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The integrated density of states (IDS) for the Schrödinger operators is defined in two ways: by using the counting function of eigenvalues of the operator restricted to bounded regions with appropriate boundary conditions or by using the spectral projection of the whole space operator. A sufficient condition for the coincidence of the two definitions above is given. Moreover, a sufficient condition for the coincidence of the IDS for the Dirichlet boundary conditions and the IDS for the Neumann boundary conditions is given. The proof is based only on the fundamental items in functional analysis, such as the min-max principle, etc.

Transglutaminase-mediated site-specific and covalent immobilization of an enzyme to chemically modified agarose was explored. Using Escherichia coli alkaline phosphatase (AP) as a model, two designed specific peptide tags containing a reactive lysine (Lys) residue with different length Gly-Ser linkers for microbial transglutaminase (MTG) were genetically attached to N- or C-termini. For solid support, agarose gel beads were chemically modified with beta-casein to display reactive glutamine (Gln) residues on the support surface. Recombinant APs were enzymatically and covalently immobilized to casein-grafted agarose beads. Immobilization by MTG markedly depended on either the position or the length of the peptide tags incorporated to AP, suggesting steric constraint upon enzymatic immobilization. Enzymatically immobilized AP showed comparable catalytic turnover (k(cat)) to the soluble counterpart and comparable operational stability with chemically immobilized AP. These results indicate that attachment of a suitable specific peptide tag to the right position of a target protein is crucial for MTG-mediated formulation of highly active immobilized proteins.

An enzymatic method for covalent and site-specific immobilization of recombinant proteins on a plastic surface was explored. Using Escherichia coli alkaline phosphatase (AP) with a specific peptide tag (MKHKGS) genetically incorporated at the N-terminus as a model (NK-AP), microbial transglutaminase (MTG)-mediated protein immobilization was demonstrated. To generate a reactive surface for MTG, a 96-well polystyrene microtiter plate was physically coated with casein, a good MTG substrate. Successful immobilization of recombinant AP to the nanolayer of casein on the surface of the microtiter plate was verified by the detection of enzymatic activity. Since little activity was observed when wild-type AP was used, immobilization of NK-AP was likely directed by the specific peptide tag. When polymeric casein prepared by MTG was used as a matrix on the plate, the loading capacity of AP was increased about 2-fold compared to when casein was used as the matrix. Transglutaminase-mediated site-specific posttranslational modification of proteins offers one way of generating a variety of protein-based solid formulations for biotechnological applications.