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This paper investigates the algebraic formulation and Nash equilibrium of competitive diffusion games by using semi-tensor product method, and gives some new results. Firstly, an algebraic formulation of competitive diffusion games is established via the semi-tensor product of matrices, based on which all the fixed points (the end of the diffusion process) are obtained. Secondly, using the algebraic formulation, a necessary and sufficient condition is presented for the verification of pure-strategy Nash equilibrium. Finally, an illustrative example is given to demonstrate the effectiveness of the obtained new results.

Background Zinc-α2-glycoprotein (ZAG) is a 42-kDa protein reported as an anti-inflammatory adipocytokine. Evidences from clinical and experimental studies revealed that brain inflammation plays important roles in epileptogenesis and seizure. Interestingly, closely relationship between ZAG and many important inflammatory mediators has been proven. Our previous study identified ZAG in neurons and found that ZAG is decreased in epilepsy and interacts with TGFβ and ERK. This study aimed to investigate the role of ZAG in seizure and explore its effect on seizure-related neuroinflammation. Methods We overexpressed AZGP1 in the hippocampus of rats via adeno-associated virus vector injection and observed their seizure behavior and EEG after pentylenetetrazol (PTZ) kindling. The level of typical inflammation mediators including TNFα, IL-6, TGFβ, ERK, and ERK phosphorylation were determined. Results The overexpression of AZGP1 reduced the seizure severity, prolonged the latency of kindling, and alleviated epileptiform discharges in EEG changes induced by PTZ. Overexpression of AZGP1 also suppressed the expression of TNFα, IL-6, TGFβ, and ERK phosphorylaton in PTZ-kindled rats. Conclusions ZAG may inhibit TGFβ-mediated ERK phosphorylation and inhibit neuroinflammation mediated by TNFα and IL-6, suggesting ZAG may suppress seizure via inhibiting neuroinflammation. ZAG may be a potential and novel therapeutic target for epilepsy.

To improve gene delivery efficiency and decrease cytotoxicity of polyamidoamine (PAMAM) polymers, P123-g-PAMAM was synthesized by modifying PAMAM with pluronic P123. The structure of the synthesized polymers was analyzed using proton nuclear magnetic resonance. The polymers were able to self-assemble with DNA, forming nanometer-scale complexes. Particle size measurement confirmed that the mean diameter of the polyplexes was 100–250 nm. The cytotoxicity and transfer efficiency were measured and compared with those of PEI and PAMAM. All of the polyplexes showed significantly low cytotoxicity in the MCF-7, HepG2, and 293T cell lines. In addition, the low level of P123 grafting to PAMAM showed significantly higher transfection efficiency than unmodified PAMAM at the optimal N/P ratio. These results suggest that P123-ylated PAMAM may prove as a useful carrier for gene delivery. Open image in new window