We study a higher order parabolic partial differential equation that arises in the context of condensed matter physics. It is a fourth order semilinear equation which nonlinearity is the determinant of the Hessian matrix of the solution. We consider this model in a bounded domain of the real plane and study its stationary solutions both when the geometry of this domain is arbitrary and when it is the unit ball and the solution is radially symmetric. We also consider the initial-boundary value problem for the full parabolic equation. We summarize our results on existence of solutions in these cases and propose an open problem related to the existence of self-similar solutions.
Apical membrane antigen-1 (AMA-1) of Plasmodium vivax Grassi et Feletti, 1890 is a promising malaria vaccine candidate. However, antigenic variation is a major problem to design a universal malaria vaccine. Hence, detailed understanding of the pvama-1 gene polymorphism can provide conductive information on this potential vaccine component. Therefore, this study investigated the extent of genetic polymorphisms at domain I (DI), DII and partial DIII of AMA-1 among Iranian P. vivax isolates. Out of 107 blood samples, 92 were analysed based on the quality of the sequencing data. The sequences were classified into 53 haplotypes. Amino acid changes were observed at 31 positions that 17 were located at DI, 11 were at DII and the rest of them (3 positions) were at DIII. Thus, codon polymorphisms at DI were found to be higher than DII. Also, five of these polymorphic codons (D242E, T374P, S389R, Y391F, I395F) were novel and have not been reported yet. Neutrality analysis by using the dN-dS difference (the difference between the rate of non-synonymous and synonymous mutations) showed a negative diversifying selection at DI, DII and across the length of both domains. The potential B-cell epitopes were found in 5 regions of the PvAMA-1 with 10 mutation sites (E145A, K188N, E189N/K/D, K190Q/E, P210S, E227V, D242E, R249H, G253E, K352E), whereas only one mutation (G288E) has been detected in intrinsically unstructured/disordered regions. Fixation index (Fst) estimation between Iranian and Indian isolates (0.0131) indicated a significant low genetic differentiation. Distribution of the polymorphic sites and IURs mapped on a three dimensional structure of PvAMA-1 showed that these regions were located at two opposite faces of the molecule. In conclusion, the results have significant value in the design and development of a malaria vaccine based on this antigen.