Background: In the post-genomic era, a central and overarching question in the analysis of protein-protein interaction networks continues to be whether biological characteristics and functions of proteins such as lethality, physiological malfunctions and malignancy are intimately linked to the topological role proteins play in the network as a mathematical structure. One of the key features that have implicitly been presumed is the existence of hubs, highly connected proteins considered to play a crucial role in biological networks. We explore the structure of protein interaction networks of a number of organisms as metric spaces and show that hubs are non randomly positioned and, from a distance point of view, centrally located. Results: By analysing how the human functional protein interaction network, the human signalling network, Saccharomyces cerevisiae, Arabidopsis thaliana and Escherichia coli protein-protein interaction networks from various databases are distributed as metric spaces, we found that proteins interact radially through a central node, high degree proteins coagulate in the centre of the network, and those far away from the centre have low degree. We further found that the distribution of proteins from the centre is in some hierarchy of importance and has biological significance. Conclusions: We conclude that structurally, protein interaction networks are mathematical entities that share properties between organisms but not necessarily with other networks that follow power-law. We therefore conclude that (i) if there are hubs defined by degree, they are not distributed randomly; (ii) zones closest to the centre of the network are enriched for critically important proteins and are also functionally very specialised for specific 'house keeping’ functions; (iii) proteins closest to the network centre are functionally less dispensable and may present good targets for therapy development; and (iv) network biology requires its own network theory modelled on actual biological evidence and that simply adopting theories from the social sciences may be misleading.

Nearly every area of mathematics, natural, social, and engineering now includes research into finding exact answers to nonlinear fractional differential equations (NFDES). In order to discover the exact solutions to the higher order Sasa-Satsuma equation in the sense of the beta derivative, the paper will discuss the modified simple equation (MSE) and exponential rational function (ERF) approaches. In general, symmetry and travelling wave solutions of the Sasa-Satsuma equation have a common correlation with each other, thus we reduce equations from wave transformations to ordinary differential equations with the help of Lie symmetries. Actually, we can say that wave moves are symmetrical. The considered procedures are effective, accurate, simple, and straightforward to compute. In order to highlight the physical characteristics of the solutions, we also provide 2D and 3D plots of the results.

This paper studies the influence of space-fractional and multiplicative noise on the exact solutions of the space-fractional stochastic dispersive modified Benjamin–Bona–Mahony equation, driven in Ito’s sense by a multiplicative Wiener process. The bifurcation of the exact solutions is investigated, and novel fractional stochastic solutions are presented. The dependence of the solutions on the initial conditions is discussed. Due to the significance of the fractional stochastic modified Benjamin–Bona–Mahony equation in describing the propagation of surface long waves in nonlinear dispersive media, the derived solutions are significantly more helpful for and influential in comprehending diverse, crucial, and challenging physical phenomena. The effect of the Wiener process and the fractional order on the exact solutions are studied.

Background: Cancer development and progression involve a complex network of pathways among which certain pathways play a pivotal role in promoting tumor growth and survival. An important pathway in this context is the PI3K/AKT pathway, which regulates crucial cellular processes including proliferation, viability, and metabolic regulation. Dysregulation of this pathway has been strongly linked to the development of various types of cancers. Consequently, it is imperative to identify the key proteins within this pathway as potential targets for impeding cancer cell proliferation and survival. Results: One of the key findings of this study was the identification of signaling proteins that dominate various forms of PI3K/Akt pathway. Furthermore, proteins play critical roles in cancer networks, acting as oncogenes that promote cancer development or as tumor suppressor genes that inhibit tumor growth. This study identified several genes, including KIT, ERBB2, PDGFRA, MET, FGFR2, and FGFR3, which are involved in various types of the PI3K/Akt pathways. Additionally, this study identified 55 proteins that are commonly found in various forms of PI3K/Akt, and these proteins play crucial roles in regulating various biological functions. Conclusions: This study highlights the importance of identifying key proteins involved in the PI3K/AKT pathway. In this study, we identified several genes involved in different pathways that play essential roles in the activation, signaling, and regulation of the pathway. Understanding the proteins participating in the PI3K/AKT pathway is vital for the development of targeted therapies, not only for cancer but also for other related diseases. By elucidating their roles and functions, this study contributes to the advancement of knowledge in the field and paves the way for the development of effective treatments targeting this pathway.

We have recently shown by formally modelling human protein interaction networks (PINs) as metric spaces and classified proteins into zones based on their distance from the topological centre that hub proteins are primarily centrally located. We also showed that zones closest to the network centre are enriched for critically important proteins and are also functionally very specialised for specific ‘house keeping’ functions. We proposed that proteins closest to the network centre may present good therapeutic targets. Here, we present multiple pieces of novel functional evidence that provides strong support for this hypothesis.

The delayed intervention techniques in real‐world problem modelling have a significant role in behavioural, social, physical, and biological engineering, biomathematical sciences, and many more disciplines. Delayed modelling of real‐world problems is a powerful tool and nonpharmaceutical technique for understanding the dynamics of disease in a population. This paper considers real‐world problems like the Lassa fever model. According to the World Health Organization (WHO), Benin, Ghana, Guinea, Liberia, Mali, Sierra Leone, Togo, Nigeria, and West Africa are the most affected countries with Lassa fever. The most dangerous situation is that eighty percent of the infected persons have no symptoms. To study the dynamics of Lassa fever, two types of populations are considered humans and rats. The human population includes susceptible, infected, and recovered. The rat population includes susceptible and infectious rodents. By introducing a delay parameter and decay exponential term into the existing model in the literature, we got the system of highly nonlinear delay differential equations (DDEs). The fundamental properties such as positivity, boundedness, existence, and uniqueness are verified for the said model. The equilibrium and reproduction number of the model are discussed. The reproduction number for the Lassa fever model is analyzed using the next‐generation matrix method. If the reproduction number is less than one, this situation helps eradicate the disease. If the reproduction number is more significant than one, then the virus will spread rapidly in human beings. We have also investigated the effect of the delay factor on reproduction numbers. The local and global stabilities for both equilibria of the model have also been presented. Furthermore, computer simulations are designed to analyze the academic behaviour of the model.