New approach to the stability and control of reaction networks

A new system-theoretic approach for studying the stability and control of chemical reaction networks (CRNs) is proposed, and analyzed. This has direct application to biological applications where biochemical networks suffer from high uncertainty in the kinetic parameters and exact structure of the rate functions. The proposed approach tackles this issue by presenting "structural" results, i.e. results that extract important qualitative information from the structure alone regardless of the specific form of the kinetics which can be arbitrary monotone kinetics, including Mass-Action. The proposed method is based on introducing a class of Lyapunov functions that we call Piecewise Linear in Rates (PWLR) Lyapunov functions. Several algorithms are proposed for the construction of these functions. Subject to mild technical conditions, the existence of these functions can be used to ensure powerful dynamical and algebraic conditions such as Lyapunov stability, asymptotic stability, global asymptotic stability, persistence, uniqueness of equilibria and exponential contraction. This shows that this class of networks is well-behaved and excludes complicated behaviour such as multi-stability, limit cycles and chaos. The class of PWLR functions is then shown to be a subset of larger class of Robust Lyapunov functions (RLFs), which can be interpreted by shifting the analysis to reaction coordinates. In the new coordinates, the problem transforms into finding a common Lyapunov function for a linear parameter varying system. Consequently, dual forms of the PWLR Lyapunov functions are presented, and the interpretation in terms of the variational dynamics and contraction analysis are given. An other class of Piecewise Quadratic in Rates Lyapunov function is also introduced. Relationship with consensus dynamics are also pointed out. Control laws for the stabilization of the proposed class of networks are provided, and the concept of control Lyapunov function is briefly discussed. Finally, the proposed framework is shown to be widely applicable to biochemical networks.Open Acces

New Advancements in Pure and Applied Mathematics via Fractals and Fractional Calculus

This reprint focuses on exploring new developments in both pure and applied mathematics as a result of fractional behaviour. It covers the range of ongoing activities in the context of fractional calculus by offering alternate viewpoints, workable solutions, new derivatives, and methods to solve real-world problems. It is impossible to deny that fractional behaviour exists in nature. Any phenomenon that has a pulse, rhythm, or pattern appears to be a fractal. The 17 papers that were published and are part of this volume provide credence to that claim. A variety of topics illustrate the use of fractional calculus in a range of disciplines and offer sufficient coverage to pique every reader's attention