Eigenvalue Problem in Two Dimensions for an Irregular Boundary II: Neumann Condition

We formulate a systematic elegant perturbative scheme for determining the eigenvalues of the Helmholtz equation (\bigtriangledown^{2} + k^{2}){\psi} = 0 in two dimensions when the normal derivative of {\psi} vanishes on an irregular closed curve. Unique feature of this method, unlike other perturbation schemes, is that it does not require a separate formalism to treat degeneracies. Degenerate states are handled equally elegantly as the non-degenerate ones. A real parameter, extracted from the parameters defining the irregular boundary, serves as a perturbation parameter in this scheme as opposed to earlier schemes where the perturbation parameter is an artificial one. The efficacy of the proposed scheme is gauged by calculating the eigenvalues for elliptical and supercircular boundaries and comparing with the results obtained numerically. We also present a simple and interesting semi-empirical formula, determining the eigenspectrum of the 2D Helmholtz equation with the Dirichlet or the Neumann condition for a supercircular boundary. A comparison of the eigenspectrum for several low-lying modes obtained by employing the formula with the corresponding numerical estimates shows good agreement for a wide range of the supercircular exponent.Comment: 26 pages, 12 figure

Interactions between staphylokinase, plasmin(ogen), and fibrin. Staphylokinase discriminates between free plasminogen and plasminogen bound to partially degraded fibrin

Staphylokinase (STA), a protein of bacterial origin, induces highly fibrin-specific thrombolysis both in human plasma in vitro and in pilot clinical trials. Using fluorescence microscopy, we investigated the spatial distribution of fluorescein isothiocyanate (FITC)-labeled STA during lysis of a plasma clot and its binding to purified fibrin clots in the presence or in the absence of plasmin(ogen). STA highly accumulated in a thin superficial layer of the lysing plasma clot following the distribution of plasminogen (Pg) during lysis. Experiments with purified fibrin clots revealed that STA binds to Pg bound to partially degraded fibrin but not to Pg bound to intact fibrin. Binding of FITC-labeled STA to various forms of plasmin(ogen) in a buffer solution was studied by measuring fluorescence anisotropy. The binding constant for Glu-Pg was estimated as 7.4 μM and for Lys-Pg as 0.28 μM; for active-site blocked plasmin the binding constant was less than 0.05 μM. The much lower affinity of STA for Glu-Pg compared with that for active site- blocked plasmin was mainly due to a lower association rate constant, as assessed by real time biospecific interaction analysis. Gel filtration of a mixture of STA with a molar excess of Glu-Pg demonstrated that STA migrated as an unbound 18-kDa protein when activation of Pg into plasmin was precluded by inhibitors of plasmin. When gel-filtered under the same conditions with plasmin, STA migrated in complex with plasmin with an apparent molecular mass of 100 kDa. Confocal fluorescence microscopy finally demonstrated that when FITC-labeled STA was added to plasma before clotting, it did not bind to fibrin fibers during the first minutes (lag phase), although Pg bound to the fibers moderately. Then, both Pg and STA started to accumulate on the fibers progressively, followed by complete lysis of the clot. In conclusion, our results imply that, when STA is added to plasma, only a small percentage associates with Pg. In contrast, STA binds strongly to plasmin and to Pg, which is bound to partially degraded fibrin. These findings add a new mechanism to the known explanations for the inefficient Pg activation by STA in plasma and specify the mechanism for fibrin-dependent activation of Pg. Chemicals/CAS: auR protein, Staphylococcus aureus, EC 3.4.24.29; Fibrin, 9001-31-4; Fibrinolytic Agents; Fluorescein-5-isothiocyanate, 3326-32-7; Metalloendopeptidases, EC 3.4.24.-; Plasmin, EC 3.4.21.7; Plasminogen, 9001-91-6; Recombinant Protein

Molecular assembly of plasminogen and tissue-type plasminogen activator on an evolving fibrin surface

A well characterized model of an intact and a degraded surface of fibrin that represents the states of fibrin during the initiation and the progression of fibrinolysis was used to quantitatively characterize the molecular interplay between tissue-type plasminogen activator (t-PA), plasminogen and fibrin. The molecular assembly of t-PA and plasminogen on these surfaces was investigated using combinations of proteins that preclude complications due to side reactions caused by generated plasmin: native plasminogen with di-isopropylphosphofluoridate-inactivated t-PA, and a recombinant human plasminogen with the active-site Ser741 mutagenized to Ala which renders the catalytic site inactive. Under these conditions, neither the affinity nor the maximal number of binding sites for plasminogen were modified by the presence of t-PA, indicating that binding sites for plasminogen pre-exist in intact fibrin and are not dependent on the presence of t-PA. In contrast, when plasminogen activation is allowed, increasing binding of plasminogen to the progressively degraded fibrin surface is directly correlated (r = 0.98) to the appearance of the fibrin E-fragment as shown using a monoclonal antibody (FDP-14) that has its epitope in the E domain of fibrin. t-PA was shown to bind with a high affinity to both the intact (Kd = 3.3 +/- 0.6 nM) and the degraded surface of fibrin (Kd = 1.2 +/- 0.4 nM). Binding of t-PA to carboxy-terminal lysine residues of degraded fibrin was shown to be efficiently competed by physiological concentrations of plasminogen (2 microM), indicating that the affinity of t-PA for these residues was lower than that of plasminogen (Kd = 0.66 +/- 0.22 microM) and unrelated to the high affinity of t-PA for specific binding sites on intact fibrin. These data confirm and establish that the generation of carboxy-terminal lysine residues on fibrin during ongoing fibrinolysis, and the binding of plasminogen to these sites, is an important pathway in the acceleration of clot dissolution