In this study, we introduce a nonlinear leukemia dynamical system for a piecewise modified ABC fractional-order derivative and analyze it for the theoretical as well computational works and examine the crossover effect of the model. For the crossover behavior of the operators, we presume a division of the period of study $[0,t_2]$ in two subclasses as ${\mathcal{I}}_1=[0,t_1]$, ${\mathcal{I}}_2=[t_1,t_2]$, for $t_1,t_2\in \mathbb{R}$ with $t_1<t_2$. In ${\mathcal{I}}_1$, the classical derivative is considered for the study of the leukemia growth while in ${\mathcal{I}}_2$ we presume modified ABC fractional differential operator. As a result, the study is initiated in the piecewise modified ABC sense of derivative for the dynamical systems. The novel constructed model is then studied for the solution existence and stability as well computational results. The symmetry in dynamics for all the three classes can be graphically observed in the presented six plots.

The study of variable order differential equations is important in science and engineering for a better representation and analysis of dynamical problems. In the literature, there are several fractional order operators involving variable orders. In this article, we construct a nonlinear variable order fractional differential system with a p-Laplacian operator. The presumed problem is a general class of the nonlinear equations of variable orders in the ABC sense of derivatives in combination with Caputo’s fractional derivative. We investigate the existence of solutions and the Hyers–Ulam stability of the considered equation. The presumed problem is a hybrid in nature and has a lot of applications. We have given its particular example as a waterborne disease model of variable order which is analysed for the numerical computations for different variable orders. The results obtained for the variable orders have an advantage over the constant orders in that the variable order simulations present the fluctuation of the real dynamics throughout our observations of the simulations.

Lectins are unique glycoproteins that react with specific sugar residues on cell surfaces resulting in agglutination. They offer enormous applications in therapeutics, diagnostics, medicine, and agriculture. Rice lectins are naturally expressed during biotic and abiotic stresses suggesting their importance in stress resistance physiology. The objective of this study was to determine the presence and relative concentration of lectins in different accessions of rice obtained from IABGR/NARC Islamabad mainly originated from Pakistan. About 210 rice accessions including 02 local varieties and 05 transgenic seeds were screened for seed lectins using a hemagglutination (HA) assay with 5% Californian bred rabbits’ erythrocytes. A protein concentration of 3–8 mg/100 mg of seed flour was measured for all the rice accessions; the highest was 8.03 mg for accession 7600, while the lowest noted was 3.05 mg for accession 7753. Out of 210 accessions, 106 showed the highest HA activity. These 106 genotypes were further screened for titer analysis and specific activity. The highest titer and specific activity were observed for accession 7271 as 1024 and 236 hemagglutination unit (HAU), respectively. The selected accessions’ relative affinity and HA capability were evaluated using blood from four different sources: human, broiler chicken, local rabbit, and Californian-breed rabbit. The highest HA activity was observed with Californian-breed rabbit RBCs. The lectin assay was stable for about 1–2 h. After the required investigations, the accessions with higher lectin concentration and HA capability could be used as a readily available source of lectins for further characterization and utilization in crop improvement programs.

An automated, qualitative and semi-quantitative micro-assay for the detection of plant lectins also known as phyto-agglutinins (glycoproteins; have exciting applications in medicines) is described as an alternative to conventional assays. The method developed in this work is based on hemagglutination (HA) assay that can simultaneously detect the presence and concentration (titer) of lectins in as many as 96 different samples without the aid of an expert eye. We used rice (Oryza sativa L.) seeds for making clarified lectin extracts and Arabic gum as positive control in phosphate buffer saline; the method is applicable to all kinds and parts of the plants. Rabbit red blood cells were used in order to carry out the HA assay in a miniaturized experiment using U-welled microplates (MPs). 25 µL of plant extract is sufficient to carry out HA micro-assay at incubation temperature of 38°C for 20 min. The method was standardized with an expert eye and automated by using MP reader. Moreover, a standard curve for the direct interpretation of lectin concentration is also developed by conversion of absorbance values into titer. The method described will save time, material, labor, and simplify the rapid semi-quantification of plant lectins.