Open Problems

Periodica Mathematica Hungarica -     

Formula 1 – A Mathematical Microworld with CAS: Analysis of Learning Opportunities and Experiences with Students

In this article we describe a mathematical microworld for investigating car motion on a racing course and its use with a group of grade 12 students. The microworld is concerned with the mathematical construction of courses and functions which describe car motion. It is implemented in the computer algebra system, Maple^®, which provides the means to represent courses and functions symbolically and graphically. We describe the learning opportunities offered by the microworld in relation to the research literature on functions. Various facets and layers of the function concept are addressed in the microworld, and we suggest how work in the microworld might help in overcoming some well-known misconceptions.This revised version was published online in September 2005 with corrections to the Cover Date.     

A high performance liquid chromatographic assay for AMP-deaminase activity in the erythrocytes of healthy subjects and patients with inherited purine disorders

A novel method for measuring AMP-deaminase activity in human erythrocytes is presented, based on the determination of the reaction product, IMP, using high performance liquid chromatography. IMP formation was found to be proportional both to the incubation time and the amount of haemolysate over a wide range. The minimal detectable AMP-deaminase activity was more than 1000 times lower than the mean activity found in healthy controls (1083 nmol/h/mg Hb). No marked difference of activity was found in the patients with the following inherited purine disorders: familial juvenile gouty nephropathy and deficiencies of adenosine deaminase, hypoxanthine-guanine phosphoribosyltransferase or adenine phosphoribosyltransferase. The activity in the erythrocytes of patients with chronic renal failure was also similar to controls. The existence of subjects with low erythrocyte AMP-deaminase activity in the population has been confirmed.     

Photosensitizers for tumor therapy: determination of bis(di-isobutyl octadecylsiloxy)silicon 2,3-naphthalocyanine (isoBOSINC) in rat tissue and serum by high-performance liquid chromatography.

Bis(di-isobutyl octadecylsiloxy)silicon 2,3-naphthalocyanine (isoBOSINC) is a synthetic potential photosensitizer for tumor therapy. A new method, which combines solvent extraction and several purification steps, has been developed to determine its presence in tissues. Separation and quantitation of isoBOSINC is done by high-performance liquid chromatography on a silica column with toluene as a mobile phase and using fluorescence detection (lambda ex = 365 nm, lambda em = 750 nm). For recovery studies, isoBOSINC was added to muscles at levels of 0.067 and 0.67 micrograms/g; the mean recoveries were 100%, with coefficients of variation of 6.1 and 6.4%, respectively. For liver samples, the amounts added were 0.67 and 6.7 micrograms/g and for serum 0.67 and 6.70 micrograms/ml. The mean recoveries for liver were 86 and 93%, with coefficients of variation of 7.7 and 4.4%, respectively. For serum, the mean recoveries were 99 and 96%, with coefficients of variation of 2.6 and 6.9%, respectively. Due to its low detection limit and selectivity, the method is appropriate for pharmacokinetic as well as tumor uptake studies following in vivo exposure to isoBOSINC. Preliminary data on tissue distribution of the photosensitizer in normal rats are also presented.     

Neutral oxygen atom density in the MESOX air plasma solar furnace facility

The density of neutral oxygen atoms in the MESOX set-up, one device of the PROMES-CNRS solar facilities, was determined by a fiber-optics catalytic probe (FOCP). Plasma was created in a flowing air within a quartz tube with the outer diameter of 5 cm by a 2.45 GHz microwave generator with the output power up to 1000 W. The flow of air was varied between 4 and 20 l/h. The O-atom density was found to increase monotonously with the increasing discharge power, and it decreased with the increasing flow rate. The degree of dissociation of oxygen molecules in the plasma column depended largely on the flow rate. At the air flow of 4 l/h it was about 80% but it decreased to about 20% at the flow of 20 l/h.     

Treatment of Malignant Melanoma by High-Peak-Power 1064 nm Irradiation Followed by Photodynamic Therapy

Irradiation of B16 pigmented melanoma subcutaneously transplanted in C57 mice with a single 650 mj pulse (10 ns) of 1064 nm light from a Q-switched Nd: YAG laser caused instantaneous bleaching of the pigmented tissue. Visual and histological examination of the resulting gray-colored tumor revealed the breakdown of melanosomes with no detectable alteration of the normal and tumor-overlying skin. Histological examination of the irradiated tumor showed some degree of vascular damage; the depth of the photodamage was not affected by the successive delivery of three consecutive light pulses. The bleached tumor grew at a modestly slower rate but the high-peak-power (HPP) laser treatment did not affect the tumor concentration of a photodynamic sensitizer Si(IV)-naphthalocyanine (isoBO-SiNc) intravenously injected 24 h before Nd : YAG irradiation. Treatment of the B16 pigmented melanoma by photodynamic therapy (PDT: 1 mg/kg isoBO-SiNc, 300 mW/cm², 520 J/cm²) from a 774 nm diode laser immediately after the 1064 nm irradiation resulted in a 16 day delay of tumor regrowth, which was markedly longer than the delay (ca 6 days) obtained after PDT under identical conditions without the preirradia-tion. Thus, pretreatment of pigmented tumors with HPP 1064 nm light appears to enhance their susceptibility to conventional PDT. The tumor response was further enhanced by repeating the combined HPP/PDT treatment at an interval of 10 days (regrowth delay: 27 days), as well as by applying hyperthermia immediately after HPP/PDT (regrowth delay: ca 34 days).     

A thiazole-containing tripodal ligand: synthesis, characterization, and interactions with metal ions and matrix metalloproteinases

A new tripodal ligand, tris[2-(((2-thiazolyl)methylidene)amino)ethyl]amine (Tatren), has been synthesized and characterized by NMR, IR, and UV-visible absorbance spectroscopy and elemental analysis. Tatren forms stable complexes with transition metal ions (Zn(2+), 1; Mn(2+), 2; Co(2+), 3) and the alkaline earth metal ions (Ca(2+), 4; Mg(2+), 5). Single-crystal X-ray structures of 1, 2, and 5 revealed six-coordinate chelate complexes with formula [M(Tatren)](ClO(4))(2) in which the metal centers are coordinated by three thiazolyl N atoms and three acyclic imine N atoms. Crystals of 1, 2, and 5 are monoclinic, P2(1)/c space group. Crystals of 4 are triclinic, P space group. The Ca(2+) complex is eight-coordinate with all N atoms of Tatren and one water molecule coordinated to the metal ion. Spectrophotometric titrations show that formation constants for the chelates of metal ions are >1 in methanol. Free Tatren inhibits the catalytic domain of matrix metalloproteinase-13 (MMP-13, collagenase-3) with K(i) = 3.5 +/- 0.6 microM. Molecular mechanics-based docking calculations suggest that one leg of Tatren coordinates to the catalytic Zn(2+) in MMPs-2, -9, and -13 with significant hydrogen bonding to backbone amide groups. High-level DFT calculations suggest that, in the absence of nonbonded interactions between Tatren and the enzyme, the most stable first coordination sphere of the catalytic Zn(2+) is achieved with three imidazolyl groups from His residues and two imine N atoms from one leg of Tatren. While complexes (1-3) do not inhibit MMP-13 to a significant extent, 4 does (K(i) = 30 +/- 10 microM). Hence, this study shows that tripodal chelating ligands of this class and their Ca(2+) complexes have potential as active-site inhibitors for MMPs.