Modeling the role of platelet-released polyphosphates in tissue factor-initiated coagulation under flow

Abstract

Background

Activated platelets release polyphosphate (polyP), a linear polymer of inorganic phosphate residues, from dense granules. Experiments performed under no-flow conditions show that polyP alters the kinetics of tissue factor (TF) pathway reactions, accelerating factor XI (FXI) activation by thrombin and FV activation by factor Xa and thrombin, and may impact inhibition by tissue factor pathway inhibitor α (TFPIα). How polyP influences this pathway in conjunction with platelet deposition under flow remains understudied.

Objectives

This study aimed to investigate how polyP-mediated acceleration of FV and FXI activation modulates thrombin generation under flow in TF-initiated coagulation.

Methods

We extended a previously validated mathematical model of platelet deposition and coagulation under flow to examine polyP-mediated effects after a small vascular injury during intravascular clotting. Simulations varied the surface density of TF exposed, wall shear rate, and plasma TFPIα concentration.

Results

Even without direct interaction between polyP and TFPIα in the model, thrombin generation is much less sensitive to TFPIα concentration with polyP, in a TF-dependent manner. Thrombin concentrations are much more robust across the range of physiological shear rates with polyP than without. Incompletely known accelerations by polyP strongly impact relative contributions of accelerations of FV and FXI activations. PolyP shifts the threshold TF density for a thrombin burst to lower TF densities and shortens the lag time to thrombin generation.

Conclusions

We identify polyP’s effects on blood coagulation beyond acceleration of FV and FXI activation: polyP can produce experimentally-observed TFPIα insensitivity without directly interacting with TFPIα and can improve the robustness of thrombin generation across shear rates.

Graphical abstract

Essentials

• •Platelet-sized polyphosphates (polyP) speed activation of FV and FXI by thrombin and FV by FXa.
• •Mathematical modeling of the TF pathway shows polyP’s impact on thrombin generation.
• •Thrombin’s apparent insensitivity to TFPI with polyP is due to faster FXI and FV activation.
• •Platelet polyP makes thrombin generation more uniform across physiological shear rates.

1. Introduction

Polyphosphates are ubiquitous across bacteria, fungi, plants, and animals [1]. Polyphosphates from bacteria and human platelets have attracted growing attention for its roles in inflammation and blood clotting. A linear polymer of inorganic phosphate residues, the length of polyphosphate differentiates the short polymers (∼60-100 phosphate units) secreted by human platelets (which we will call polyP) from the much longer polymers (up to several thousand phosphate units) released by microorganisms. These length differences are functionally significant, as different polyphosphate sizes differentially modulate the blood clotting system [2].

Upon platelet activation, platelets release polyP from their dense granules [3]. Early clinical observations linked defects in platelet dense granules to bleeding tendencies [4,5]. Later studies provided a potential explanation for that link: defects in dense granules can lead to polyP levels ∼one-tenth of normal [6]. Moreover, persons with Hermansky-Pudlak syndrome have platelets lacking polyP and exhibit reduced clotting activity restored by the addition of polyP, highlighting the procoagulant role of polyP [7].

More recent findings highlight the role of polyphosphates in both the contact and tissue factor (TF) pathways. In general, polyphosphates PolyP are highly anionic, binding tightly to proteins which initiate the contact pathway of coagulation. The first reported role for polyphosphates in blood clotting demonstrated that they are strongly procoagulant and trigger the contact pathway [8]. Subsequent studies showed that long-chain polyphosphates, as opposed to short-chain polyP released by platelets, can potently activate the contact pathway and trigger thrombosis and inflammation [9].

Platelet-secreted polyP also enhances the activation of key clotting factors in the TF and common coagulation pathways. Studies have shown that polyP accelerates the proteolytic activation of factor V (FV) by both factor Xa (FXa) and thrombin, resulting in an accelerated thrombin burst during plasma clotting reactions [8,10]. PolyP is also reported to decrease the activity of the coagulation inhibitor TF pathway inhibitor α (TFPIα), possibly indirectly through its acceleration of FV activation [8]. Additionally, polyP accelerates factor XI (FXI) activation by thrombin (by ∼3000-fold), further supporting a role for polyP in amplifying thrombin generation [11,12].

These observations have prompted increased interest in polyP as a therapeutic target [13], as inhibiting polyP function in vivo may provide a strategy for developing antithrombotic or anti-inflammatory agents, potentially with reduced bleeding risk compared with conventional anticoagulant therapies [14]. Also, since added polyP can shorten clotting times in plasma from patients with hemophilia A or B, polyP may be useful as a hemostatic agent [15]. In hemophilia plasma, the procoagulant effect of polyP was additive with recombinant activated factor VII (FVIIa), suggesting it could complement existing therapies [15].

To better understand the potential therapeutic roles of polyP, it is important to examine how polyP influences the coupled reactions of coagulation under flow. In vitro studies of polyP are generally performed under static, purified conditions, capturing key biochemical effects but overlooking the complexity of the interconnected reactions that regulate coagulation under flow. To this end, Yeon et al. [16] developed a mathematical model of the contact pathway of coagulation that examined the effect of surface-immobilized polyP localized on the walls of microfluidic channels during clot formation. Their simulations predicted that localized short-chain polyP can accelerate clotting of flowing blood plasma at subphysiological shear rates. Their model treated all coagulation reactions as occurring in the moving plasma phase. However, critical coagulation reactions occur on the surfaces of activated platelets that have accumulated at the injury site, as included in our previous modeling studies [17,18]. Our studies indicated that without such surface binding, activated coagulation factors are rapidly cleared by flow, reducing their local concentrations and limiting thrombin generation. Motivated by these observations, we extend our existing, experimentally validated mathematical model of platelet deposition and TF-mediated coagulation under flow to investigate the role of polyP in modulating thrombin generation. Our model extensions take two forms: an “implicit” model, where the catalytic rate constants for FXI and FV activations are set to the polyP-accelerated values from the start of the simulations, and an “explicit” model, where these catalytic rates depend dynamically on the bound polyP concentrations.

2. Methods

Previous versions of the model on which this paper is based have been published [[17], [18], [19], [20], [21], [22], [23], [24]] and their predictions experimentally validated [[25], [26], [27], [28]]. Here, we sketch that model and indicate how we extend it to include platelet-released polyP.

2.1. Review of our previously published mathematical model of coagulation and platelet deposition

We consider an arteriole-sized vessel in which an injury has uncovered a small patch of subendothelium, exposing TF and collagen to flowing blood, triggering coagulation (Figure 1A [19]) and platelet deposition (Figure 1B [23]), respectively. We focus on events in a thin fluid layer above the injury, which we call the “reaction zone” (RZ) (Figure 1B) and whose initial height (∼1-2 μm) we estimated by a boundary layer analysis accounting for the near-wall flow velocity and the diffusivities of the relevant protein and cellular species [17].

Figure 1.

Schematic of coagulation reactions and representation of polyP dynamics. (A) The coagulation enzyme-cofactor complexes form on the subendothelium and APSs. Standard notation is used for most of the coagulation proteins. We introduce the notation FVh to refer to the partially activated forms of FV produced by FXa’s action or released from platelet granules. Thrombomodulin (TM) on endothelial cells (ECs) is a cofactor for thrombin in producing the inhibitor activated protein C (APC). Other major inhibitors are antithrombin (AT) and TFPIα. Surface-bound enzyme complexes TF:VIIa, VIIIa:IXa, Va:Xa, and TM:IIa and other surface-bound species are shown in boxes. Also shown are: cell or chemical activation (dashed magenta arrows); movement in fluid or along a surface (solid blue arrows); enzyme action in a forward direction (solid black arrows); the feedback action of enzymes (dashed black arrows); binding to or unbinding from surface (light green double-headed arrows); chemical inhibitors (red, orange, and purple shapes); and release of species such as FV from platelet stores (grey arrows with faded tails). The large circles highlight specific reactions that are accelerated by the presence of platelet-derived polyP. Adapted from [19]. (B) Vascular injury exposes collagen (purple strands). Platelets from upstream (blue) adhere to the collagen in the reaction zone (RZ, black box), become activated (spiky yellow), and aggregate to form a platelet plug. Platelets and proteins are washed away by flow. Adapted from [23]. All model reactions occur in the RZ [23]. (C) In the explicit model, polyP released from activated platelets binds to APSs or is cleared by flow. In the implicit model, the presence of polyP is modeled by increasing the catalytic rate constants for FXI and FV activations to the polyP-accelerated values. (D–F) In the explicit model, dependence of the catalytic rate constant $k^{cat}$ on bound polyP concentration for 3 choices of k^∗ and 3 coagulation reactions: (D) FXI activation by thrombin, (E) FV activation by FXa, and (F) FV activation by thrombin. APS, activated platelet surface; FIIa, thrombin FIXa, activated factor IX; FV, factor V; FVa, activated factor V; FVIIa, activated factor VII; FVIIIa, activated factor VIII; FXa, activated factor X; polyP, polyphosphate; TF, tissue factor; TFPIα, tissue factor pathway inhibitor α.

We consider all protein and cellular species within the RZ to vary over time but to be uniformly distributed in space and we describe each species by its time-dependent concentration. We use an ordinary differential equation to express the rate of change of each concentration due to reactions and transport into and out of the RZ. The overall model consists of coupled nonlinear ordinary differential equation for all species.

The model includes the number densities of 3 populations of platelets: unactivated fluid-phase platelets, activated platelets bound to the subendothelium, and activated platelets bound to other platelets in the thrombus. It includes mechanisms by which platelets become activated and bind to the subendothelium or other platelets along with mechanisms by which fluid-phase platelets move with the flow (Figure 1B). Bound platelets are considered immobile.

The model includes concentrations of zymogens, pro-cofactors, and the corresponding enzymes and active cofactors of the TF pathway (Figure 1A) and of the major chemical inhibitors of coagulation. It distinguishes clotting proteins in part by their chemical identity. Because the initiation of coagulation occurs on the subendothelial surface whereas critical reactions in the amplification phase of coagulation occur on activated platelet surfaces (APSs), the model further distinguishes proteins by whether they are in the plasma, bound to the subendothelium, or bound to an APS. The model does not however generally distinguish between reactions rates for platelet-bound species vs fluid-phase species.

The model allows enzymes activated on the subendothelium to reach an APS by unbinding from the subendothelium, moving through the plasma, then binding to a receptor on the APS. While in the plasma, species may be carried downstream by the flow, removing them from the RZ.

Because APSs are the sites for critical procoagulant reactions, platelets play a prominent role in the model. Since each activated platelet has a limited ability to support these reactions, and, initially no activated platelets are present in the RZ, the increasing availability of APSs as platelets accumulate there strongly influences the progression of the coagulation reactions. Platelets also have an anticoagulant role: each platelet’s adhesion to the injured vessel wall covers part of the subendothelium and blocks access to TF:FVIIa on that portion [17].

Progressive increases in platelet adhesion block increasing fractions of the total TF:FVIIa activity [25].

We assume coagulation proteins interact as depicted in Figure 1 and as outlined here:

• 1.FVII and FVIIa bind to TF on the subendothelium. FXa activates FVII in plasma or in TF:FVII. FXa binds TF:FVII directly from the plasma without first binding to the subendothelium.
• 2.TF:FVIIa binds and activates FIX and FX from the plasma.
• 3.Thrombin in plasma or on an APS, and FXa on an APS, activate FV and FVIII. Prescribed numbers of unactivated and partially activated variants of FV are secreted from platelet α-granules upon platelet activation. We denote the partially activated form of FV as FVh (for half-activated) and the fully activated form as FVa [24]. FXa converts FV to FVh; thrombin converts FV and FVh to FVa.
• 4.Platelet-bound FVIIIa and FIXa bind to form the tenase complex FVIIIa:FIXa, which activates platelet-bound FX, producing platelet-bound FXa.
• 5.Platelet-bound FVh and FVa bind platelet-bound FXa to form the prothrombinase complexes FVh:FXa (“PROh”) and FVa:FXa (“PRO”), respectively. These complexes can activate platelet-bound prothrombin into thrombin, which is immediately released into the plasma.
• 6.FXI is activated by thrombin in plasma and on an APS. FIX is activated by FXIa in plasma and on an APS.
• 7.
  The model includes the inhibitors/inactivators antithrombin, activated protein C (APC), and TF pathway inhibitor (TFPIα).

  • •
    By binding to FIXa, FXa, FXIa, and thrombin, antithrombin permanently inactivates them.
  • •
    APC binds to FVa and FVIIIa in the plasma and on an APS, permanently inactivating them but cannot bind to FVa or FVIIIa already incorporated in platelet-bound prothrombinase or tenase. APC is produced adjacent to the RZ by a complex of thrombin and thrombomodulin.
  • •
    TFPIα in plasma binds to FXa. TFPIα: FXa can then inhibit TF:FVIIa. TFPIα also binds to and inhibits FXa and FVh in the plasma or on an APS, as well as to the FVh in PROh on an APS.
• 8.TF:FVIIa activity decreases as platelets adhere to the subendothelium, physically blocking access to the subendothelium patch to which they are attached.

2.2. Novel model extensions to include effects of polyP

As previously mentioned, our novel extensions take two forms: the “implicit” model and the “explicit” model. In the implicit model, we change only the rate constants for thrombin activation of FXI to FXIa, FV and FVh to FVa, FVh within PROh to FVa within PRO, and for FXa activation of FV to FVh, to reflect the acceleration of those reactions observed in the presence of platelet polyP. This effectively assumes that a saturating concentration of polyP is present from the start of each simulation. In the implicit model, we replace the low catalytic rate constant $k_L^{cat}$ relevant when polyP is absent, with a high-rate constant $k_H^{cat}$ relevant when polyP is present.

The explicit model includes the release of a specified quantity of polyP by each platelet as the platelet is activated. We assume that the released polyP polymers bind to specific sites on APSs or are washed away by the flow (Figure 1C), and we track the concentrations of fluid-phase polyP (P^f) and platelet-bound polyP (P^b). We assume that the effect of polyP on the coagulation reactions depends in a saturable way on the concentration of P^b, as described by Equation (1) (Figure 1D–F):

$$k^{\text{cat}}=(k_H^{\text{cat}}-k_L^{\text{cat}})\frac{P^{\text{b}}}{k^{\ast }+P^{\text{b}}}+k_L^{\text{cat}}$$ (1)

where $k_L^{cat}$ and $k_H^{cat}$ are described above, and k^∗ is the concentration of P^b at which the catalytic rate is halfway between $k_L^{cat}$ and $k_H^{cat}$ and therefore determines the sensitivity of $k^{cat}$ to changes in P^b. The total concentration of polyP (P^f + P^b) increases at the rate of polyP release by platelets and decreases at the rate of removal of fluid-phase polyP by flow (Supplementary Equations S1 and S2). PolyP bound to an APS is unsusceptible to removal by flow and increases as platelet-binding sites become more available, which occurs as platelets accumulate.

We assume that the kinetics of polyP binding to and unbinding from platelets are fast relative to the rates of release of polyP from platelet stores and of its removal by flow. We therefore make the quasi-steady-state assumption that P^f and P^b equilibrate rapidly according to the current total polyP and polyP binding site concentrations (Supplementary Material S2). Consequently, the explicit model requires the addition of one differential equation for the total polyP concentration and algebraic formulas for P^f and P^b (Supplementary Equations S6–S8). The explicit model extension introduces 3 unknown parameters: the number n_bs of binding sites per platelet for polyP polymers, the dissociation constant K_D for polyP binding with these sites, and the concentration k^∗ defined above. The model’s behavior for ranges of these parameters is examined.

For both the explicit and implicit models, parameters are set to experimentally measured literature values whenever possible. Otherwise, parameters are constrained by experimental measurements. For example, for activation of FXI by thrombin we set $k_L^{cat}$ to be as in [29] and $k_H^{cat}$ to be as in [12]. For FV activations, $k_L^{cat}$ is known but the magnitude of the enhancement of FV activation in the presence of polyP has not yet been experimentally measured. We find that a 10-fold enhancement of FV activations in the presence of polyP reproduces the time to 10 nM thrombin as measured in [8] (Supplementary Figure S1). We set the default $k_H^{cat}$ values for FV activations in the presence of polyP to be 10-fold the respective $k_L^{cat}$ values (Table [3,12,[29], [30], [31], [32]]).

Table.

Implicit and explicit model parameters and kinetic constants.

Parameter	Description	Value (Units)	Note
Implicit and explicit model parameters
$k_H^{cat}$	FXI to FXIa by thrombin	0.05 s−1	[12]
$k_L^{cat}$		1.29 × 10-4 s−1	[29]
$k_H^{cat}$	FV to FVa, FVh to FVa by thrombin	2.3 s−1	Default (10-fold)
$k_L^{cat}$		0.23 s−1	[30]
$k_H^{cat}$	FV to FVh by FXa	0.46 s−1	Default (10-fold)
$k_L^{cat}$		0.046 s−1	[31]
Explicit model-specific parameters
nrel	Number of polyP polymers released per activated platelet	6 × 104	[3]
nbs	Number of polyP polymers that can bind per activated plateleta	-	Unknown
k∗	Sensitivity of kcat to bound polyP concentration	-	Unknown
KD	Dissociation constant for polyP binding to APSs	-	Unknown

(Upper) Parameters common to both the implicit and explicit models. (Lower) Explicit-model-specific parameters.

APS, activated platelet surface; FV, factor V; FVa, activated factor V; FVh, partially activated factor V; FXa, activated factor X; FXI, factor XI; FXIa, activated factor XI; polyP, polyphosphate.

^aHow polyP binds to platelets is unknown. It is possible that platelet-released cationic platelet factor 4 might play a role [32].

3. Results

Results from our simulations highlight changes, over the course of coagulation, in concentrations of proteins that polyP affects directly—FVh, FVa, FXIa—and others that polyP influences indirectly, including FVIIIa:FIXa, FVa:FXa, and thrombin. For thrombin we define two output metrics: lag time t_lag and the thrombin concentration at 10 minutes [Thrombin]_10min, which are widely used in the literature [[33], [34], [35]]. Lag time t_lag is defined as the time at which [Thrombin] reaches 1 nM, a concentration at which thrombin activates platelets. [Thrombin]_10min is the thrombin concentration 10 minutes after TF exposure. We refer to curves of [Thrombin]_10min vs TF density as threshold curves. A typical threshold curve exhibits a small interval of TF density over which [Thrombin]_10min increases sharply from low levels (<0.1 nM) with little physiological effect to high levels (>10 nM) with strong effect. We define the rapid increase in thrombin generation that occurs once the TF threshold is achieved as the thrombin burst. We begin by showing results from the implicit model.

3.1. In the implicit model, polyP enhances thrombin generation, FXIa formation, and FVa formation

In Figure 2, we examine how varying TF density affects [Thrombin]_10min and t_lag with and without polyP. We conducted a series of simulations at flow shear rates of 100, 500, and 1500 s⁻¹, with [TFPIα] = 0.5 nM, varying only TF density between simulations and computing the resulting thrombin concentration time courses. Figure 2A shows that for shear rate 100 s⁻¹, the addition of polyP decreases t_lag for each TF density and can lead to a thrombin burst when none occurs without polyP (TF ∼0.05-0.4 fmol/cm²). Figure 2B shows that the threshold curves [Thrombin]_10min are shifted to the left when polyP is present. Even when a thrombin burst occurs without polyP, adding polyP increases [Thrombin]_10min by up to several orders of magnitude for low to moderate TF densities (TF ∼0.4-2 fmol/cm²). At high TF densities, however, polyP hardly affects [Thrombin]_10min. PolyP’s effects on t_lag and [Thrombin]_10min are consistent across shear rates as shown in Figure 2C–F. As shear rate increases, curves for t_lag and the thrombin generation thresholds shift to the right. However, this shift is much smaller in the presence of polyP than in its absence. This indicates that thrombin generation is robust across shear rates in the presence of polyP.

Figure 2.

PolyP makes thrombin generation more robust across physiological shear rates in the implicit model. For each shear rate, polyP shortens the time to 1 nM thrombin (t_lag) and increases [Thrombin]_10min in the implicit model. (A, C, E) Time to 1 nM thrombin and (B, D, F) [Thrombin]_10min with polyP (accelerated FXI and FV activation) and without polyP (baseline FXI and FV activation) at [TFPIα] = 0.5 nM across shear rates (γ) of 100 s⁻¹ (A and B, solid lines), 500 s⁻¹ (C and D, dash-dotted lines), and 1500 s⁻¹ (E and F, dotted lines). [Thrombin]_10min and t_lag are computed for each simulation of coagulation (specified TF and shear rate). TF densities are constant within but are varied between simulations. FV, factor V; FXI, factor XI; polyP, polyphosphate; TF, tissue factor; TFPIα, tissue factor pathway inhibitor α.

In Supplementary Figure S2, we validate our model by examining how the presence of polyP affects the time courses of thrombin as well as those of FXIa and FVa. Thrombin generation occurs sooner and increases more rapidly with polyP because polyP accelerates FXI and FV activation. Accelerated FXI activation produces dramatically elevated FXIa concentrations, in turn enhancing FIX activation and FVIIIa:FIXa (tenase) complex formation on APSs. On the other hand, accelerated FV activation produces relatively brief increases in FVh and PROh concentrations, followed by larger and more sustained increases in FVa and PRO concentrations. These results collectively echo the experimentally observed roles of polyP in enhancing thrombin generation by accelerating FXI and FV activation [8,11]. The results of Supplementary Figure S2 are qualitatively similar across super-threshold TF densities.

3.2. Relative contributions in the implicit model of polyP-accelerated FXI and FV activation to thrombin formation

In Figure 3, we show the extent to which polyP-accelerated FXI activation by thrombin and polyP-accelerated FV activation by thrombin and FXa act separately and in combination to promote thrombin generation. Figure 3A and B show that accelerating either FXI activation alone or FV activation alone decreases t_lag (Figure 3A) and increases [Thrombin]_10min (Figure 3B). Each produces slightly more than half the effect of accelerating them together for TF densities between ∼0.05 and 0.1 fmol/cm². Therefore, polyP-accelerated FXI and FV activation contribute roughly equally to accelerating thrombin generation in the implicit model, and their combined effect is generally sub-additive. At higher TF densities (>0.3 fmol/cm²), accelerated FV activation contributes slightly more than accelerated FXI activation.

Figure 3.

Relative contributions of FXI and FV activation to thrombin generation with polyP in the implicit model. (A) Time to 1 nM thrombin (t_lag) and (B) [Thrombin]_10min with varying levels of acceleration of FXI activation by thrombin and FV activation by FXa and thrombin. (C) Time to 1 nM thrombin and (D) [Thrombin]_10min with accelerated FXI activation by thrombin and varying levels of acceleration of FV activation by FXa and thrombin. In (C) and (D), the curves corresponding to slow activation of all FV reactions (solid yellow) and to fast activation of FVh to FVa only (dashed purple) are on top of one another. “Fast” indicates that the respective reaction has been accelerated due to the presence of polyP ($k_H^{cat}$ in Table), while “slow” indicates the baseline rate of activation without polyP ($k_L^{cat}$ in Table). [TFPIα] = 0.5 nM; shear rate = 100 s⁻¹. FV, factor V; FVh, partially activated factor V; FXa, activated factor X; FXI, factor XI; polyP, polyphosphate; TF, tissue factor.

Figure 3C and D show the effects of accelerating the different FV activation reactions. Accelerating either FV to FVa or FV to FVh alone decreases t_lag across TF densities (Figure 3C) and increases [Thrombin]_10min (Figure 3D), although FV to FVa activation leads to slightly more thrombin formation, mainly for TF densities ∼0.2-0.6 fmol/cm². Therefore, accelerated activation of FV to FVa and FV to FVh each contributes roughly equally to the acceleration of thrombin generation seen when polyP is included in the implicit model.

Supplementary Figure S3 shows simulations from Figure 3 without polyP-accelerated FXI activation. Faster activation of FV to FVa alone contributes more to the acceleration of thrombin generation than faster activation of FV to FVh alone. However, the effect of accelerating FVh to FVa activation by thrombin remains negligible. Results obtained by repeating these simulations using a 100-fold increase in FV activation by FXa and thrombin, with and without accelerated FXI activation by thrombin, are shown in Supplementary Figures S4 and S5, respectively. These figures indicate that accelerated FV activation alone contributes more strongly than accelerated FXI activation to accelerating thrombin generation, with FV to FVh conversion having the largest effect.

3.3. In the implicit model, polyP reduces the sensitivity of thrombin generation to TFPIα concentration in a TF-dependent manner

In Figure 4, we examine polyP’s influence on TFPIα function using the implicit model. At TF = 1 fmol/cm², including polyP-induced acceleration of FXI and FV activation makes the system much less sensitive to the TFPIα concentration. Figure 4A shows that, with polyP, t_lag increases slowly from ∼4 to ∼9 minutes at 0 and 2.5 nM TFPIα, respectively, but without polyP, t_lag rises quickly and steeply, from ∼6 to >30 minutes at 0 and 2.5 nM TFPIα, respectively. Figure 4B shows that [Thrombin]_10min is insensitive to TFPIα when polyP is present, remaining nearly constant and high (∼300 nM) across all TFPIα levels shown. In contrast, without polyP, [Thrombin]_10min drops by more than three orders of magnitude with increasing TFPIα (from ∼150 to 0.01 nM at 0 and 2.5 nM TFPIα, respectively). These results demonstrate that, although no direct effect of polyP on TFPIα function was included in the simulations, t_lag and [Thrombin]_10min were much less sensitive to TFPIα concentration with polyP-accelerated FXI and FV activation than without them. Similar behavior is observed at TF density 0.3 fmol/cm² (Supplementary Figure S6).

Figure 4.

PolyP diminishes the sensitivity of thrombin generation to TFPI function in the implicit model. (A–F) Time to 1 nM thrombin, t_lag (left), and [Thrombin]_10min (right) vs TFPIα concentration: (A and B) with polyP (black) and without polyP (red); (C and D) with varying levels of acceleration of FXI activation by thrombin and FV activation by FXa and thrombin; and (E and F) with accelerated FXI activation by thrombin and varying levels of acceleration of FV activation by FXa and thrombin. In (E) and (F), the curves corresponding to slow activation of all FV reactions (solid yellow) and fast activation of FVh to FVa only (dashed purple) overlap. “Fast” indicates that the respective reaction has been accelerated due to the presence of polyP ($k_H^{cat}$ in Table), while “slow” indicates the baseline rate of activation without polyP ($k_L^{cat}$ in Table). TF density = 1 fmol/cm²; shear rate = 100 s⁻¹. FV, factor V; FVa, activated factor V; FVh, partially activated factor V; FXa, activated factor X; FXI, factor XI; polyP, polyphosphate; TF, tissue factor; TFPIα, tissue factor pathway inhibitor α.

Figure 4C and D show that accelerating either FXI or FV activation reduces the sensitivity of t_lag and [Thrombin]_10min, respectively, to TFPIα. When only FV activation is accelerated, t_lag remains short across TFPIα concentrations, increasing slowly, and [Thrombin]_10min decreases gradually at higher TFPIα concentrations. When only FXI activation is accelerated, t_lag rises more quickly with TFPIα, and thrombin levels decline more sharply beyond 1 nM TFPIα. Therefore, accelerating FV activation alone reduces the sensitivity of t_lag and [Thrombin]_10min to TFPIα more than does accelerating FXI activation alone.

Figure 4E and F show the effects of accelerating the various FV activation reactions, with fast FXI activation. Accelerating either thrombin’s conversion of FV to FVa or FXa’s conversion of FV to FVh decreases the sensitivity of both [Thrombin]_10min and t_lag to TFPIα, with the former producing the larger reduction. However, accelerating only thrombin’s conversion of FVh to FVa had little effect on either metric.

While panels (B), (D) and (F) show little sensitivity of [Thrombin]_10min to TFPIα, a sufficiently high TFPIα concentration would suppress thrombin production. Supplementary Figure S7 shows simulations without this accelerated activation of FXI and indicates that the reduced sensitivity of thrombin generation to TFPIα persists. Supplementary Figure S8 shows the same simulations but with a 100-fold rather than 10-fold increase in FV activation rates, along with accelerated FXI activation. It shows that accelerated FV activation contributes even more strongly than accelerated FXI activation to the reduced sensitivity of thrombin generation to TFPIα.

The sensitivity of the thrombin generation metrics t_lag and [Thrombin]_10min to TFPIα, described above for TF density 1 fmol/cm², is further decreased when polyP is present for higher TF densities (not shown). Figure 5 demonstrates that the sensitivity of thrombin generation to TFPIα depends both on the presence or absence of polyP and on the TF density. Figure 5A, B show that with polyP, both thrombin generation metrics become increasingly sensitive to TFPIα as the TF density is decreased below 1 fmol/cm², with the greatest changes in sensitivity occurring for TF <0.5 fmol/cm². Figure 5C, D show that without polyP, the thrombin metrics become increasingly insensitive to TFPIα as TF is increased from 1 to 3 fmol/cm², although even for TF 3 fmol/cm² the metrics without polyP are more sensitive to TFPIα than they are for TF 1 fmol/cm² with polyP. Quantifying TFPIα insensitivity in terms of the average change in lag time (ie, the slope of the secant line), due to an increase in TFPIα concentration from 0 to 0.5 nM confirms our findings (Supplementary Figure S9).

Figure 5.

PolyP modulates the sensitivity of thrombin generation to TFPIα in a TF-dependent manner. (A–D) Time to 1 nM thrombin, t_lag, (left) and [Thrombin]_10min (right) vs TFPIα concentration: (A, B) with polyP for TF densities 0.1, 0.3, 0.5, 0.7, 0.9, and 1 fmol/cm², and (C, D) without polyP for TF densities 0.5, 1, 1.5, 2, 2.5, and 3 fmol/cm². Shear rate = 100 s⁻¹. polyP, polyphosphate; TF, tissue factor; TFPIα, tissue factor pathway inhibitor α.

3.4. Explicit model parameters determine the impact of polyP on thrombin generation

Recall that the explicit model tracks polyP release from platelets and binding to APSs and that the catalytic rates of FXI and FV activation depend dynamically on the platelet-bound polyP concentration, P^b (Equation 1). The impact of the new explicit model parameters—the number of binding sites (n_bs) for polyP on each APS, the dissociation constant (K_D) for polyP binding to these sites, and the sensitivity (k^∗) of the catalytic rate constants to P^b (Equation 1)—on t_lag and [Thrombin]_10min are shown in Figure 6A, B, respectively. Figure 6 shows that the availability of binding sites for polyP on APSs (larger n_bs) and tighter binding of polyP to APSs (lower K_D) strongly impacts the ability of polyP to accelerate thrombin production by preventing polyP from being washed away by flow. Increasing the sensitivity of catalytic rate constants to bound polyP also strongly increases acceleration of thrombin production by polyP. Varying the explicit model parameters shifts both metrics between the extreme cases observed in the implicit model with and without polyP.

Figure 6.

Explicit model parameters determine whether thrombin generation resembles the implicit model with or without polyP. (A) Time to 1 nM thrombin (t_lag) and (B) [Thrombin]_10min for the implicit model with polyP (black), the implicit model without polyP (red), and different explicit model parameter sets (dotted). In (A) and (B), the curves corresponding to the explicit model parameter set, k^∗ = 10 μM, K_D = 10,000 nM, and n_bs = 1 (dotted yellow), and to the implicit model with polyP (solid red), are on top of one another. [TFPIα] = 0.5 nM; shear rate = 100 s⁻¹. n_bs, number of binding sites; polyP, polyphosphate; TF, tissue factor; TFPIα, tissue factor pathway inhibitor α.

The parameters’ effects on t_lag and [Thrombin]_10min are TF dependent. For low TF (0.7 fmol/cm²), lag times from 6 to 15 min are seen, while for TF 1.5 fmol/cm², a much smaller range, 5 to 9 min, is observed, depending on the polyP kinetic parameters. Consequently, [Thrombin]_10min also varies strongly with these parameters and TF levels, ranging between 0.5 and 250 nM for TF 0.7 fmol/cm² and 100 to 300 nM for TF 1.5 fmol/cm².

3.5. The ratio R = (n_bs/K_D)/k^∗ determines explicit model dynamics

The extent of acceleration of thrombin generation is captured roughly in the parameter ratio R = (n_bs/K_D)/k^∗ (nM⁻²); it is proportional to n_bs and inversely proportional to K_D and k^∗. In Figure 7, we examine nine parameter sets for which R is 0.1, 1, or 10 nM⁻² (3 sets for each R value). For each TF density, parameter sets with the same ratio yield nearly identical t_lag (Figure 7A) and [Thrombin]_10min (Figure 7B). Increasing R accelerates thrombin production (Figure 7C). For R = 0.1, t_lag decreases and [Thrombin]_10min increases only slightly compared to the case without polyP. However, increasing R to 1 and then to 10 strongly reduces t_lag and strongly increases [Thrombin]_10min, reaching about half the acceleration achieved in the implicit model with polyP. For even larger values of R, such as R = 100 (not shown), t_lag and [Thrombin]_10min are close to those of the implicit model with polyP, suggesting that the ratio R captures the dependence of thrombin metrics on the explicit model parameters.

Figure 7.

Effect of the explicit model parameter ratio, R, on thrombin generation, fluid-phase, and bound polyP concentrations. (A) Time to 1 nM thrombin t_lag and (B) [Thrombin]_10min for the implicit model with polyP (black) and without polyP (red), and for the explicit model parameter sets chosen to yield different values of the ratio R = (n_bs/K_D)/k^∗ (nM⁻²). In (A), the red curve overlaps the brown curves (R = 0.1). [TFPIα] = 0.5 nM; shear rate = 100 s⁻¹. (C–E) Time courses of species concentrations: (C) Thrombin, (D) Fluid-phase polyP (P^f), and (E) Platelet-bound polyP (P^b) for R = (n_bs/K_D)/k^∗ values of 0.1, 1, or 10 nM⁻². Note that the y-axis scales are different in each panel. Larger R values lead to earlier and more rapid thrombin generation. The values of R strongly influence polyP concentrations from ∼400 to ∼700 s, where the polyP curves for different values of R separate (panels D and E). TF density = 1 fmol/cm²; [TFPIα] = 0.5 nM; shear rate = 100 s⁻¹. Vertical dashed lines in (D) and (E) represent times at which thrombin reaches 100 nM and correspond to solid lines of the same color. Supplementary Figure S10 shows that the rates of polyP release and removal by flow are almost equal during each simulation, so most polyP released is carried away by flow. It also shows that after ∼200 s, almost all polyP in the reaction zone is platelet-bound. n_bs, number of binding sites; polyP, polyphosphate; TF, tissue factor; TFPIα, tissue factor pathway inhibitor α.

Figure 7C–E show that R also determines the time courses of thrombin, fluid-phase polyP (P^f) and platelet-bound polyP (P^b), for R = 0.1, 1, or 10 nM⁻², using the same values of n_bs = 10000 and K_D = 10 nM, while k^∗ is varied (0.1, 1.0, or 10.0 μM). In all cases, P^b increases sharply over the first 100 s to about 200 nM as platelets accumulate and release polyP. P^b then decreases from ∼100 s to ∼360 s (R = 10), 500 s (R = 1), and 580 s (R = 0.1), increases quickly again, and then increases much slower at a concentration of about 40 nM. The three P^b curves attain similar levels because n_bs and K_D are the same. Total polyP concentrations typically peak at ∼200 nM, consisting nearly entirely of polyP bound to APSs. As the ratio increases from 0.1 to 10, the second increase in polyP concentrations becomes more rapid and coincides with the rapid increase in thrombin from 1 to 5 nM to >100 nM. This likely reflects thrombin-mediated enhancement of platelet activation and accumulation.

4. Discussion

To gain quantitative insight into the role of polyP in thrombin production, we extended an existing, validated mathematical model of flow-mediated platelet deposition and TF-initiated coagulation. Experiments show that polyP accelerates specific coagulation reactions, including FXI and FV activation. Our modeling framework allows these effects to be examined within a full model of the TF pathway under flow. The framework accounts for concurrent platelet accumulation and more varied reaction conditions than in experiments, which are often performed under no-flow conditions and examine only a subset of coagulation reactions.

Because quantitative measurements of polyP-enhanced FV activation by thrombin and FXa have not been reported [8,10], we chose a 10-fold increase in their catalytic rates to reproduce the experimentally observed reduction in lag time to 10 nM thrombin in the presence of polyP [8]. Simulations using a 100-fold increase in FV activation by FXa and thrombin alter TFPIα insensitivity and the relative importance of accelerated FV and FXI activations, motivating future experimental studies to precisely quantify how much polyP accelerates FV activation. Since flow is critical, these experiments could consist of a purified system for the TF pathway under flow and attempt to measure how FVa concentrations grow over time and their equilibrium values.

Additionally, experiments in the presence of polyP have shown apparent insensitivity to TFPIα levels but could not distinguish whether this is due to direct inhibition of TFPIα or is an indirect effect of accelerated coagulation reactions [8]. In our simulations, polyP indeed greatly decreases the sensitivity of thrombin generation to TFPIα. This change is driven by faster FV and FXI activation, with faster thrombin-mediated conversion of FV to FVa being the dominant contribution when FV activation is accelerated 10-fold. We emphasize that the model includes no direct action of polyP on TFPIα. The model further reveals that polyP’s ability to reduce TFPIα’s anticoagulant effect depends strongly on TF density. Namely, while polyP makes thrombin generation less sensitive to TFPIα at each TF density, it also makes thrombin production at lower TF densities about as sensitive to TFPIα as thrombin production without polyP is at higher TF densities. Thus, polyP nonuniformly affects TFPIα’s apparent activity in a TF-dependent manner. Experiments to investigate TFPIα insensitivity in the presence of polyP should be done under flow in a purified system for the tissue-factor-pathway including polyP across variable TF densities.

Our model also predicts a novel potential function for polyP. Namely, polyP in our model promotes robust thrombin generation across physiological shear rates, with thrombin metrics being less sensitive to changes in shear rate with polyP than without. Future experiments could evaluate thrombin generation in microfluidic or flow chamber assays with and without polyP across shear rates.

Using the model, we are able to quantify the relative contributions of faster FXI and FV activation to thrombin formation, which to our knowledge has not been previously attempted. When polyP accelerates FV activation 10-fold, polyP-accelerated FXI and FV activation individually contribute approximately equally to accelerating thrombin generation, while their combined effect is subadditive. Moreover, when FXI activation is enhanced, accelerated conversion of FV to FVa by thrombin and FV to FVh by FXa contribute roughly equally to the acceleration of thrombin generation, whereas accelerated conversion of FV to FVh by thrombin has a negligible impact. The last occurs because PROh and PRO activate prothrombin with the same kinetics [36]. Identifying the relative contributions of accelerations of FV and FXI activations clarifies the mechanism by which polyP enhances thrombin production; this can guide whether to target the polyP-mediated enhancement of FV or FXI for polyP-mediated therapies. These results further underscore the need to determine the level of enhancement of FV activation in the presence of polyP.

In our explicit model, we allow for the possibility that the availability of polyP can be ratelimiting. The impact of platelet-released polyP on thrombin generation depends in particular on the parameter ratio R = (n_bs/K_D)/k^∗ (nM⁻²). R increases with the number of polyP binding sites (n_bs) on platelets and decreases with the dissociation constant (K_D) for polyP binding and the sensitivity parameter (k^∗) in the catalytic rates for FXI and FV activation. Because the value of R influences thrombin generation and polyP concentrations, experimentally measuring the parameters that comprise this ratio is warranted, although outside the scope of this study.

A limitation of our explicit model is that it can only investigate coagulation under flow. Removal of fluid-phase polyP by flow likely produces the lower concentrations of polyP than have been measured experimentally under static conditions. PolyP remaining in the aggregate is therefore critical for polyP to maintain its procoagulant functions. In our model, polyP does so by binding to APSs. A previous study [37] suggests that polyP nanoparticles may bind to platelet surfaces via divalent metal ions. It is possible that platelet-released polyP polymers bind to platelets by similar means.

Future model explorations could incorporate additional experimentally reported effects of polyP whose quantitative details are currently unknown. These include FV activation by FXIa, polyP enhanced FXI autoactivation, and FXIa-mediated TFPIα inactivation, all likely most relevant for the contact pathway [10,11,38]. Including contact pathway dynamics, while slow compared to the TF pathway, is a second future model exploration that would provide an additional means by which polyphosphates accelerate coagulation. Long-chain polyphosphates, as released by bacteria, provide a negatively charged surface which could activate FXII, initiating the contact pathway. A third future model exploration includes additional proteins, such as fibrinogen or platelet factor 4, which interact with platelet-released polyphosphates. Fibrin(ogen) in particular also has the potential to inhibit thrombin. However, we expect flow-mediated thrombin removal to dominate thrombin sequestration by fibrin(ogen), as we saw to be the case for direct thrombin inhibitors [17,18,21,22,39]. Thus, omission of fibrin(ogen) is not expected to qualitatively alter the results of this study.

Our mathematical modeling results provide a coherent interpretation of how (platelet-released) polyP modulates thrombin generation under flow in TF-initiated coagulation. By accelerating FXIa and FVa formation, polyP promotes earlier and stronger thrombin generation. PolyP also leads to more robust thrombin generation under varying shear rates. PolyP indirectly reduces the sensitivity of thrombin generation to TFPIα in a TF-dependent manner, an effect predominantly due to polyP-accelerated thrombin-mediated activation of FV. This work provides mechanistic insight into how polyP may be exploited as either a hemostatic agent or an antithrombotic target.