ON THE EQUIVALENCE OF DELAYED ARGUMENTS AND TRANSFER EQUATIONS FOR MODELING DYNAMIC SYSTEMS

Development and improvement of mathematical methods used in modeling biological systems represents a topical issue of mathematical biology. In this paper, we considered a general form of a system of first-order delayed differential equations, traditionally used for describing the function of biological systems of different hierarchical levels. The main feature of this class of models is that some inherent processes (for example, elongation of DNA, RNA, and protein synthesis) are described in a subtle form and can be explicitly specified only through delayed arguments. In this paper, we propose an algorithm for rewriting systems with constant delayed arguments in an equivalent form that represents a system of partial differential equations with transfer equations. The algorithm is universal, since it does not impose any special conditions on the form of the right-hand parts of systems with delayed arguments. The proposed method is a multivariant algorithm. That is, based on one system of differential equations with delayed arguments, the algorithm allows writing out a number of special systems of partial differential equations, which are equivalent to the original system with delayed argument in the entire solution set. The results obtained indicate that delayed arguments and transfer equations are equivalent mathematical tools for describing all types of dynamic processes of energy and/or matter transfer in biological, chemical, and physical systems, indicating a deep-level similarity between properties of dynamic systems, regardless of their origin. At the same time, those processes that are subtle when retarded argument is used can be explicitly described in the form of transfer equations using systems of partial differential equations. This property is extremely important for the modeling of molecular genetic systems in which processes of DNA, RNA, and protein synthesis proceed at variable rates and need to be considered in certain problems, what can easily be done in models constructed using the mathematical tool of partial derivatives

The molecular view of mechanical stress of brain cells, local translation, and neurodegenerative diseases

The assumption that chronic mechanical stress in brain cells stemming from intracranial hypertension, arterial hypertension, or mechanical injury is a risk factor for neurodegenerative diseases was put forward in the 1990s and has since been supported. However, the molecular mechanisms that underlie the way from cell exposure to mechanical stress to disturbances in synaptic plasticity followed by changes in behavior, cognition, and memory are still poorly understood. Here we review (1) the current knowledge of molecular mechanisms regulating local translation and the actin cytoskeleton state at an activated synapse, where they play a key role in the formation of various sorts of synaptic plasticity and long-term memory, and (2) possible pathways of mechanical stress intervention. The roles of the mTOR (mammalian target of rapamycin) signaling pathway; the RNA-binding FMRP protein; the CYFIP1 protein, interacting with FMRP; the family of small GTPases; and the WAVE regulatory complex in the regulation of translation initiation and actin cytoskeleton rearrangements in dendritic spines of the activated synapse are discussed. Evidence is provided that chronic mechanical stress may result in aberrant activation of mTOR signaling and the WAVE regulatory complex via the YAP/TAZ system, the key sensor of mechanical signals, and influence the associated pathways regulating the formation of F actin filaments and the dendritic spine structure. These consequences may be a risk factor for various neurological conditions, including autistic spectrum disorders and epileptic encephalopathy. In further consideration of the role of the local translation system in the development of neuropsychic and neurodegenerative diseases, an original hypothesis was put forward that one of the possible causes of synaptopathies is impaired proteome stability associated with mTOR hyperactivity and formation of complex dynamic modes of de novo protein synthesis in response to synapse-stimulating factors, including chronic mechanical stress