We demonstrate that base mismatches of caspase-3 DNA sequences can be detected by surface plasmon resonance (SPR) following signal amplification by polymerase from Thermus aquaticus (Taq). The concentration of magnesium ions and the respective dNTPs for polymerase binding to the oligonucleotides on the sensing surface were optimized. Taq polymerase binds to double-stranded DNA that is self-assembled on the gold surface of the biosensor to induce an SPR signal. Experiments are presented on the effect of Mg(II) and dNTP concentrations on the activity of the polymerase on the sensing surface. The detection limits are 50 pM, 0.1 nM, 0.7 nM, 7 nM, and 20 nM for correctly matched, single-base mismatched, two-base mismatched, three-base mismatched and four-base mismatched DNA of caspase-3, respectively. This is attributed to the optimized experimental conditions, with samples containing 2 μM of Mg(II) and 0.3 mM of dNTP.
A directed projection graph of the [Formula: see text]-dimensional hypercube on the two-dimensional plane is successfully created. Any [Formula: see text]-variable Boolean function can be easily transformed to an induced subgraph of the projection. Therefore, the discussions on [Formula: see text]-variable Boolean functions only need to focus on a two-dimensional planar graph. Some mathematical theories on the projection graph and the induced subgraph are established, and some properties and characteristics of a balanced linearly separable Boolean function (BLSBF) are uncovered. In particular, the sub-hypercube decompositions of BLSBF is easily represented on the projection, and meanwhile, the enumeration scheme for counting the number of [Formula: see text]-variable BLSBFs is developed by using equivalence classification and conformal transformation. With the aid of the directed projection grap constructed in this paper, one can further study many difficult problems in some fields such as Boolean functions and artificial neural networks.
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