Plasminogen Activator Complex Research Articles

Production of monoclonal antibodies against the plasminogen activator inhibitor-1/tissue type plasminogen activator complex

Production of monoclonal antibodies against the plasminogen activator inhibitor-1/tissue type plasminogen activator complex

Invasion of brain tissue by primary glioma: Evidence for the involvement of urokinase-type plasminogen activator as an activator of type iv collagenase

The immunocharacterization of a metalloproteinase isolated from rat glioma cell conditioned medium is described and confirms that the enzyme is identical to type IV collagenase. Free, active plasminogen activator (PA) and PA-PAI complexes were identified as being secreted by the same cells. Using affinity-purified metalloproteinase we demonstrate that the enzyme can be partially activated by u-PA but not by plasmin in vitro. On the basis of these findings and previous published work we propose a scheme for the proteolytic degradation of normal brain tissue during tumour invasion.

Diurnal patterns of serotonin, 5-hydroxyindoleacetic acid, tryptophan and fibrinolytic activity in blood of depressive patients and healthy volunteers

Diurnal changes of serotonin-related factors in whole blood and fibrinolytic activity were determined in depressed patients and healthy controls. Whole blood serotonin concentration of depressed patients showed marked changes throughout daytime, with maximum values in the evening and lowest values in the morning, whereas its metabolite 5-HIAA followed a contrary pattern. The circadian rhythm of 5-HT and 5-HIAA in the control group was quite different from depressed patients. Plasma levels of tPA decreased from 12:30 to 16:30. Concentrations of free plasminogen activator inhibitor (PAI-1) and complex of tPA-PAI-1 decreased from 8:30 to 16:30. Plasma levels of total PAI-1 decreased from 8:30 to 16:30. Plasma levels of the fibrinolytic parameters may be lower in depressive patients than in normal contols. These results support the changes in the circadian rhythm of serotonin and its related substances in the blood of depressive patients.

Cardiovascular Disease

The results of multiple clinical trials performed during the 1980s make it clear in 1990 that thrombolytic therapy should be considered for all patients with evolving myocardial infarction. The major trials that have been completed or that are in progress include the following: the Thrombolysis in Myocardial Infarction (TIMI) Trial Phase I published in 1985¹found that tissue plasminogen activator (TPA) was superior to streptokinase in dissolving coronary artery thrombi. The AIMS study²reported that the recently Food and Drug Administration—approved thrombolytic agent anisoylated streptokinase plasminogen activator complex can be given as a bolus with reperfusion results similar to those achieved with streptokinase. The TIMI Trial Phase II A published in November 1988³showed that, contrary to prevailing views, there was no advantage to performing angioplasty immediately after thrombolytic therapy over performing the procedure 18 to 24 hours later. The Second International Study of Infarct Survival⁴

Cardiovascular disease

The results of multiple clinical trials performed during the 1980s make it clear in 1990 that thrombolytic therapy should be considered for all patients with evolving myocardial infarction. The major trials that have been completed or that are in progress include the following: the Thrombolysis in Myocardial Infarction (TIMI) Trial Phase I published in 1985¹found that tissue plasminogen activator (TPA) was superior to streptokinase in dissolving coronary artery thrombi. The AIMS study²reported that the recently Food and Drug Administration—approved thrombolytic agent anisoylated streptokinase plasminogen activator complex can be given as a bolus with reperfusion results similar to those achieved with streptokinase. The TIMI Trial Phase II A published in November 1988³showed that, contrary to prevailing views, there was no advantage to performing angioplasty immediately after thrombolytic therapy over performing the procedure 18 to 24 hours later. The Second International Study of Infarct Survival⁴

Summary of U.S. clinical trials program for evaluation of anistreplase

Because the reestablishment of coronary blood flow is believed to be central to the benefit of thrombolytic therapy, the U.S. evaluation program of anistreplase has focused on reperfusion, patency, and reocclusion comparisons with approved thrombolytic regimens, including (1) intracoronary streptokinase (completed) (TEAM-I), (2) intravenous streptokinase (completed) (TEAM-II), and (3) intravenous tissue plasminogen activator complex or alteplase (tPA, ongoing) (TEAM-III). The TEAM trials have established that anistreplase possesses similar reperfusion efficacy to intracoronary streptokinase, if given within 4 h, but is much easier to administer; comparisons with currently approved intravenous agents, streptokinase, and tissue plasminogen activator (tPA) have been completed and are being analyzed or are underway; reocclusion and reinfarction rates are low; safety is good, and similar to that of streptokinase; and mortality rates have been low with treatment, consistent with beneficial reports from randomized trials. Based on overall efficacy, ease of administration, and safety, anistreplase has been shown to be a promising agent for treatment of acute myocardial infarction.

Effect of fibrin-targeting on clot lysis with urokinase-type plasminogen activator

The effect of fibrin-targeting of urokinase-type plasminogen activator (u-PA) on its fibrinolytic potency was studied using recombinant fusion proteins of u-PA with the NH 2-terminal region of tissue-type plasminogen activator (t-PA/u-PA) and chemical complexes of u-PA with F(ab′) 2 fragments of a fibrin specific monoclonal antibody (u-PA/MA-15C5-F(ab′) 2). Two chain derivatives of a low M r variant of u-PA comprising amino acids Leu 144-Leu 411 (tcu-PA-32k), obtained by cleavage of recombinant single-chain u-PA (rscu-PA-32k) with thrombin (rtcu-PA-32k/T) or plasmin (rtcu-PA-32k/P) were investigated. The plasmin-derived two chain u-PA moieties, rtcu-PA-32k/P, rt-PA/tcu-PA-32k/P and rtcu-PA-32k/MA-15C5-F(ab′) 2/P had high specific activities in amidolytic and fibrin plate assays (130,000 and 150,000 IU/mg u-PA, 43,000 and 71,000 IU/mg u-PA and 32,000 and 56,000 IU/mg u-PA respectively). The thrombin-derived two chain u-PA moieties had a very low amidolytic activity, corresponding to ≤ 1 percent of that of their plasmin-derived counterparts. On fibrin plates, however, rtcu-PA-32k/T had a negligible activity, whereas rt-PA/tcu-PA-32k/T and rtcu-PA-32k/MA-15C5-F(ab′) 2/T had specific activities of 12,000 and 25,000 IU/mg u-PA respectively. The catalytic efficiency for plasminogen activation of rtcu-PA-32k/MA-15C5-F(ab′) 2/T is 4,000-fold lower than that of rtcu-PA-32k/MA-15C5-F(ab') 2/P, but its concentration required for 50 percent lysis in 2 hours of a 125I-fibrin labeled plasma clot in human plasma (C 50) is only 25-fold higher. The catalytic efficiency of rt-PA/tcu-PA-32k/T is 1,600-fold lower and the C 50 100-fold higher than that of rt-PA/tcu-PA-32k/P. The catalytic efficiency and the fibrinolytic potential of rtcu-PA-32k/T are negligible as compared to that of rtcu-PA-32k/P. These observations may be explained by conversion of the thrombin derived two chain u-PA moeities to their plasmin-derived analogues at the fibrin surface. This conversion appears to be most efficient for the antibody conjugate which has a high fibrin-affinity, less efficient for the t-PA/u-PA chimera which has only moderate fibrin-affinity, and negligible for the unconjugated u-PA moiety which has no fibrin-affinity. These findings illustrate the importance of plasmin-mediated positive feedback mechanisms in u-PA mediated clot lysis.

Fluctuations of euglobulin lysis time, tissue plasminogen activator, and free and total plasminogen activator inhibitor levels in plasma in daytime

Blood was taken from healthy 15 males and 10 females at 9:30 h, 10:00 h and 12:30 h. The euglobulin fraction was prepared and clotted by the addition of human thrombin. The clot lysis time shortened significantly from 9:30 h to 10:00 h (p<0.01) and further to 12:30 h (p<0.01). Plasma levels of tissue plasminogen activator (t-PA) antigens did not change from 9:30 h to 10:00 h, but slightly and significantly to 12:30 h (p<0.01). Plasma levels of free plasminogen activator inhibitor-1 (PAI-1) and complex of t-PA-PAI-1 decreased from 9:30 h to 10:00 h (p<0.02) and to 12:30 h (p<0.01). Plasma levels of total PAI-1 (free plus complex) decreased from 9:30 h to 10:00 h (p<0.01) and to 12:30 h (p<0.01). These results suggest that a major factor contributing to the enhanced fibrinolytic activity of the euglobulin fraction may be a level of PAI-1 (free and total).

Regulation of Streptokinase-Human Plasmin Complex by the Plasma Proteinase Inhibitors α2-Antiplasmin and α2-Macroglobulin Is Species Specific and Temperature Dependent

Streptokinase-plasmin complex (SkPl) was prepared with human plasminogen. Regulation of SkPl and plasmin by the plasma proteinase inhibitors, alpha 2-antiplasmin (alpha 2AP) and alpha 2-macroglobulin (alpha 2M), was studied as a function of temperature in plasminogen-depleted human plasma, mouse plasma, and solutions of purified proteins. The reaction of plasmin with proteinase inhibitors in human plasma was complete. alpha 2AP was the predominant inhibitor. The fraction of alpha 2M-plasmin recovered was not affected significantly by incubation temperature. In contrast, the reaction of SkPl with human proteinase inhibitors was markedly temperature dependent. The apparent second-order rate constant for the reaction of SkPl with purified alpha 2AP at 37 degrees C (1.5 x 10(2) mol/L-1 s-1) was greater than 150-fold higher than the constant derived at 4 degrees C. In human plasma and in solutions containing mixtures of purified human proteins, alpha 2AP was the principal inhibitor of SkPl. Elevating the temperature enhanced the reaction of SkPl with alpha 2AP and alpha 2M comparably. Equivalent results were obtained when incubations were performed in platelet-rich plasma (PRP) or whole blood. In murine plasma, SkPl reacted readily with the proteinase inhibitors. The principal inhibitor of SkPl was alpha 2M. Maximum reaction between SkPl and murine alpha 2M was observed at 37 degrees C; however, significant reaction also occurred at 4 degrees C. alpha 2 AP was the predominant inhibitor of plasmin in mouse plasma. Reaction of alpha 2AP with SkPl in murine plasma was significant only after the alpha 2M was inactivated with methylamine. These results were not affected by platelets or whole blood cells. We conclude that the thrombolytic efficacy of streptokinase reflects not only the nature of the plasminogen activator complex but also the function of the proteinase inhibitors.

Regulation of streptokinase-human plasmin complex by the plasma proteinase inhibitors alpha 2-antiplasmin and alpha 2-macroglobulin is species specific and temperature dependent

Streptokinase-plasmin complex (SkPl) was prepared with human plasminogen. Regulation of SkPl and plasmin by the plasma proteinase inhibitors, alpha 2-antiplasmin (alpha 2AP) and alpha 2-macroglobulin (alpha 2M), was studied as a function of temperature in plasminogen- depleted human plasma, mouse plasma, and solutions of purified proteins. The reaction of plasmin with proteinase inhibitors in human plasma was complete. alpha 2AP was the predominant inhibitor. The fraction of alpha 2M-plasmin recovered was not affected significantly by incubation temperature. In contrast, the reaction of SkPl with human proteinase inhibitors was markedly temperature dependent. The apparent second-order rate constant for the reaction of SkPl with purified alpha 2AP at 37 degrees C (1.5 x 10(2) mol/L-1 s-1) was greater than 150-fold higher than the constant derived at 4 degrees C. In human plasma and in solutions containing mixtures of purified human proteins, alpha 2AP was the principal inhibitor of SkPl. Elevating the temperature enhanced the reaction of SkPl with alpha 2AP and alpha 2M comparably. Equivalent results were obtained when incubations were performed in platelet-rich plasma (PRP) or whole blood. In murine plasma, SkPl reacted readily with the proteinase inhibitors. The principal inhibitor of SkPl was alpha 2M. Maximum reaction between SkPl and murine alpha 2M was observed at 37 degrees C; however, significant reaction also occurred at 4 degrees C. alpha 2 AP was the predominant inhibitor of plasmin in mouse plasma. Reaction of alpha 2AP with SkPl in murine plasma was significant only after the alpha 2M was inactivated with methylamine. These results were not affected by platelets or whole blood cells. We conclude that the thrombolytic efficacy of streptokinase reflects not only the nature of the plasminogen activator complex but also the function of the proteinase inhibitors.

Measurement of Plasminogen Activator Inhibitor 1 In Biologic Fluids With a Murine Monoclonal Antibody-Based Enzyme-Linked Immunosorbent Assay

An enzyme-linked immunosorbent assay for plasminogen activator inhibitor-1 (PAI-1) in biologic fluids was developed on the basis of two murine monoclonal antibodies raised against PAI-1 purified from HT-1080 fibrosarcoma cells. The lower limit of sensitivity of the assay in plasma is 2 ng/mL. The assay is 12 times less sensitive toward the PAI-1/human tissue-type plasminogen activator (t-PA) complex as compared with free PAI-1. The intraassay, interassay, and interdilution coefficients of variation are 5.2%, 8.0%, and 7.1%, respectively. The level of PAI-1 in platelet-poor plasma of healthy subjects is 18 +/- 10 ng/mL (mean +/- SD, n = 45). In platelet-rich plasma after freezing and thawing, 92% of PAI-1 antigen is released from platelets, whereas only 8% is found in the corresponding platelet-poor plasma. In platelet-poor plasma from healthy subjects, a linear correlation (r = 0.80) was found between PAI activity and PAI-1 antigen. In plasma approximately two thirds of the PAI-1 antigen was functionally active, whereas only 5% of the PAI-1 antigen released from platelets was active. During pregnancy a progressive increase of PAI-1 antigen levels up to three- to sixfold the control value was observed. In plasma of patients with recurrent deep vein thrombosis, PAI-1 levels were 44 +/- 20 ng/mL (mean +/- SD, n = 7), during a clinically silent phase. Four of these patients had a level above 38 ng/mL (mean +/- 2 SD of normal). The present assay, based on stable and reproducible reagents, allows the specific determination of PAI-1 antigen in biologic fluids. It may facilitate interlaboratory comparisons and be useful for further investigations of the role of PAI-1 in clinical conditions associated with impaired fibrinolysis and/or a tendency to thrombosis and investigations of the role of PAI-1 in platelets.

Measurement of plasminogen activator inhibitor 1 in biologic fluids with a murine monoclonal antibody-based enzyme-linked immunosorbent assay

An enzyme-linked immunosorbent assay for plasminogen activator inhibitor-1 (PAI-1) in biologic fluids was developed on the basis of two murine monoclonal antibodies raised against PAI-1 purified from HT- 1080 fibrosarcoma cells. The lower limit of sensitivity of the assay in plasma is 2 ng/mL. The assay is 12 times less sensitive toward the PAI- 1/human tissue-type plasminogen activator (t-PA) complex as compared with free PAI-1. The intraassay, interassay, and interdilution coefficients of variation are 5.2%, 8.0%, and 7.1%, respectively. The level of PAI-1 in platelet-poor plasma of healthy subjects is 18 +/- 10 ng/mL (mean +/- SD, n = 45). In platelet-rich plasma after freezing and thawing, 92% of PAI-1 antigen is released from platelets, whereas only 8% is found in the corresponding platelet-poor plasma. In platelet-poor plasma from healthy subjects, a linear correlation (r = 0.80) was found between PAI activity and PAI-1 antigen. In plasma approximately two thirds of the PAI-1 antigen was functionally active, whereas only 5% of the PAI-1 antigen released from platelets was active. During pregnancy a progressive increase of PAI-1 antigen levels up to three- to sixfold the control value was observed. In plasma of patients with recurrent deep vein thrombosis, PAI-1 levels were 44 +/- 20 ng/mL (mean +/- SD, n = 7), during a clinically silent phase. Four of these patients had a level above 38 ng/mL (mean +/- 2 SD of normal). The present assay, based on stable and reproducible reagents, allows the specific determination of PAI-1 antigen in biologic fluids. It may facilitate interlaboratory comparisons and be useful for further investigations of the role of PAI-1 in clinical conditions associated with impaired fibrinolysis and/or a tendency to thrombosis and investigations of the role of PAI-1 in platelets.

Historical overview of antithrombotic and thrombolytic therapy

Early attempts to treat thromboembolic disorders focused on the prevention of thrombus formation, along with the prevention of extension and/or embolization of an existing thrombus. Antithrombotic therapy for these conditions included intravenous and oral anticoagulants and later antiplatelet agents, all of which remain in use today. Subsequent research sought a thrombolytic agent that could lyse a clot directly. Intravenously administered streptokinase and urokinase were found to act on the inactive proenzyme plasminogen to produce the active enzyme plasmin that could safely and effectively lyse fresh fibrin clots. These agents have proved effective for the treatment of myocardial infarction and pulmonary embolism, as well as peripheral arterial and venous thromboembolic conditions. Recent efforts have been directed toward a search for clot-specific agents that would avoid the systemic lytic state associated with the use of urokinase and streptokinase. Three clot-specific plasminogen activators are currently being evaluated: recombinant tissue-specific plasminogen activator, anisoylated streptokinase plasminogen activator complex, and pro-urokinase.

Acylated Derivatives of Streptokinase-Plasminogen Activator Complex as Thrombolytic Agents in a Dog Model of Aged Venous Thrombosis

A procedure is described for the establishment of experimental clots, aged in vivo for 72 hr, in the jugular vein of beagle dogs. Two acylated derivatives of streptokinase-human (lys) plasminogen activator complex with greatly differing deacylation rates under physiological conditions were compared as thrombolytic agents in the model. These were BRL 26921 (deacylation half-life, 40 min) and BRL 33575 (deacylation half-life c. 17 hr). The pharmacokinetic clearance rate of BRL 33575 from the circulation was studied and gave a clearance half-life of about 7 hr. BRL 33575 was found to be the superior agent in lysing 72 hr aged clots, being effective at a single bolus dose of 420 micrograms/kg or in three equal divided doses of 140 micrograms/kg given at 12 hr intervals. The single dose regime gave moderate systemic plasminogen activation, and the effect was significantly reduced with the divided dose regime. Infusion of freshly formed streptokinase-human plasminogen activator complex at 420 micrograms/kg over 15 hr gave little thrombolysis despite marked systemic plasminogen activation.

Comparison of the hypotensive effects of streptokinase-(human) plasmin activator complex and BRL 26921 (p-anisoylated streptokinase-plasminogen activator complex) in the dog after high dose, bolus administration

In view of current interest in the possibility of rapid, high-dose administration of thrombolytic agents by the intravenous route in patients with coronary thrombosis (AMI), a study of this technique was carried out in the dog. Streptokinase-(human) plasmin activator complex (SK-Pm) and BRL 26921 (p-anisoylated streptokinase-(human) plasminogen activator complex) were each given at equivalent doses (28,500 IU/kg and 800 IU/kg respectively) to groups of beagle dogs by rapid injection over 10 sec and their effects on blood pressure, plasmin formation and kallikrein production were compared over the next 3h. SK-Pm produced, within 1–3 min, a pronounced hypotensive effect that was kinetically related to a rapid and steep rise in systemic plasmin and kallikrein concentrations. BRL 26921 had no hypotensive effect, the rise in plasmin production was slower and the rate and extent of kallikrein formation was significantly less than in the SK-Pm group.

A characterization of native streptokinase and altered streptokinase isolated from a human plasminogen activator complex.

ADVERTISEMENT RETURN TO ISSUEPREVArticleNEXTCharacterization of native streptokinase and altered streptokinase isolated from a human plasminogen activator complexWilliam J. Brockway and Francis J. CastellinoCite this: Biochemistry 1974, 13, 10, 2063–2070Publication Date (Print):May 1, 1974Publication History Published online1 May 2002Published inissue 1 May 1974https://doi.org/10.1021/bi00707a010RIGHTS & PERMISSIONSArticle Views111Altmetric-Citations77LEARN ABOUT THESE METRICSArticle Views are the COUNTER-compliant sum of full text article downloads since November 2008 (both PDF and HTML) across all institutions and individuals. These metrics are regularly updated to reflect usage leading up to the last few days.Citations are the number of other articles citing this article, calculated by Crossref and updated daily. Find more information about Crossref citation counts.The Altmetric Attention Score is a quantitative measure of the attention that a research article has received online. Clicking on the donut icon will load a page at altmetric.com with additional details about the score and the social media presence for the given article. Find more information on the Altmetric Attention Score and how the score is calculated. Share Add toView InAdd Full Text with ReferenceAdd Description ExportRISCitationCitation and abstractCitation and referencesMore Options Share onFacebookTwitterWechatLinked InReddit PDF (2 MB) Get e-Alerts Get e-Alerts

Direct evidence for the generation of an active site in the plasminogen moiety of the streptokinase-human plasminogen activator complex

Summary A plasmin active site titrant, p-nitrophenyl p -(amidinothiomethyl)benzoate (AmSMeBzoNph), has been prepared with a radioactive label in the acid moiety and utilized to directly demonstrate the generation of an active site in plasminogen upon incubation with streptokinase to form the human plasminogen activator complex. When a 1:1 stoichiometric complex of streptokinase and human plasminogen are incubated for 2 minutes in the presence of a 10 fold excess of [ 14 C]-AmSMeBzoNph, 0.7–0.8 moles of p-nitrophenol are released per mole of plasminogen added and 0.7–0.8 moles of the [ 14 C]-substrate are covalently incorporated into the complex. Sodium dodecyl sulfate polyacrylamide gel electrophoresis demonstrated that only streptokinase and human plasminogen were present and all the radioactivity was found in the plasminogen band. When a 1:1 stoichiometric complex of streptokinase and human plasminogen are incubated for 2 minutes in the presence of a 10 fold excess of unlabeled p-nitrophenyl-p′-guanidino benzoate (GdnBzoNph), followed by the same amount of [ 14 C]-AmSMeBzoNph, no label is incorporated into the complex. This evidence supports the contention that an active site is developed in human plasminogen upon complex formation with streptokinase.

Identification of Modified Streptokinase as the Activator of Bovine and Human Plasminogen

Abstract This paper reports observations made of the events which occur when two apparently enzymatically inactive proteins, streptokinase and plasminogen, interact in the correct ratio to form a complex which is an activator of plasminogen. The unique physical and functional properties of this plasminogen activator complex were demonstrated by: (a) observation of the complex in polyacrylamide gel systems and (b) by its capacity to convert bovine or guinea pig plasminogen to plasmin. When the complex was dissociated on a sodium dodecyl sulfate polyacrylamide gel system, two components having physical properties similar to those of the original inactive components were recovered. However, following dissociation, the component which was similar to plasminogen had plasmin activity, and the component which was similar to streptokinase had bovine plasminogen activator activity. The completeness of the separation of the two components was examined revealing that approximately 1 mole of streptokinase contaminated every 1600 moles of the dissociated plasmin component and 1 mole of plasmin contaminated every 1000 moles of the dissociated streptokinase component. The activator activity of the dissociated streptokinase component decreased less than 10% upon a reduction of the plasmin contaminant by 97%. It was therefore concluded that the dissociated streptokinase component bore the site for bovine plasminogen activator activity. Studies of the dissociated streptokinase component indicated that there was no demonstrable change in molecular weight. From these data and the demonstration that the dissociated streptokinase activated human plasminogen as well as bovine plasminogen, a model describing the mechanism of streptokinase activation of human plasminogen was constructed. In this model streptokinase interacts with the plasmin contaminating human plasminogen to form a modified streptokinase which in turn acts on human plasminogen in a positive feedback.

The streptokinase–human plasminogen activator complex. Composition and identity of a subcomponent with activator activity

Reactions between purified plasminogen and streptokinase, labelled with (131)I and (125)I respectively, were investigated by polyacrylamide-gel discontinuous electrophoresis. A molecular complex consisting of both (131)I-labelled plasminogen and (125)I-labelled streptokinase migrated between plasminogen and streptokinase. This complex contained bovine plasminogen activator activity. The relative quantities of (131)I-labelled plasminogen and (125)I-labelled streptokinase in this complex were markedly affected by reaction conditions. A fragment that retained 50% or more of the parent activator activity was released from the complex after exposure to mercaptoethanol. This subcomponent had an estimated molecular weight of 70000, and contained both (131)I-labelled plasminogen and (125)I-labelled streptokinase.

Quantitative Determination of Plasminogen

SummaryA modification of the caseinolytic assay for plasminogen is described. This assay system is characterized by the following features :1. Urokinase is used as activator achieving a complete activation of the plasminogen whereas with streptokinase caseinolytically inactive plasminogen-activator complexes are formed.2. All incubation times are reduced to the minimum which is still compatible with accuracy.3. Results are expressed in percent of a standard of ten normal plasmas.4. In this two-stage assay-system (activation of plasminogen to plasmin, digestion of casein by plasmin) both stages proceed simultaneously in the same system, thus the plasmin formed is stabilized “in statu nascendi” by the casein.5. Several conditions (stability of plasminogen in frozen plasma, use of anticoagulants, reproducibility) are defined.