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.

The nonlinearity of regulation in biological networks

The extent to which the components of a biological system are (non)linearly regulated determines how amenable they are to therapy and control. To better understand this property termed "regulatory nonlinearity", we analyzed a suite of 137 published Boolean network models, containing a variety of complex nonlinear regulatory interactions, using a probabilistic generalization of Boolean logic that George Boole himself had proposed. Leveraging the continuous-nature of this formulation, we used Taylor decomposition to approximate the models with various levels of regulatory nonlinearity. A comparison of the resulting series of approximations of the biological models with appropriate random ensembles revealed that biological regulation tends to be less nonlinear than expected, meaning that higher-order interactions among the regulatory inputs tend to be less pronounced. A further categorical analysis of the biological models revealed that the regulatory nonlinearity of cancer and disease networks could not only be sometimes higher than expected but also be relatively more variable. We show that this variation is caused by differences in the apportioning of information among the various orders of regulatory nonlinearity. Our results suggest that there may have been a weak but discernible selection pressure for biological systems to evolve linear regulation on average, but for certain systems such as cancer, on the other hand, to simultaneously evolve more nonlinear rules.

Computation in bacterial communities

Bacteria across many scales are involved in a dynamic process of information exchange to coordinate activity and community structure within large and diverse populations. The molecular components bacteria use to communicate have been discovered and characterized, and recent efforts have begun to understand the potential for bacterial signal exchange to gather information from the environment and coordinate collective behaviors. Such computations made by bacteria to coordinate the action of a population of cells in response to information gathered by a multitude of inputs is a form of collective intelligence. These computations must be robust to fluctuations in both biological, chemical, and physical parameters as well as to operate with energetic efficiency. Given these constraints, what are the limits of computation by bacterial populations and what strategies have evolved to ensure bacterial communities efficiently work together? Here the current understanding of information exchange and collective decision making that occur in microbial populations will be reviewed. Looking toward the future, we consider how a deeper understanding of bacterial computation will inform future direction in microbiology, biotechnology, and biophysics.