Limit cycles in models of circular gene networks regulated by negative feedback loops

Impact of Negative Feedbacks on De Novo Pyrimidines Biosynthesis in Escherichia coli

Earlier studies aimed at investigating the metabolism of endogenous nucleoside triphosphates in synchronous cultures of E. coli cells revealed an auto-oscillatory mode of functioning of the pyrimidine and purine nucleotide biosynthesis system, which the authors associated with the dynamics of cell division. Theoretically, this system has an intrinsic oscillatory potential, since the dynamics of its functioning are controlled through feedback mechanisms. The question of whether the nucleotide biosynthesis system has its own oscillatory circuit is still open. To address this issue, an integral mathematical model of pyrimidine biosynthesis was developed, taking into account all experimentally verified negative feedback in the regulation of enzymatic reactions, the data of which were obtained under in vitro conditions. Analysis of the dynamic modes of the model functioning has shown that in the pyrimidine biosynthesis system, both the steady-state and oscillatory functioning modes can be realized under certain sets of kinetic parameters that fit in the physiological boundaries of the investigated metabolic system. It has been demonstrated that the occurrence of the oscillatory nature of metabolite synthesis depended on the ratio of two parameters: the Hill coefficient, h_UMP1-the nonlinearity of the UMP effect on the activity of carbamoyl-phosphate synthetase, and the parameter r characterizing the contribution of the noncompetitive mechanism of UTP inhibition to the regulation of the enzymatic reaction of UMP phosphorylation. Thus, it has been theoretically shown that the E. coli pyrimidine biosynthesis system possesses its own oscillatory circuit whose oscillatory potential depends to a significant degree on the mechanism of regulation of UMP kinase activity.

Stratifications and foliations in phase portraits of gene network models

Periodic processes of gene network functioning are described with good precision by periodic trajectories (limit cycles) of multidimensional systems of kinetic-type differential equations. In the literature, such systems are often called dynamical, they are composed according to schemes of positive and negative feedback between components of these networks. The variables in these equations describe concentrations of these components as functions of time. In the preparation of numerical experiments with such mathematical models, it is useful to start with studies of qualitative behavior of ensembles of trajectories of the corresponding dynamical systems, in particular, to estimate the highest likelihood domain of the initial data, to solve inverse problems of parameter identification, to list the equilibrium points and their characteristics, to localize cycles in the phase portraits, to construct stratification of the phase portraits to subdomains with different qualities of trajectory behavior, etc. Such an à priori geometric analysis of the dynamical systems is quite analogous to the basic section "Investigation of functions and plot of their graphs" of Calculus, where the methods of qualitative studies of shapes of curves determined by equations are exposed. In the present paper, we construct ensembles of trajectories in phase portraits of some dynamical systems. These ensembles are 2-dimensional surfaces invariant with respect to shifts along the trajectories. This is analogous to classical construction in analytic mechanics, i. e. the level surfaces of motion integrals (energy, kinetic moment, etc.). Such surfaces compose foliations in phase portraits of dynamical systems of Hamiltonian mechanics. In contrast with this classical mechanical case, the foliations considered in this paper have singularities: all their leaves have a non-empty intersection, they contain limit cycles on their boundaries. Description of the phase portraits of these systems at the level of their stratifications, and that of ensembles of trajectories allows one to construct more realistic gene network models on the basis of methods of statistical physics and the theory of stochastic differential equations.

The finding and researching algorithm for potentially oscillating enzymatic systems

Many processes in living organisms are subject to periodic oscillations at different hierarchical levels of their organization: from molecular-genetic to population and ecological. Oscillatory processes are responsible for cell cycles in both prokaryotes and eukaryotes, for circadian rhythms, for synchronous coupling of respiration with cardiac contractions, etc. Fluctuations in the numbers of organisms in natural populations can be caused by the populations' own properties, their age structure, and ecological relationships with other species. Along with experimental approaches, mathematical and computer modeling is widely used to study oscillating biological systems. This paper presents classical mathematical models that describe oscillatory behavior in biological systems. Methods for the search for oscillatory molecular-genetic systems are presented by the example of their special case - oscillatory enzymatic systems. Factors influencing the cyclic dynamics in living systems, typical not only of the molecular-genetic level, but of higher levels of organization as well, are considered. Application of different ways to describe gene networks for modeling oscillatory molecular-genetic systems is considered, where the most important factor for the emergence of cyclic behavior is the presence of feedback. Techniques for finding potentially oscillatory enzymatic systems are presented. Using the method described in the article, we present and analyze, in a step-by-step manner, first the structural models (graphs) of gene networks and then the reconstruction of the mathematical models and computational experiments with them. Structural models are ideally suited for the tasks of an automatic search for potential oscillating contours (linked subgraphs), whose structure can correspond to the mathematical model of the molecular-genetic system that demonstrates oscillatory behavior in dynamics. At the same time, it is the numerical study of mathematical models for the selected contours that makes it possible to confirm the presence of stable limit cycles in them. As an example of application of the technology, a network of 300 metabolic reactions of the bacterium Escherichia coli was analyzed using mathematical and computer modeling tools. In particular, oscillatory behavior was shown for a loop whose reactions are part of the tryptophan biosynthesis pathway.

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.

Evolution and extinction can occur rapidly: a modeling approach

Fossil record of Earth describing the last 500 million years is characterized by evolution discontinuity as well as recurring global extinctions of some species and their replacement by new types, the causes of which are still debate. We developed a model of evolutionary self-development of a large ecosystem. This model of biota evolution based on the universal laws of living systems functioning: reproduction, dependence of reproduction efficiency and mortality on biota density, mutational variability in the process of reproduction and selection of the most adapted individuals. We have shown that global extinctions and phases of rapid growth and biodiversity stasis can be a reflection of the emergence of bistability in a self-organizing system, which is the Earth’s biota. Bistability was found to be characteristic only for ecosystems with predominant sexual reproduction. The reason for the transition from one state to another is the selection of the most adapted individuals. That is, we explain the characteristics of the Earth’s fossil record during the last 500 million years by the internal laws of Earth’s ecosystem functioning, which appeared at a certain stage of evolution as a result of the emergence of life forms with an increased adaptive diversification associated with sexual dimorphism.