Mathematical and computational models of RNA nanoclusters and their applications in data-driven environments

Nonlocal models in biology and life sciences: Sources, developments, and applications

Mathematical modeling is one of the fundamental techniques for understanding biophysical mechanisms in developmental biology. It helps researchers to analyze complex physiological processes and connect like a bridge between theoretical and experimental observations. Various groups of mathematical models have been studied to analyze these processes, and the nonlocal models are one of them. Nonlocality is important in realistic mathematical models of physical and biological systems when local models fail to capture the essential dynamics and interactions that occur over a range of distances (e.g., cell-cell, cell-tissue adhesions, neural networks, the spread of diseases, intra-specific competition, nanobeams, etc.). This review illustrates different nonlocal mathematical models applied to biology and life sciences. The major focus has been given to sources, developments, and applications of such models. Among other things, a systematic discussion has been provided for the conditions of pattern formations in biological systems of population dynamics. Special attention has also been given to nonlocal interactions on networks, network coupling and integration, including brain dynamics models that provide an important tool to understand neurodegenerative diseases better. In addition, we have discussed nonlocal modeling approaches for cancer stem cells and tumor cells that are widely applied in the cell migration processes, growth, and avascular tumors in any organ. Furthermore, the discussed nonlocal continuum models can go sufficiently smaller scales, including nanotechnology, where classical local models often fail to capture the complexities of nanoscale interactions, applied to build biosensors to sense biomaterial and its concentration. Piezoelectric and other smart materials are among them, and these devices are becoming increasingly important in the digital and physical world that is intrinsically interconnected with biological systems. Additionally, we have reviewed a nonlocal theory of peridynamics, which deals with continuous and discrete media and applies to model the relationship between fracture and healing in cortical bone, tissue growth and shrinkage, and other areas increasingly important in biomedical and bioengineering applications. Finally, we provided a comprehensive summary of emerging trends and highlighted future directions in this rapidly expanding field.

Coupled Multiphysics Modelling of Sensors for Chemical, Biomedical, and Environmental Applications with Focus on Smart Materials and Low-Dimensional Nanostructures

Low-dimensional nanostructures have many advantages when used in sensors compared to the traditional bulk materials, in particular in their sensitivity and specificity. In such nanostructures, the motion of carriers can be confined from one, two, or all three spatial dimensions, leading to their unique properties. New advancements in nanosensors, based on low-dimensional nanostructures, permit their functioning at scales comparable with biological processes and natural systems, allowing their efficient functionalization with chemical and biological molecules. In this article, we provide details of such sensors, focusing on their several important classes, as well as the issues of their designs based on mathematical and computational models covering a range of scales. Such multiscale models require state-of-the-art techniques for their solutions, and we provide an overview of the associated numerical methodologies and approaches in this context. We emphasize the importance of accounting for coupling between different physical fields such as thermal, electromechanical, and magnetic, as well as of additional nonlinear and nonlocal effects which can be salient features of new applications and sensor designs. Our special attention is given to nanowires and nanotubes which are well suited for nanosensor designs and applications, being able to carry a double functionality, as transducers and the media to transmit the signal. One of the key properties of these nanostructures is an enhancement in sensitivity resulting from their high surface-to-volume ratio, which leads to their geometry-dependant properties. This dependency requires careful consideration at the modelling stage, and we provide further details on this issue. Another important class of sensors analyzed here is pertinent to sensor and actuator technologies based on smart materials. The modelling of such materials in their dynamics-enabled applications represents a significant challenge as we have to deal with strongly nonlinear coupled problems, accounting for dynamic interactions between different physical fields and microstructure evolution. Among other classes, important in novel sensor applications, we have given our special attention to heterostructures and nucleic acid based nanostructures. In terms of the application areas, we have focused on chemical and biomedical fields, as well as on green energy and environmentally-friendly technologies where the efficient designs and opportune deployments of sensors are both urgent and compelling.

Machine Learning for Shape Memory Graphene Nanoribbons and Applications in Biomedical Engineering

Shape memory materials have been playing an important role in a wide range of bioengineering applications. At the same time, recent developments of graphene-based nanostructures, such as nanoribbons, have demonstrated that, due to the unique properties of graphene, they can manifest superior electronic, thermal, mechanical, and optical characteristics ideally suited for their potential usage for the next generation of diagnostic devices, drug delivery systems, and other biomedical applications. One of the most intriguing parts of these new developments lies in the fact that certain types of such graphene nanoribbons can exhibit shape memory effects. In this paper, we apply machine learning tools to build an interatomic potential from DFT calculations for highly ordered graphene oxide nanoribbons, a material that had demonstrated shape memory effects with a recovery strain up to 14.5% for 2D layers. The graphene oxide layer can shrink to a metastable phase with lower constant lattice through the application of an electric field, and returns to the initial phase through an external mechanical force. The deformation leads to an electronic rearrangement and induces magnetization around the oxygen atoms. DFT calculations show no magnetization for sufficiently narrow nanoribbons, while the machine learning model can predict the suppression of the metastable phase for the same narrower nanoribbons. We can improve the prediction accuracy by analyzing only the evolution of the metastable phase, where no magnetization is found according to DFT calculations. The model developed here allows also us to study the evolution of the phases for wider nanoribbons, that would be computationally inaccessible through a pure DFT approach. Moreover, we extend our analysis to realistic systems that include vacancies and boron or nitrogen impurities at the oxygen atomic positions. Finally, we provide a brief overview of the current and potential applications of the materials exhibiting shape memory effects in bioengineering and biomedical fields, focusing on data-driven approaches with machine learning interatomic potentials.

A MOLECULAR DYNAMICS STUDY OF NANOWIRE RESONATOR BIO-OBJECT DETECTION

This paper presents a comprehensive analysis, carried out by the molecular dynamics (MD) simulations, of the vibrations of silicon nanowire (SiNW) resonators, having diverse applications including biological and medical fields. The chosen approach allows us to obtain a better understanding of the nanowire (NW) materials’ characteristics, providing a more detailed insight into the behavior of nanostructures, especially when the topic of interest is relevant to their dynamics, interatomic interactions, and atoms trajectories’ prediction. We first simulate a SiNW to study its frequency of vibrations using MD simulations. Then, we add a molecule of human immunodeficiency virus as an example to investigate the potential of the SiNW resonator for the detection of tiny bio-objects. The developed technique and its application to the detection of tiny objects, such as viruses, are discussed in the context of several key effects pertinent to the design of SiNW.

Mathematical Models with Nonlocal Initial Conditions: An Exemplification from Quantum Mechanics

Nonlocal models are ubiquitous in all branches of science and engineering, with a rapidly expanding range of mathematical and computational applications due to the ability of such models to capture effects and phenomena that traditional models cannot. While spatial nonlocalities have received considerable attention in the research community, the same cannot be said about nonlocality in time, in particular when nonlocal initial conditions are present. This paper aims at filling this gap, providing an overview of the current status of nonlocal models and focusing on the mathematical treatment of such models when nonlocal initial conditions are at the heart of the problem. Specifically, our representative example is given for a nonlocal-in-time problem for the abstract Schrödinger equation. By exploiting the linear nature of nonlocal conditions, we derive an exact representation of the solution operator under assumptions that the spectrum of Hamiltonian is contained in the horizontal strip of the complex plane. The derived representation permits us to establish the necessary and sufficient conditions for the problem’s well-posedness and the existence of its solution under different regularities. Furthermore, we present new sufficient conditions for the existence of the solution that extend the existing results in this field to the case when some nonlocal parameters are unbounded. Two further examples demonstrate the developed methodology and highlight the importance of its computer algebra component in the reduction procedures and parameter estimations for nonlocal models. Finally, a connection of the considered models and developed analysis is discussed in the context of other reduction techniques, concentrating on the most promising from the viewpoint of data-driven modelling environments, and providing directions for further generalizations.