Surface-mediated enzymatic reactions: simulations of tissue factor activation of factor X on a lipid surface

Different modeling approaches in the simulation of extrinsic coagulation factor X activation: Limitations and areas of applicability

Proteolytic reactions on the phospholipid membrane surface, so-called "membrane-dependent" reactions, play central role in the process of blood clotting. One particularly important example is FX activation by the extrinsic tenase (VIIa/TF). Here we constructed three mathematical models of FX activation by VIIa/TF: (A) a homogeneous "well-mixed" model, (B) a two-compartment "well-mixed" model, (C) a heterogeneous model with diffusion, to investigate the impact and importance of inclusion of each complexity level. All models provided good description of the reported experimental data and were equivalently applicable for <40 μM of phospholipids. Model C provided better predictions than A, B in the presence of TF-negative phospholipid microparticles. Models predicted that for high TF surface density (STF ) and FX deficiency the FX activation rate was limited by the rate of FX binding to the membrane. For low STF and excess of FX the reaction rate was limited by the tenase formation rate. The analysis of the substrate delivery pathways revealed that FX bound to VIIa/TF predominantly from solution for STF  >2.8 × 10-3  nmol/cm2 and from the membrane for lower STF . We proposed the experimental setting to distinguish between the collision-limited and non-collision-limited binding. The analysis of models in flow and non-flow conditions revealed that the model of a vesicle in flow might be substituted by model C in the absence of the substrate depletion. Together, this study was the first which provided the direct comparison of more simple and more complex models. The reaction mechanisms were studied in a wide range of conditions.

Blood flow and mass transfer regulation of coagulation

Blood flow regulates coagulation and fibrin formation by controlling the transport, or mass transfer, of zymogens, co-factors, enzymes, and inhibitors to, from, and within a growing thrombus. The rate of mass transfer of these solutes relative to their consumption or production by coagulation reactions determines, in part, the rate of thrombin generation, fibrin deposition, and thrombi growth. Experimental studies on the influence of blood flow on specific coagulation reactions are reviewed here, along with a theoretical framework that predicts how flow influences surface-bound coagulation binding and enzymatic reactions. These flow-mediated transport mechanisms are also used to interpret the role of binding site densities and injury size on initiating coagulation and fibrin deposition. The importance of transport of coagulation proteins within the interstitial spaces of thrombi is shown to influence thrombi architecture, growth, and arrest.

A cellular automaton model of crystalline cellulose hydrolysis by cellulases

BACKGROUND

Cellulose from plant biomass is an abundant, renewable material which could be a major feedstock for low emissions transport fuels such as cellulosic ethanol. Cellulase enzymes that break down cellulose into fermentable sugars are composed of different types - cellobiohydrolases I and II, endoglucanase and β-glucosidase - with separate functions. They form a complex interacting network between themselves, soluble hydrolysis product molecules, solution and solid phase substrates and inhibitors. There have been many models proposed for enzymatic saccharification however none have yet employed a cellular automaton approach, which allows important phenomena, such as enzyme crowding on the surface of solid substrates, denaturation and substrate inhibition, to be considered in the model.

RESULTS

The Cellulase 4D model was developed de novo taking into account the size and composition of the substrate and surface-acting enzymes were ascribed behaviors based on their movements, catalytic activities and rates, affinity for, and potential for crowding of, the cellulose surface, substrates and inhibitors, and denaturation rates. A basic case modeled on literature-derived parameters obtained from Trichoderma reesei cellulases resulted in cellulose hydrolysis curves that closely matched curves obtained from published experimental data. Scenarios were tested in the model, which included variation of enzyme loadings, adsorption strengths of surface acting enzymes and reaction periods, and the effect on saccharide production over time was assessed. The model simulations indicated an optimal enzyme loading of between 0.5 and 2 of the base case concentrations where a balance was obtained between enzyme crowding on the cellulose crystal, and that the affinities of enzymes for the cellulose surface had a large effect on cellulose hydrolysis. In addition, improvements to the cellobiohydrolase I activity period substantially improved overall glucose production.

CONCLUSIONS

Cellulase 4D simulates the enzymatic hydrolysis of cellulose to glucose by surface and solution phase-acting enzymes and accounts for complex phenomena that have previously not been included in cellulose hydrolysis models. The model is intended as a tool for industry, researchers and educators alike to explore options for enzyme engineering and process development and to test hypotheses regarding cellulase mechanisms.

Modelling thrombin generation in human ovarian follicular fluid

A mathematical model is constructed to study thrombin production in human ovarian follicular fluid. The model results show that the amount of thrombin that can be produced in ovarian follicular fluid is much lower than that in blood plasma, failing to reach the level required for fibrin formation, and thereby supporting the hypothesis that in follicular fluid thrombin functions to initiate cellular activities via intracellular signalling receptors. It is also concluded that the absence of the amplification pathway to thrombin production in follicular fluid is a major factor in restricting the amount of thrombin that can be produced. Titration of the initial concentrations of the various reactants in the model lead to predictions for the amount of tissue factor and phospholipid that is required to maintain thrombin production in the follicle, as well as to the conclusion that tissue factor pathway inhibitor has little effect on the time that thrombin generation is sustained. Numerical experiments to determine the effect of factor V, which is at a much reduced level in follicular fluid compared to plasma, and thrombomodulin, illustrate the importance for further experimental work to determine values for several parameters that have yet to be reported in the literature.

Simulating fibrin clotting time

The clotting time (CT) of fibrinogen mixed with thrombin decreased, then increased with increasing fibrinogen levels. By contrast, log CT decreased monotonically with respect to the log level of activating enzyme (thrombin or reptilase). Here, the CT was determined over a large range of fibrinogen concentration (to 100 mg ml(-1)) at a fixed level of enzyme. A new parameter, [Fib]min, the minimal fibrinogen concentration required for thrombin or reptilase-instigated phase change (coagulation), was determined as [Fib]min = 0.2 +/- 0.05 microM fibrinogen. A dynamic simulation program (Stella) was employed to organize simulations based on simple and complex coagulation mechanisms, which generated CT values. The successful simulation aimed at forming [Fib]min and "recognized" the binding of unreacted fibrinogen with intermediate fibrin protofibrils. The "virtual data" mimicked the biphasic experimental CT values over a wide range of concentrations. Fibrinogen appeared to act in three modalities: as a thrombin substrate; as a precursor of fibrin; and as a competitor for fibrin protofibrils. The optimized simulation may provide a basis for predicting CT in more complex systems, such as pathological plasmas or whole blood or at high concentrations encountered with fibrin sealant.

Mathematical Modeling and Computer Simulation in Blood Coagulation

Over the last two decades, mathematical modeling has become a popular tool in study of blood coagulation. The in silico methods were able to yield interesting and significant results in the understanding of both individual reaction mechanisms and regulation of large sections of the coagulation cascade. The objective of this paper is to review the development of theoretical research in hemostasis and thrombosis, to summarize the main findings, and outline problems and possible prospects in the use of mathematical modeling and computer simulation approaches. This review is primarily focused on the studies dealing with: (1) the membrane-dependent reactions of coagulation; (2) regulation of the coagulation cascade, including effects of positive and negative feedback loops, diffusion of coagulation factors, and blood flow.

Positive Feedbacks of Coagulation

Tissue factor (TF), the initiator of coagulation, continuously circulates in the plasma, and the clotting system “idles,” generating very low levels of active clotting enzymes, clotting products, and by-products. Given the enormous amplification potential of the clotting cascade, rigorous control is required to ensure that such low-level stimulation does not cause massive system amplification and response. We propose that among the various mechanisms of regulation, activation thresholds may play a major role. These arise when positive-feedback reactions, of which there are several in the clotting system, are regulated by inhibitors. Such thresholds act like switches, so that small stimuli and/or nonproductive local conditions will generate no response, whereas larger stimuli or the existence of local prothrombotic conditions will produce a full, explosive response. We review here the evidence for system idling, the structures of the various feedback mechanisms of clotting, the mechanisms by which they can produce threshold behavior, and the possible role of thresholds in system regulation.

Two subpopulations of thrombin-activated platelets differ in their binding of the components of the intrinsic factor X-activating complex

Binding of fluorescein-labeled coagulation factors IXa, VIII, X, and allophycocyanin-labeled annexin V to thrombin-activated platelets was studied using flow cytometry. Upon activation, two platelet subpopulations were detected, which differed by 1-2 orders of magnitude in the binding of the coagulation factors and by 2-3 orders of magnitude in the binding of annexin V. The percentage of the high-binding platelets increased dose dependently of thrombin concentration. At 100 nm of thrombin, platelets with elevated binding capability constituted approximately 4% of total platelets and were responsible for the binding of approximately 50% of the total bound factor. Binding of factors to the high-binding subpopulation was calcium-dependent and specific as evidenced by experiments in the presence of excess unlabeled factor. The percentage of the high-binding platelets was not affected by echistatin, a potent aggregation inhibitor, confirming that the high-binding platelets were not platelet aggregates. Despite the difference in the coagulation factors binding, the subpopulations were indistinguishable by the expression of general platelet marker CD42b and activation markers PAC1 (an epitope of glycoprotein IIb/IIIa) and CD62P (P-selectin). Dual-labeling binding studies involving coagulation factors (IXa, VIII, or X) and annexin V demonstrated that the high-binding platelet subpopulation was identical for all coagulation factors and for annexin V. The high-binding subpopulation had lower mean forward and side scatters compared with the low-binding subpopulation ( approximately 80% and approximately 60%, respectively). In its turn, the high-binding subpopulation was not homogeneous and included two subpopulations with different scatter values. We conclude that activation by thrombin induces the formation of two distinct subpopulations of platelets different in their binding of the components of the intrinsic fX-activating complex, which may have certain physiological or pathological significance.

Tissue factor pathway inhibitor: a possible mechanism of action

We have analyzed several mathematical models that describe inhibition of the factor VIIa-tissue factor complex (VIIa-TF) by tissue factor pathway inhibitor (TFPI). At the core of these models is a common mechanism of TFPI action suggesting that only the Xa-TFPI complex is the inhibitor of the extrinsic tenase activity. However, the model based on this hypothesis could not explain well all the available experimental data. Here, we show that a good quantitative description of all experimental data could be achieved in a model that contains two more assumptions. The first assumption is based on the hypothesis originally proposed by Baugh et al. [Baugh, R.J., Broze, G.J. Jr & Krishnaswamy, S. (1998) J. Biol. Chem. 273, 4378-4386], which suggests that TFPI could inhibit the enzyme-product complex Xa-VIIa-TF. The second assumption proposes an interaction between the X-VIIa-TF complex and the factor Xa-TFPI complex. Experiments to test these hypotheses are suggested.

The role of membrane patch size and flow in regulating a proteolytic feedback threshold on a membrane: possible application in blood coagulation

Positive feedback controls in proteolytic systems are characterized by thresholds which are regulated by the concentration of the initial stimulus and the kinetic parameters for enzyme generation and inhibition. Significant complexity is added when a positive feedback is localized on a membrane in contact with a flowing medium, such as seen in the early steps of blood coagulation. A partial differential equation model of an archetypal feedback loop is examined in which a proteolytic enzyme catalyzes its own formation from a zymogen on a membrane in contact with a flowing medium. As predicted from prior solution-phase and membrane-phase analyses, the threshold conditions for activation of the system are regulated by the kinetics of enzyme generation and inhibition and by the density of reactant-binding sites on the membrane; but the present analysis also establishes how the feedback threshold is controlled by the flow rate of the adjacent medium and the physical size of the membrane patch on which the feedback loop is localized. For given systems of particular kinetic properties, lower flow rates or larger active patches of membrane can result in the activation threshold being exceeded, whereas higher flow rates or smaller membrane patches can prevent initiation. In addition to numerical simulation, a simplified non-flowing model is analyzed to formulate an approximate mathematical statement of the dependence of the minimum activatable patch size on the kinetic and other parameters.

Regulation of tissue plasminogen activator activity by cells. Domains responsible for binding and mechanism of stimulation

A number of cell types have previously been shown to bind tissue plasminogen activator (tPA), which in some cases can remain active on the cell surface resulting in enhanced plasminogen activation kinetics. We have investigated several cultured cell lines, U937, THP1, K562, Molt4, and Nalm6 and shown that they bind both tPA and plasminogen and are able to act as promoters of plasminogen activation in kinetic assays. To understand what structural features of tPA are involved in cell surface interactions, we performed kinetic assays with a range of tPA domain deletion mutants consisting of full-length glycosylated and nonglycosylated tPA (F-G-K1-K2-P), DeltaFtPA (G-K1-K2-P), K2-P tPA (BM 06.022 or Reteplase), and protease domain (P). Deletion variants were made in Escherichia coli and were nonglycosylated. Plasminogen activation rates were compared with and without cells, over a range of cell densities at physiological tPA concentrations, and produced maximum levels of stimulation up to 80-fold with full-length, glycosylated tPA. Stimulation for nonglycosylated full-length tPA dropped to 45-60% of this value. Loss of N-terminal domains as in DeltaFtPA and K2P resulted in a further loss of stimulation to 15-30% of the full-length glycosylated value. The protease domain alone was stimulated at very low levels of up to 2-fold. Thus, a number of different sites are involved in cell interactions especially within finger and kringle domains, which is similar to the regulation of tPA activity by fibrin. A model was developed to explain the mechanism of stimulation and compared with actual data collected with varying cell, plasminogen, or tPA concentrations and different tPA variants. Experimental data and model predictions were generally in good agreement and suggest that stimulation is well explained by the concentration of reactants by cells.

Regulation of extrinsic pathway factor Xa formation by tissue factor pathway inhibitor

Tissue factor (TF) pathway inhibitor (TFPI) regulates factor X activation through the sequential inhibition of factor Xa and the VIIa.TF complex. Factor Xa formation was studied in a purified, reconstituted system, at plasma concentrations of factor X and TFPI, saturating concentrations of factor VIIa, and increasing concentrations of TF reconstituted into phosphatidylcholine:phosphatidylserine membranes (TF/PCPS) or PC membranes (TF/PC). The initial rate of factor Xa formation was equivalent in the presence or absence of 2.4 nM TFPI. However, reaction extent was small (<20%) relative to that observed in the absence of TFPI, implying the rapid inhibition of VIIa.TF during factor X activation. Initiation of factor Xa formation using increasing concentrations of TF/PCPS or TF/PC in the presence of TFPI yielded families of progress curves where both initial rate and reaction extent were linearly proportional to the concentration of VIIa.TF. These observations were consistent with a kinetic model in which the rate-limiting step represents the initial inhibition of newly formed factor Xa. Numerical analyses of progress curves yielded a rate constant for inhibition of VIIa.TF by Xa.TFPI (>10(8) M-1.s-1) that was substantially greater than the value (7.34 +/- 0.8 x 10(6) M-1.s-1) directly measured. Thus, VIIa.TF is inhibited at near diffusion-limited rates by Xa.TFPI formed during catalysis which cannot be explained by studies of the isolated reaction. We propose that the predominant inhibitory pathway during factor X activation may involve the initial inhibition of factor Xa either bound to or in the near vicinity of VIIa.TF on the membrane surface. As a result, VIIa.TF inhibition is unexpectedly rapid, and the concentration of active factor Xa that escapes regulation is linearly dependent on the availability of TF.

Antithrombotic efficacy of inactivated active site recombinant factor VIIa is shear dependent in human blood

Several studies have indicated a profound role for factor VII(a) [FVII(a)] in venous and arterial thrombogenesis. In the present study, we quantified the inhibitory efficacy of dansyl-glutamyl-glycyl-arginyl-recombinant FVIIa (DEGR-rFVIIa) on acute thrombus formation. Thrombus formation was elicited by immobilized tissue factor (TF) in a parallel-plate perfusion chamber device at blood flow conditions characterized by wall shear rates of 100 S-1 (veins) and 650 S-1 (medium-sized healthy arteries). Native human blood was drawn directly from an antecubital vein by a pump into a heparin-coated mixing device in which DEGR-rFVIIa (0.09 to 880 nmol/L final plasma concentration) or buffer was mixed homogeneously with flowing blood. Subsequently, the blood was passed over a plastic coverslip coated with TF and phospholipids in the parallel-plate perfusion chamber. Fibrin deposition, platelet-fibrin adhesion, and platelet thrombus volume triggered by this surface were measured by morphometry. DEGR-rFVIIa inhibited thrombus formation in a dose-dependent manner, but the efficacy was shear rate dependent. At a wall shear rate of 100 S-1, the IC50 (50% inhibition) was 30 nmol/L, whereas at 650 S-1, the IC50 was 0.6 nmol/L. Binding studies to immobilized TF under flow conditions using surface plasmon resonance revealed a significantly higher on-rate for DEGR-rFVIIa and FVIIa than for FVII, 2.8 x 10(5), 2.6 x 10(5), and 1.8 x 10(5) M-1 S-1, respectively. This indicates that a contributing factor to the shear-dependent efficacy may be a differential importance of on-rates at arterial and venous blood flow conditions.

Physical modulation of intracellular signaling processes by locational regulation

Recent observations in the field of signal transduction suggest that where a protein is located within a cell can be as important as its activity measured in solution for activation of its downstream pathway. The physical organization of the cell can provide an additional layer of control upon the chemical reaction networks that govern ultimately perceived signals. Using the cytosol and plasma membrane as relevant compartmental distinctions, we analyze the effect of relocation on the rate of association with a membrane-associated target. We quantify this effect as an enhancement factor E in terms of measurable parameters such as the number of available targets, molecular diffusivities, and intrinsic reaction rate constants. We then employ two simple yet relevant example models to illustrate how relocation can affect the dynamics of signal transduction pathways. The temporal profiles and phase behavior of these models are investigated. We also relate experimentally observable aspects of signal transduction such as peak activation and the relative time scales of stimulus and response to quantitative aspects of the relocation mechanisms in our models. In our example schemes, nearly complete relocation of the cytosolic species in the signaling pair is required to generate meaningful activation of the model pathways when the association rate enhancement factor E is as low as 10; when E is 100 or greater, only a small fraction of the protein must be relocated.

Initiation of blood coagulation: the tissue factor/factor VIIa complex

Tissue factor (TF), a transmembrane glycoprotein, functions as an essential activator of the serine protease factor VIIa. This enzymatic complex is considered to be the principal initiator of in vivo coagulation. Recent studies emphasize the role of the TF/VIIa complex in a number of pathophysiological processes, such as Gram-negative sepsis, coronary artery disease and neointimal hyperplasia after angioplasty. Monocytes/macrophages are important contributors to some of these diseases and there have been new insights into the biology of TF regulation in monocytes. In the light of its structural similarity to cytokine receptors, there has been frequent speculation that TF has a role in intracellular signaling, a suggestion that is supported by some recent studies that propose a true receptor function for TF.