitored at two APC concentrations (0.5 and 2.0 nM) in the presence of excess PC:PS vesicles (20 μM). The lower concentration of APC was used to initiate a slow inactivation reaction that would enable multiple time points to monitor the sequence of proteolytic events and the residual cofactor activity (as measured by a one stage clotting assay). In studies with 0.5 nM APC, within one minute there was a 50 % loss in cofactor activity ( Figure 3, Panel A, black circles) which continued in a monophasic decay. According to previous studies, measurement of FVa cofactor activity as measured in clotting assays is sensitive to APC cleavage of Arg 506 [41].
Western blot analyses (representative blot in Figure 3, Panel B) monitoring the disappearance of intact FVa HC (residues 1-709) display the rapid proteolysis of FVa heavy chain proteolyzed at either site within the first or third minutes with 2.0 or 0.5 nM APC, respectively ( Figure 3, Panel C, red and black circles, respectively). These results were each fit to monophasic exponential decays (fit not shown), and the initial slope of the fitted curve was determined after normalizing the densitometry to the maximal value (RMV). At 0.5 nM APC, the initial rate of disappearance of the heavy chain was 200 pM/s (based on 20 nM starting concentration); at 2.0 nM APC, the rate of disappearance was nearly three-fold faster (558 pM/s). The disappearance of the heavy chain is initially concurrent with accumulation of the FVa 1-506 fragment ( Figure 3, Panel D). The generation of the FVa 1-506 fragment reaches a maximum within 30 sec (2.0 nM APC) or 90 sec (0.5 nM APC) ( Figure 3, Panel D, red and black circles, respectively). At this point most of the FVa HC has been proteolyzed primarily at Arg 506 , and the secondary cleavage at Arg 306 in the FVa 1-506 species becomes the dominant proteolytic event. Analysis of the appearance of the inactivation fragment FVa 307-506 (resulting from cleavage at both Arg 306 and Arg 506 , Figure 3, Panel E), indicates that as the FVa 1-506 fragment is approaching its maximum level it is being further proteolyzed to (See image on previous page.) Figure 1 Schematic Representation of Factor Va and the Mechanism of the APC-Generated Partially Proteolyzed Species. Human factor Va (FVa), consisting of the non-covalently associated 105 kDa heavy chain (FVa HC ) and 74/72 kDa light chain (two carbohydrate variants) (FVa LC ), has maximal cofactor activity in the prothrombinase complex. FVa contains three APC cleavage sites: Arg 306 , Arg 506 , and Arg 679 ; cleavage at Arg 679 plays a minimal role in altering cofactor activity and so will not be considered further. APC cleaves FVa in a random, non-sequential manner at Arg 306 and Arg 506 leading to the formation of partially proteolyzed species ( the FVa 307-506 fragment. In order to isolate the kinetics of proteolysis at Arg 306 , FVa i 506 was used as the starting material in inactivation studies with 0.5 nM APC ( Figure 3, Panel F, open circles). This membrane bound partially proteolyzed species has already lost most of its cofactor activity in a clotting assay [30,31,33]. Tracking of the final inactivation fragment, FVa 307-506 , by Western blotting reveals a monophasic accumulation of this fragment (fit not shown) ( Figure 3, Panel F).

Mathematical Simulations
Using the non-sequential reaction mechanism proposed by Hockin et al. (Table 1, Figure 2) [33] and kinetic constants used to describe their bovine system of proteins, in silico simulations of APC inactivation of FVa were performed and predicted time courses for various FVa inactivation species compared to the empirical data presented in Figure 3. Mathematical simulations show good correlation with empirical measurements of FVa cofactor activity and the densitometric analysis of FVa HC (Figure 3, Panel A and C, dashed lines vs. filled circles, respectively). Hockin et al. reported a good fit using this model between simulated and empirical time courses of APC catalyzed changes in cofactor activity in the bovine system [33]. The present study's correlation between clotting assay measurements, in silico FVa cofactor activity, and densitometric analysis of FVa HC are consistent with the findings of earlier studies that FVa cofactor activity is based on the amount of intact FVa HC present [41].
However, the predicted generation and disappearance of the FVa 1-506 fragment are significantly delayed compared to the empirical data for this inactivation fragment (    Though the bovine based Hockin et al. [33] rate constants clearly captured the disappearance of FVa HC as measured through activity and fragment analysis, they did not accurately predict the presentation of either the FVa 1-506 or FVa 307-506 fragments in the human system. The rapid clearance of the FVa 1-506 fragment and faster appearance of the smaller FVa 307-506 fragment led us to speculate that their model rate constant for the cleavage at Arg 306 was not appropriate for the human system of proteins. From a comparison of the empirical data to the initial in silico simulation we estimated that the rate of cleavage occurring at Arg 306 was three times faster than the model predictions. To account for this the rate constant (  [33]. The rate constant for cleavage at Arg 306 , k 4 , was adjusted to 0.192 s -1 for all subsequent studies. Our empirically driven adjustment to k 4 results in a second order rate constant for the overall process of cleavage at Arg 306 of 2.74 x 10 7 M -1 s -1 . Previously published second order rate constants for the APC cleavage at Arg 306 in human FVa (1.55 x 10 6 M -1 s -1 [42], 1.7 x 10 6 M -1 s -1 [31], and 2.5 x 10 6 M -1 s -1 [41]) are an order of magnitude lower than our currently derived estimate and lower than that used by Hockin et al. (9.1 x 10 6 M -1 s -1 [33]) to describe the bovine system. Studies using human proteins on APC inactivation of FVa reported second order rate constants for the cleavage at Arg 506 ranging from 3.0-4.3 x 10 7 M -1 s -1 [31,41]; a more recent study by Nicolaes et al. [42] has reported the second order rate constant to be 1.17 x 10 8 M -1 s -1 . This value is quite similar to the value (1.42 x 10 8 M -1 s -1 ) used in the current study.
An alternative approach to fitting these data sets would have been to use an optimization method in which all rate constants may be simultaneously adjusted to generate a best fit to all the experimental data. Failure of model predictions to fit the empirical data can stem from the collective effect of small errors in the rate constants used. It is not uncommon for laboratories to report 2-3 fold differences in rate constants of biochemical reactions and interactions measured in vitro. These ranges are accepted because the findings would fall within the margin of error expected for slightly variable experiments conducted in different laboratories, or for different techniques used to measure a rate constant. Such variation can have significant effects on model outputs [43]. However, lack of fit can arise from a fundamental mistake or omitted interaction in the reaction description. Therefore, we have not taken this approach as we developed this model.

APC Inactivation of Factor Va: Effect of Factor Xa Under Non-saturating Conditions
It has been widely reported that FXa shields FVa from degradation by APC through its reversible association in the prothrombinase (FXa • FVa) complex [34][35][36][37][38]44]. The mechanistic basis of this protection has been conceptualized as a mutually exclusive competition for association with membrane bound FVa, so that when FVa is associated with FXa it is not a substrate for APC [35], at least at the Arg 506 site [37]. In order to develop a model description that can predict the extent of the protective effect exerted by FXa against APC proteolysis of FVa over a range of reactant concentrations, empirical experiments were performed at saturating and non-saturating concentrations of FXa.

Empirical Analyses
In the first set of experiments, prothrombinase was preformed under non-saturating concentrations with FVa in excess (0.2 nM FVa and 0.1 nM FXa) on 20 μM PC:PS vesicles. Under these conditions, if the K D = 0.5 nM for the (FXa • FVa) complex (estimate based on previous studies [45]), only 13 % of the FVa would be expected to be bound to FXa. Inactivation reactions were initiated with 0, 0.5, or 2.0 nM APC. For these experiments, the amount of thrombin generated in the 0 nM APC control was set at 100 % prothrombinase activity, with thrombin generation levels represented relative to the prothrombinase activity in the 0 nM APC control. The loss of prothrombinase activity following the addition of APC at either concentration ( Figure 4, Panel A, filled circles) appeared to follow a monophasic decay (fits not shown).

Mathematical Simulations
To simulate this experiment, six additional equations were required ( . Our initial mathematical simulations were carried out utilizing a K D of 0.5 nM [45], resulting in an initial concentration of 26 pM for prothrombinase, with off and on rates of 0.2 s -1 and 4.0 x 10 8 M -1 s -1 , respectively (Table 2, Eqn 11) [46,47]. The mathematical predictions for the loss of prothrombinase activity in the presence of either 0.5 or 2.0 nM APC showed an approximately 2-fold faster loss in prothrombinase activity than observed in the empirical experiments ( Figure 4, Panel A, dashed lines vs. filled circles).
To generate a better fit to the empirical data we first examined the dissociation constant of FXa and FVa on PC:PS vesicles, which by literature reports varies from as low as 83 pM [31] to as high as 1 nM [48]. Simulations to test the wide range of reported K D values were first conducted by altering the k off rate constants for the prothrombinase complex, while maintaining the same k on rate constant of 4.0 x 10 8 M -1 s -1 . For each K D value tested, the starting concentrations of free and bound species (FVa, FXa, and FXa • FVa) were recalculated and used as the initial conditions for a mathematical simulation. An adjustment made to the off rate for the prothrombinase complex resulting in a K D of 0.1 nM (58.8 pM prothrombinase initially) generated an in silico prediction that more closely represented the empirical results seen with both 0.5 and 2.0 nM APC (Figure 4, Panel A, black and red, respectively, dotted lines vs filled circles).
In addition to altering the k off rate constant while maintaining a k on rate constant to reach a desired overall K D , changes to both isolated rate constants were explored as a potential solution. One alternative that we explored was adjusting the k on rate constant to 1.5 x 10 8 M -1 s -1 and setting the k off rate constant to values that yielded the same set of K D values. Figure 4, Panel B presents these model results at 0.5 and 2.0 nM APC (black and red, respectively) for K D 's of 0.5 nM (dashed lines) and 0.1 nM (dotted lines) for prothrombinase assembly. In contrast to simulations based with a k on value of 4.0 x 10 8 M -1 s -1 this analysis indicates that a K D of 0.5 nM yields a better fit to the data.
These results highlight an important feature of dynamic systems like this one: the magnitude of the observed protective effect may depend not only on the concentrations of the three protein components and the K D 's characterizing their interactions, but also on the rate constants defining the competing K D 's.

APC Inactivation of Factor Va: Effect of Factor Xa Under Saturating Conditions
The previous analysis under non-saturating conditions indicates that either a dissociation constant of 0.1 nM or 0.5 nM for the prothrombinase complex can recapitulate our empirical data depending on the values assigned to k on and k off . From the point of view of empirically measuring k on and k off values the differences in these assigned values (e.g.~2.5 fold between 1.5 or 4.0 x 10 8 M -1 s -1 ) are small and within the margin of error for this type of analysis. In order to further explore which dissociation constant and set of rate constants is appropriate for the empirical data, we tested APC inactivation of FVa under saturating conditions with FXa present in excess (Figure 4, Panel C and Figure 5).  [49] and the practical requirement for a concentration suitable for Western blot analysis. Active site blocked FXa was used to prevent FXa proteolysis of FVa [50].

Empirical Analyses
Reactions where the FVa population is highly associated with FXa* showed 70 % of FVa HC remained after 1 minute (Figure 4, Panel C and Figure 5, Panel A, red circles) compared to~10 % without FXa* (Figure 3, Panel C, red circles). Comparison of initial rates showed a greater than 7-fold reduction in the initial rate of 2.0 nM APC proteolysis of FVa due to its FXa association (77 pM/s vs. 558 pM/s). Consistent with this overall suppression of heavy chain proteolysis, generation of the FVa 1-506 fragment was slower in the presence of FXa*, reaching a maximum in 2 min vs~30 s in its absence ( Figure 5, Panel B vs. Figure 3, Panel D, red circles). Its clearance was also suppressed, taking~4 min to decline to 50 % of its maximum as opposed to~1 min without FXa* ( Figure 5, Panel B vs. In the mathematical model as constructed APC proteolysis at any site is permitted only with FVa species not bound to FXa (or FXa*). The more rapid clearance of FVa 1-506 observed empirically suggests two possible problems with the model construct: 1) the binding affinity between FVa i 506 and FXa is weaker than the model value of 1 nM based on studies with bovine FVa [32] or 2) APC can cleave at Arg 306 when the FVa i 506 species is bound to FXa. Previous studies with recombinant human proteins have reported K D values ranging from 3.9 nM [31] to~1.35 nM [51]. To evaluate the likelihood that a weaker binding affinity was the culprit, mathematical simulations were conducted where the K D for the FXa • FVa i 506 complex was varied; to fit the empirical data the K D for this complex would have to be greater than 10 nM (data not shown) and as such this adjustment was discarded as a viable option.
The alternative explanation for the empirical data presenting the clearance of the FVa 1-506 fragment is that when bound to FXa, the FVa i 506 species is susceptible to APC cleavage at Arg 306 . To explore this hypothesis, an additional reaction was added to the mathematical construct: APC cleaving (FXa • FVa i 506 ) at Arg 306 . To establish whether there was a value for this rate constant that would improve the fit, simulations varying its value were run using the model value (2.7 X 10 7 M -1 s -1 ) for the cleavage of Arg 306 in free FVa as a point of reference. Representative examples are presented in Figure 5. The mathematical analysis suggests that if the Arg 306 in (FXa • FVa i 506 ) is a target for APC, the magnitude of the rate constant regulating the event is~10 to 20 % that for free FVa.

Structural Analysis
Analysis of the proposed structural models of the prothrombinase complex allows for the possibility that Arg 306 is susceptible to APC in the FXa • FVa i 506 complex [52][53][54]. The prothrombinase complex models differ in the orientation of FXa relative to FVa, and the binding interface between enzyme and cofactor. What is consistent between the three models is that Arg 506 in FVa is occluded by the binding of FXa, rendering this site inaccessible to APC cleavage ( Figure 6). Additionally Arg 306 is surface exposed and not masked by FXa in any of the models. As such, the structural models suggest that this arginine site could be accessible to APC cleavage even when FVa or FVa i species are complexed to FXa. However, mathematical simulations allowing the cleavage to occur in FVa while bound to FXa overestimate both the rate of FVa HC disappearance and the rate of generation of FVa 307-679/709 fragment observed empirically (data not shown). The data was best fit by modeling only the FXa • FVa i 506 complex as susceptible to cleavage at Arg 306 . Thus the kinetic and structural modeling in concert with the empirical data suggests two effects: binding of FVa to FXa blocks both cleavage at Arg 306 and Arg 506 despite the unmasked surface exposure of Arg 306 ; and that binding of FXa to FVa i 506 improves the geometry at Arg 306 rendering the site vulnerable to APC.

APC Inactivation of Factor Va: Effect of Prothrombin
In addition to the protection afforded to FVa by FXa in the prothrombinase complex, the substrate of the complex, prothrombin (PT), has also been shown to inhibit APC inactivation of FVa [39,40,58]. Guinto and Esmon reported that prothrombin protects the intact FVa molecule by competing with APC for binding to FVa [58]; more recent studies have reported a K D of between 500-700 nM for FVa and prothrombin [39,40].

Empirical Analyses
In order to quantify the degree to which prothrombin protects FVa from APC, reactions were carried out at  Contrary to inactivation experiments in the absence of prothrombin (Figure 3, Panel B), there is a more noticeable generation of the FVa 307-679/709 fragment (Figure 7, Panel A inset) that enables more reliable densitometric quantitation (Figure 7, Panel C).

Mathematical Simulations
An additional reaction describing the reversible formation of the FVa • PT complex with a K D of 700 nM [39] was added to the ordinary differential equation (ODE) network ( Table 2, Eqn 17). The initial concentration of preformed FVa • PT complex, given 20 nM FVa and 1.4 μM PT, was calculated to be 13.3 nM. Simulated time courses for FVa heavy chain fragments generated by APC (2.0 nM or 20.0 nM, red and blue lines, respectively) proteolysis did not correlate well with the empirical data, in fact suggesting that there would be little to no protection of FVa by 1.4 μM prothrombin if the K D was 700 nM (Figure 7, Panel A, dashed with dotted lines vs. filled circles; Note, red dashed with dotted line is underneath blue dashed line). Test simulations varying the K D were performed by altering the k off constant for the equilibrium describing the prothrombin and FVa interaction. Even when a K D of 50 nM was assigned to the interaction,~10-fold less than published estimates [40] the resulting simulations did not capture the observed level of protection (data not shown).
In the computational model one of the assumptions is the presence of unlimited binding sites located on a single surface so that competition between proteins for occupancy of the phospholipid surface is not a factor. At 20 μM PC:PS vesicles, and setting the phospholipid/protein binding site ratio between 30 and 60 [59][60][61], available binding sites fall in a range between 220 and 440 nM. The simultaneous solution of the equilibrium expressions for prothrombin, FVa, and APC (defined as non-catalytic) binding to phospholipid membrane with K D 's of 230 nM [62], 2.72 nM [63], and 500 nM [64], respectively, shows that between 77-78 % of binding sites are occupied by prothrombin, over 96 % of the FVa is bound, and only approximately 11 % of the APC is able to bind to the surface whether 2.0 or 20.0 nM APC is present. By using this bound fraction of APC as the catalytically relevant population, the mathematical simulations better captured the kinetics of FV HC proteolysis in the presence of 1.4 μM prothrombin (Figure 7, Panel A, solid lines vs. filled circles). Similarly, comparison of empirical and simulated time courses for the three generated fragments (Figure 7, Panels B-D, filled circles vs. solid lines) also show improved fits. Our findings indicate that there may in fact be a tighter association between FVa and PT, in line with the findings of Yegneswaran et al. (Figure 7. Panel A, filled circles vs. dashed lines) [40].
To verify that the impaired FVa proteolysis observed under the empirical reaction conditions was not solely due to competition for phospholipid binding sites, mathematical simulations were constructed using the adjusted APC concentration, but without permitting the formation of the FVa • PT complex. This model construct failed to predict the time courses for FVa HC disappearance and the fragments FVa 1-506 and FVa 307-679/709 (Figure 7, Panel A, B and C, filled circles vs. dotted lines). Thus the computational studies indicate that the prothrombin dependent suppression of FVa proteolysis by APC observed in our empirical reactions is due both to binding site competition (prothrombin membrane association limiting APC access to the membrane surface) and to the FVa • PT complex not being a substrate for APC. Comparisons of the empirical data for FVa HC proteolysis (Figure 7, Panel A) with the simulations representing unlimited binding sites (dashed with dotted lines), limited binding site adjustment but no FVa • PT interaction (dotted line) and the limited binding site adjustment with FVa • PT complex formation (K D = 700 nM) (solid line) indicate that under these conditions approximately half of the observed PT dependent suppression is due to FVa • PT complex formation.

APC Inactivation of Factor Va: Effect of Prothrombin and Saturating Levels of Factor Xa Empirical Analysis
In order to measure the cumulative protection prothrombin and saturating levels of active site blocked FXa have on the APC inactivation of FVa, reactions were carried out at physiological concentrations of FVa (20 nM) and prothrombin (1.4 μM), with saturating levels of FXa* (30 nM) on phospholipid vesicles (20 μM). Figure 8 summarizes the time course data for FVa HC proteolysis by 2.0 nM APC for reactions constructed with FVa, FVa + FXa*, FVa + PT, and FVa + FXa* + PT. The combined presence of prothrombin and FXa* provided maximum protection against APC inactivation. Figure 9 presents the empirical data comparing the proteolysis of FVa in the presence of 30 nM FXa* and   Figure 9, Panel A shows a representative Western blot for the reactions with 2.0 nM APC. Visual inspection of Western blots revealed extremely low generation of the inactivation fragments FVa 1-506 , FVa 307-679/709 , and FVa 307-506 . Productions of these fragments were consistently barely above detection limits and thus made measurement of these fragments highly variable, and as a result were not considered further.
Addition of 2.0 nM APC to the reaction system led to only 39 % of the FVa HC being proteolyzed after 20 minutes (Figure 9, Panel B, red circles). The initial rate of disappearance was 14 pM/s, nearly 41-fold lower than the inactivation rate in the absence of both prothrombin and FXa (558 pM/s; Figure 3, Panel A, red circles). Inactivation studies with 10 times more APC resulted in a nearly 8-fold increase in the initial rate of inactivation (100 pM/s) (Figure 9, Panel B, blue circles).

Mathematical Simulations
To incorporate the formation of the ternary prothrombinase • substrate complex, four additional reactions are required ( Table 3, . Based on studies indicating that prothrombinase complexes formed with partially proteolyzed FVa species have a K M similar to the complex formed with intact FVa [31,51], we set the K M of prothrombin for any prothrombinase complex to be the same and consistent with previous modeling studies [46,47]. We included two additional reactions (Table 3, Eqns 23-24) to allow for the dissociation of the ternary complex formed with some of the partially proteolyzed FVa species. Solving the simultaneous equilibrium of FVa, FXa*, and PT in a lipid independent system, approximately 18.9 nM FVa is found in a prothrombinase complex (FVa • FXa or FVa • FXa • PT), approximately 720 pM FVa is in the FVa • PT complex, and approximately 360 pM FVa is unassociated.
Simulated time courses for the disappearance of the FVa HC were generated for the experiments where 2.0 nM or 20.0 nM APC was added. Based on the findings from the APC inactivation studies of FVa in the presence of prothrombin, APC effective concentrations were adjusted to 11 % of the total to account for the limited binding sites (Figure 9, Panel B, solid lines). The simulations' initial rates of proteolysis of the FVa HC (6 pM/s) were less than half the measured empirical rates (14 pM/s). The mathematical simulations greater level of protection extends through 20 minutes with higher amounts of FVa HC anticipated compared to empirical results (2.0 nM APC: 76 % vs. 61 %, respectively; 20.0 nM APC: 26 % vs. 11 %, respectively) ( Figure 9, Panel B). Empirical studies were also conducted with 50 μM PC:PS vesicles and did not result in any significant changes in the initial rate or overall time course of FVa HC proteolysis by 2.0 nM APC (data not shown), suggesting that under the current conditions, phospholipid membrane accessibility is not impeding the inactivation reaction.
Potential explanations for this disparity include: 1) accumulated effect of small errors in rate constants; 2) improved binding of APC to the surface; 3) a misestimate of the affinity of the prothrombinase • prothrombin interaction; and 4) the affinity of human APC for FVa is slightly higher, e.g. the K D is less than 7 nM. Additional experiments will be required to distinguish between these possibilities. It is important to note that phospholipid composition plays an important role in proteinmembrane interactions which can extend to effects on enzyme activity (Reviewed in [65]). Several studies have directly highlighted the effect of membrane composition on the APC inactivation of FVa [66][67][68].

Conclusions
We have constructed an ODE based model of APC inactivation of the prothrombinase complex. This model includes 24 chemical reactions and interactions with 14 unique rate constants which describe the flux in concentrations of 24 species (Figure 10). We did so in stepwise fashion, analyzing the time course of FVa inactivation in empirical reaction systems with increasing number of interacting components and generating corresponding model constructs of each reaction system. Reaction mechanisms, rate constants and equilibrium constants informing these model constructs were initially derived from various research groups reporting on APC inactivation of FVa in isolation [22,[31][32][33], in the presence of FXa [14,[34][35][36][37][38], and in the presence of prothrombin [39,40,58]. Model predictions were assessed against empirical time course data measuring the appearance and disappearance of multiple FVa degradation intermediates as well as prothrombinase activity changes, with experiments done multiple times, with plasma proteins derived from multiple preparations. Current coagulation models that incorporate the protein C pathway [69][70][71][72][73][74][75][76][77][78][79] provide an often simplistic one step inactivation reaction of FVa by APC, without incorporating the feedback inhibition of the APC • FVa 1-306 • FVa LC complex, the ability of partially proteolyzed FVa species to form catalytically active prothrombinase species, and the formation of FVa • PT species. They either do not provide empirical data verifying their model construct or rely on a single global output to validate the entire model construct. Wagenvoord et al. have outlined the limitations of mathematical models of complex reaction networks validated by a single analyte [80]. This study highlights the complexity of the inactivation process and is a crucial step towards creating a module of equations describing the PC pathway that can be integrated into existing comprehensive mathematical models describing the interplay between procoagulant and anticoagulant processes during tissue factor initiated coagulation.

Reagents and Proteins
Human prothrombin and Factor X were isolated according to methods as described [81]. Factor Xa was prepared as previously described using Russel's Viper Venom Factor Xa Activator [82,83]. Human Factor V was purified from citrated plasma according to previously described procedures [83]. Activated human protein C was purchased from Haematologic Technologies, Essex Junction, VT. 1,2-Dioleolyl-sn-Glycero-3-Phospho-L-Serine (PS) and 1,2-Dioleoyl-sn-Glycero-3-Phosphocholine (PC) were purchased from Avanti Polar Lipids, Inc (Alabaster, AL) and phospholipid vesicles (PC:PS) composed of 75 % PC and 25 % PS were prepared as described [60,84]. Spectrozyme TH and recombinant hirudin were purchased from American Diagnostica, Inc (Greenwich, CT) and EDTA was purchased from Sigma (St Louis, MO). D-Phe-Pro-ArgCH 2 Cl (FPRck) was prepared in house. Monoclonal anti-fV (αHFV#17) was obtained from the Biochemistry Antibody Core Laboratory (University of Vermont) and a goat anti-mouse IgG conjugated to HRP was purchased from Southern Biotech (Birmingham, AL). Active site blocked FXa (FXa*) was produced according to the previously published method [85].
Multiple preparations of FV/FVa and different lots of APC were utilized for these studies.  APC for 20 minutes at 37°C, after 20 minutes an additional 5.0 nM APC was added. Following the inactivation reaction, the FVa i 506 containing reaction was placed on ice and incubated with 1.0 mM di-isopropyl phosphate to inactivate the APC.

Inactivation of Factor Va by Activated Protein C
Solutions of factor Va (20 nM) in HBS-PEG-Ca with 20 μM PC:PS were prepared at 37°C. At the zero time point 0.5 or 2.0 nM APC was added to the FVa solution. At designated intervals samples were removed, and either quenched in denaturing sample preparation buffer (0.31 M Tris[hydroxymethyl]aminomethane, 10 % sodium dodecyl sulfate, 50 % glycerol, 0.5 % Bromophenol Blue, pH 6.8) and analyzed either for proteolytic cleavage using SDS-PAGE and subsequent immunoblot analysis (Western blotting), or for co-factor activity using a onestage clotting assay in factor V deficient plasma. Inactivation reactions were diluted 40-fold in an appropriate volume of factor V deficient plasma and assayed immediately. Clotting activity is represented as a percentage of the clotting time observed for unproteolyzed FVa.

Inactivation of Factor Va by Activated Protein C in the Presence of Factor Xa
Activity analysis: Rates of APC inactivation of FVa (0.2 nM) in the presence of PC:PS vesicles (20 μM) and FXa (0.1 nM) were assessed with the prothrombinase complex preformed for five minutes at 37°C in HBS-PEG-Ca. At the zero time point APC (0-2.0 nM) was added to the prothrombinase complex solution. At designated intervals aliquots were removed and added to a minimal amount of concentrated prothrombin (1.0 μM final). After 90 s, aliquots were quenched with the addition of 0.5 M EDTA (25 mM final) and subsequently monitored for thrombin activity against the chromogenic substrate SpecTH. Thrombin concentrations were determined by reference to a standard curve and the level of thrombin generated used as a relative measurement of prothrombinase concentration compared to measurements with intact FVa.

Proteolytic analysis:
Prothrombinase complex was preformed on 20 μM PC: PS vesicles in HBS-PEG-Ca for four minutes at 37°C using 30 nM fluorescein-active site blocked FXa (FXa*) and 20 nM FVa. A sample aliquot was removed, APC added (2.0 nM), aliquots taken at selected time points and quenched into denaturing sample preparation buffer, and subsequently analyzed by Western blotting.

Western blotting and densitometric analyses
All FVa samples were run under reducing conditions. FVa heavy chain and inactivation fragments were analyzed by fractionation on 4-12 % SDS-PAGE acrylamide slab gels followed by electrophoretic transfer of proteins to Immobilon-FL membranes (Millipore Inc., Billerica, MA), and detected using a mouse monoclonal antibody (αHFV#17) that binds to an epitope between residues 307-506 of the factor Va heavy chain and a goat anti-mouse IgG conjugated to HRP (Southern Biotech). Luminescence was detected using a Fuji LAS-4000 in conjunction with Ima-geCapture software (Fujifilm, Tokyo, Japan). Densitometry was carried out using MultiGauge software (Fujifilm, Tokyo, Japan). For each species, densitometric values at each time point were normalized to their maximal value during the time course.

Mathematical modeling
The description using ordinary differential equations of the reaction pathway for bovine APC inactivation of bovine FVa reported by Hockin et al. [33] was used as a starting framework (Figure 2; Table 1). The scheme describes a random order cleavage process for Arg 306 and Arg 506 in which the cleavages are independent of each other and characterized by substantially different rate constants. Our simulator employs a fourth order Runga-Kutta algorithm to generate a series of timedependent concentration profiles for all reactants, intermediates, and products.
Factor Va cofactor activity will be represented by the cumulative presence of FVa HC species present at any point in time. For measurements of prothrombinase activity, the sum of all FVa species was added to the weighted (20 %) sum of FVa i 306 , FVa i 506 , and FVa i 306/506 . Factor Va inactivation fragments will be represented by including respective fragments both associated and dissociated from the FVa LC .
To generate an appropriate simulation of the empirical experiment where FXa and FVa are pre-incubated together on the phospholipid surface to form the prothrombinase complex prior to the addition of APC, the concentrations of free FXa and FVa, and the prothrombinase complex were solved. For example, studies with 200 pM FVa and 100 pM FXa, for each potential K D (e.g. 0.1 nM or 0.5 nM, respectively), the values of the three initial conditions ranged from 41-74 pM for free FXa, 141-174 pM for free FVa, and 59-26 pM for prothrombinase. The concentrations of the three species then served as the initial conditions in the simulations in conjunction with either APC concentration.
In silico simulations were carried out with the initial concentrations used in the empirical experiments and the relevant output calculated and compared to the appropriate analyte measured in the empirical experiments.