Periodic, Quasi-periodic and Chaotic Dynamics in Simple Gene Elements with Time Delays

Stability, bifurcation, and chaos in a class of scalar quartic polynomial delay systems

In this paper, a class of scalar quartic polynomial delay systems is investigated. We found rich dynamics in this system through numerical simulation, including chaotic attractors, chaotic saddles, and intermittent chaos. Moreover, this chaotic quartic system may serve as an approximation, through Taylor expansion, for a wide class of scalar delay differential equations. Thus, these nonlinear systems may exhibit chaotic behaviors, and the studies in our paper may provide an insight into the emergence of chaos in other time-delay nonlinear systems. We also conduct a detailed theoretical analysis of the system, including the stability of equilibria and Hopf bifurcation analysis based on the theory of normal form and center manifold. Additionally, a numerical analysis is provided to give numerical evidence for the existence of chaos.

Chaos in gene regulatory networks: Effects of time delays and interaction structure

In biological system models, gene expression levels are typically described by regulatory feedback mechanisms. Many studies of gene network models focus on dynamical interactions between components, but often overlook time delays. Here we present an extended model for gene regulatory networks with time delayed negative feedback, which is described by delay differential equations. We analyze nonlinear properties of the model in terms of chaos and compare the conditions with the benchmark homogeneous gene regulatory network model. Chaotic dynamics depend strongly on the inclusion of time delays, but the minimum motifs that show chaos differ when both original and extended models are considered. Our results suggest that, for a particular higher order extension of the gene network, it is possible to observe chaotic dynamics in a two-gene system without adding any self-inhibition. This finding can be explained as a result of modification of the original benchmark model induced by previously unmodeled dynamics. We argue that the inclusion of additional parameters in regulatory gene circuit models substantially enhances the likelihood of observing non-periodic dynamics.

Quantifying Mutational Response to Track the Evolution of SARS-CoV-2 Spike Variants: Introducing a Statistical-Mechanics-Guided Machine Learning Method

The emergence of SARS-CoV-2 and its variants that critically affect global public health requires characterization of mutations and their evolutionary pattern from specific Variants of Interest (VOIs) to Variants of Concern (VOCs). Leveraging the concept of equilibrium statistical mechanics, we introduce a new responsive quantity defined as "Mutational Response Function (MRF)" aptly quantifying domain-wise average entropy-fluctuation in the spike glycoprotein sequence of SARS-CoV-2 based on its evolutionary database. As the evolution transits from a specific variant to VOC, we find that the evolutionary crossover is accompanied by a dramatic change in MRF, upholding the characteristic of a dynamic phase transition. With this entropic information, we have developed an ancestral-based machine learning method that helps predict future domain-specific mutations. The feedforward binary classification model pinpoints possible residues prone to future mutations that have implications for enhanced fusogenicity and pathogenicity of the virus. We believe such MRF analyses followed by a statistical mechanics augmented ML approach could help track different evolutionary stages of such species and identify a critical evolutionary transition that is alarming.

Cancer and Chaos and the Complex Network Model of a Multicellular Organism

In the search of theoretical models describing cancer, one of promising directions is chaos. It is connected to ideas of "genome chaos" and "life on the edge of chaos", but they profoundly differ in the meaning of the term "chaos". To build any coherent models, notions used by both ideas should be firstly brought closer. The hypothesis "life on the edge of chaos" using deterministic chaos has been radically deepened developed in recent years by the discovery of half-chaos. This new view requires a deeper interpretation within the range of the cell and the organism. It has impacts on understanding "chaos" in the term "genome chaos". This study intends to present such an interpretation on the basis of which such searches will be easier and closer to intuition. We interpret genome chaos as deterministic chaos in a large module of half-chaotic network modeling the cell. We observed such chaotic modules in simulations of evolution controlled by weaker variant of natural selection. We also discuss differences between free and somatic cells in modeling their disturbance using half-chaotic networks.

Stability and Bifurcation Analysis of a Diffusive miR-9/Hes1 Network With Time Delay

In this paper, a model of miR-9/Hes1 interaction network involving one time delay and diffusion effect under the Neumann boundary conditions is studied. First of all, the stability of the positive equilibrium and the existence of local Hopf bifurcation and Turing-Hopf bifurcation are investigated by analyzing the associated characteristic equation. Second, a algorithm for determining the direction, stability and period of the corresponding bifurcating periodic solutions is presented. The obtained results suggest that the quiescent progenitors (high steady-state Hes1) can be easily excited into oscillation by time delay whereas the differentiated state (low steady-state Hes1) is basically unaffected, and the integrated effect of delay and diffusion can induce the occurrence of spatially inhomogeneous patterns. More importantly, spatially homogeneous/inhomogeneous periodic solutions can exist simultaneously when the diffusion coefficients of Hes1 mRNA and Hes1 protein are appropriately small, conversely, there is only spatially homogeneous periodic solutions. Intriguingly, both temporal patterns and spatial-temporal patterns show that time delay can prompt Hes1 protein to shift from the high concentration state to the low concentration one ("ON" → "OFF"), where Hes1 protein shows low level whereas miR-9 shows high level. Finally, some numerical examples are presented to verify and visualize theoretical results.

The onset of dissipative chaos driven by nonequilibrium conditions

Dissipative chaos appears widely in various nonequilibrium systems; however, it is not clear how dissipative chaos originates from nonequilibrium. We discuss a framework based on the potential-flux approach to study chaos from the perspective of nonequilibrium dynamics. In this framework, chaotic systems possess a wide basin on the potential landscape, in which the rotational flux dominates the system dynamics, and chaos occurs with the appearance of this basin. In contrast, the probability flux is particularly associated with the detailed balance-breaking in nonequilibrium systems. This implies that the appearance of dissipative chaos is driven by nonequilibrium conditions.

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.

Developing an Embedding, Koopman and Autoencoder Technologies-Based Multi-Omics Time Series Predictive Model (EKATP) for Systems Biology research

It is very important for systems biologists to predict the state of the multi-omics time series for disease occurrence and health detection. However, it is difficult to make the prediction due to the high-dimensional, nonlinear and noisy characteristics of the multi-omics time series data. For this reason, this study innovatively proposes an Embedding, Koopman and Autoencoder technologies-based multi-omics time series predictive model (EKATP) to predict the future state of a high-dimensional nonlinear multi-omics time series. We evaluate this EKATP by using a genomics time series with chaotic behavior, a proteomics time series with oscillating behavior and a metabolomics time series with flow behavior. The computational experiments demonstrate that our proposed EKATP can substantially improve the accuracy, robustness and generalizability to predict the future state of a time series for multi-omics data.

Nonlinear delay differential equations and their application to modeling biological network motifs

Biological regulatory systems, such as cell signaling networks, nervous systems and ecological webs, consist of complex dynamical interactions among many components. Network motif models focus on small sub-networks to provide quantitative insight into overall behavior. However, such models often overlook time delays either inherent to biological processes or associated with multi-step interactions. Here we systematically examine explicit-delay versions of the most common network motifs via delay differential equation (DDE) models, both analytically and numerically. We find many broadly applicable results, including parameter reduction versus canonical ordinary differential equation (ODE) models, analytical relations for converting between ODE and DDE models, criteria for when delays may be ignored, a complete phase space for autoregulation, universal behaviors of feedforward loops, a unified Hill-function logic framework, and conditions for oscillations and chaos. We conclude that explicit-delay modeling simplifies the phenomenology of many biological networks and may aid in discovering new functional motifs.

Network motif models focus on small sub-networks in biological systems to quantitatively describe overall behavior but they often overlook time delays. Here, the authors systematically examine the most common network motifs via delay differential equations (DDE), often leading to more concise descriptions.

GENAVOS: A New Tool for Modelling and Analyzing Cancer Gene Regulatory Networks Using Delayed Nonlinear Variable Order Fractional System

Gene regulatory networks (GRN) are one of the etiologies associated with cancer. Their dysregulation can be associated with cancer formation and asymmetric cellular functions in cancer stem cells, leading to disease persistence and resistance to treatment. Systems that model the complex dynamics of these networks along with adapting to partially known real omics data are closer to reality and may be useful to understand the mechanisms underlying neoplastic phenomena. In this paper, for the first time, modelling of GRNs is performed using delayed nonlinear variable order fractional (VOF) systems in the state space by a new tool called GENAVOS. Although the tool uses gene expression time series data to identify and optimize system parameters, it also models possible epigenetic signals, and the results show that the nonlinear VOF systems have very good flexibility in adapting to real data. We found that GRNs in cancer cells actually have a larger delay parameter than in normal cells. It is also possible to create weak chaotic, periodic, and quasi-periodic oscillations by changing the parameters. Chaos can be associated with the onset of cancer. Our findings indicate a profound effect of time-varying orders on these networks, which may be related to a type of cellular epigenetic memory. By changing the delay parameter and the variable order functions (possible epigenetics signals) for a normal cell system, its behaviour becomes quite similar to the behaviour of a cancer cell. This work confirms the effective role of the miR-17-92 cluster as an epigenetic factor in the cancer cell cycle.

Toward New Modalities in VEP-Based BCI Applications Using Dynamical Stimuli: Introducing Quasi-Periodic and Chaotic VEP-Based BCI

Visual evoked potentials (VEPs) to periodic stimuli are commonly used in brain computer interfaces for their favorable properties such as high target identification accuracy, less training time, and low surrounding target interference. Conventional periodic stimuli can lead to subjective visual fatigue due to continuous and high contrast stimulation. In this study, we compared quasi-periodic and chaotic complex stimuli to common periodic stimuli for use with VEP-based brain computer interfaces (BCIs). Canonical correlation analysis (CCA) and coherence methods were used to evaluate the performance of the three stimulus groups. Subjective fatigue caused by the presented stimuli was evaluated by the Visual Analogue Scale (VAS). Using CCA with the M2 template approach, target identification accuracy was highest for the chaotic stimuli (M = 86.8, SE = 1.8) compared to the quasi-periodic (M = 78.1, SE = 2.6, p = 0.008) and periodic (M = 64.3, SE = 1.9, p = 0.0001) stimulus groups. The evaluation of fatigue rates revealed that the chaotic stimuli caused less fatigue compared to the quasi-periodic (p = 0.001) and periodic (p = 0.0001) stimulus groups. In addition, the quasi-periodic stimuli led to lower fatigue rates compared to the periodic stimuli (p = 0.011). We conclude that the target identification results were better for the chaotic group compared to the other two stimulus groups with CCA. In addition, the chaotic stimuli led to a less subjective visual fatigue compared to the periodic and quasi-periodic stimuli and can be suitable for designing new comfortable VEP-based BCIs.

Strong-chaos-caused negative mobility in a periodic substrate potential

We numerically investigate negative mobility of an inertial Brownian particle moving in a periodic double-well substrate potential in the presence of a time-periodic force and a constant bias. For the deterministic case, we find from the average velocity that the varying shape parameter and driving forces can cause negative mobility, differential negative mobility, and giant positive mobility. We analyze these findings via the bifurcation diagram and maximal Lyapunov exponent and find that certain chaos can give rise to negative mobility. For the presence of a Gaussian color noise, the results suggest that the noise intensity can enhance or result in negative and positive mobilities, whereas correlation time can enhance, weaken, or even eliminate them. On the basis of the time series, phase-space map, and power spectrum of various attractors, we unveil how these mobilities connect to strong chaotic attractors (SCAs), including both stable attractor and unstable attractors, and propose an underlying mechanism that SCAs can result in the negative mobility, whereas other attractors do not. Our findings may be potentially useful for research on anomalous transports of the particles and on designs of various devices, such as atomic chains, crystals with dislocations, and superconducting nanowires, etc.

On the dynamical aspects of local translation at the activated synapse

Background

The key role in the dynamic regulation of synaptic protein turnover belongs to the Fragile X Mental Retardation Protein, which regulates the efficiency of dendritic mRNA translation in response to stimulation of metabotropic glutamate receptors at excitatory synapses of the hippocampal pyramidal cells. Its activity is regulated via positive and negative regulatory loops that function in different time ranges, which is an absolute factor for the formation of chaotic regimes that lead to disrupted proteome stability. The indicated condition may cause a number of neuropsychiatric diseases, including autism and epilepsy. The present study is devoted to a theoretical analysis of the local translation system dynamic properties and identification of parameters affecting the chaotic potential of the system.

Results

A mathematical model that describes the maintenance of a specific pool of active receptors on the postsynaptic membrane via two mechanisms – de novo synthesis of receptor proteins and restoration of protein function during the recycling process – has been developed. Analysis of the model revealed that an increase in the values of the parameters describing the impact of protein recycling on the maintenance of a pool of active receptors in the membrane, duration of the signal transduction via the mammalian target of rapamycin pathway, influence of receptors on the translation activation, as well as reduction of the rate of synthesis and integration of de novo synthesized proteins into the postsynaptic membrane – contribute to the reduced complexity of the local translation system dynamic state. Formation of these patterns significantly depends on the complexity and non-linearity of the mechanisms of exposure of de novo synthesized receptors to the postsynaptic membrane, the correct evaluation of which is currently problematic.

Conclusions

The model predicts that an increase of “receptor recycling” and reduction of the rate of synthesis and integration of de novo synthesized proteins into the postsynaptic membrane contribute to the reduced complexity of the local translation system dynamic state. Herewith, stable stationary states occur much less frequently than cyclic states. It is possible that cyclical nature of functioning of the local translation system is its “normal” dynamic state.

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

BACKGROUND

The regulatory feedback loops that present in structural and functional organization of molecular-genetic systems and the phenomenon of the regulatory signal delay, a time period between the moment of signal reception and its implementation, provide natural conditions for complicated dynamic regimes in these systems. The delay phenomenon at the intracellular level is a consequence of the matrix principle of data transmission, implemented through the rather complex processes of transcription and translation.However, the rules of the influence of system structure on system dynamics are not clearly understood. Knowledge of these rules is particularly important for construction of synthetic gene networks with predetermined properties.

RESULTS

We study dynamical properties of models of simplest circular gene networks regulated by negative feedback mechanisms. We have shown existence and stability of oscillating trajectories (cycles) in these models. Two algorithms of construction and localization of these cycles have been proposed. For one of these models, we have solved an inverse problem of parameters identification.

CONCLUSIONS

The modeling results demonstrate that non-stationary dynamics in the models of circular gene networks with negative feedback loops is achieved by a high degree of non-linearity of the mechanism of the autorepressor influence on its own expression, by the presence of regulatory signal delay, the value of which must exceed a certain critical value, and transcription/translation should be initiated from a sufficiently strong promoter/Shine-Dalgarno site. We believe that the identified patterns are key elements of the oscillating construction design.

Chaotic vibration of microtubules and biological information processing

A new nonlinear phenomenon has been studied theoretically on one of the main cytoskeletal element of eukaryotic cells, namely chaos in microtubules vibrations. The general model of microtubules is used to draw phase portraits and Lyapunov spectra. The examination of numerical results reveals that the velocity of the chaotic wave could be the physical parameter that governs chaos. The energy released after the hydrolysation of guanosine triphosphate is converted to active turbulence leading to chaos. The high values of the Lyapunov exponents give hints that there are strong dissipations yielding in the lessening of the velocity of chaotic wave propagation in the microtubules. Moreover, the role of chaos in information processing has been established in microtubules. The energy coming from hydrolysis of guanosine triphosphate stimulates the tubulin leading it to probe its environment and collect information. The net sum of Lyapunov exponents is found to be positive in this stage of the process. Also, the collected information is compressed with a negative sum of Lyapunov exponents. Eventually, the compressibility rate has been estimated to be η=67.2%, and 1.11 bit is lost.

Genotype Components as Predictors of Phenotype in Model Gene Regulatory Networks

Models of gene regulatory networks (GRN) have proven useful for understanding many aspects of the highly complex behavior of biological control networks. Randomly generated non-Boolean networks were used in experimental simulations to generate data on dynamic phenotypes as a function of several genotypic parameters. We found that predictive relationships between some phenotypes and quantitative genotypic parameters such as number of network genes, interaction density, and initial condition could be derived depending on the strength of the topological (positional) genotype on specific phenotypes. We quantitated the strength of the topological genotype effect (TGE) on a number of phenotypes in multi-gene networks. For phenotypes with a low influence of topological genotype, derived and empirical relationships using quantitative genotype parameters were accurate in phenotypic outcomes. We found a number of dynamic network properties, including oscillation behaviors, that were largely dependent on genotype topology, and for which no such general quantitative relationships were determinable. It remains to be determined if these results are applicable to biological gene regulatory networks.

Capturing the Forest but Missing the Trees: Microstates Inadequate for Characterizing Shorter-Scale EEG Dynamics

The brain is known to be active even when not performing any overt cognitive tasks, and often it engages in involuntary mind wandering. This resting state has been extensively characterized in terms of fMRI-derived brain networks. However, an alternate method has recently gained popularity: EEG microstate analysis. Proponents of microstates postulate that the brain discontinuously switches between four quasi-stable states defined by specific EEG scalp topologies at peaks in the global field potential (GFP). These microstates are thought to be “atoms of thought,” involved with visual, auditory, salience, and attention processing. However, this method makes some major assumptions by excluding EEG data outside the GFP peaks and then clustering the EEG scalp topologies at the GFP peaks, assuming that only one microstate is active at any given time. This study explores the evidence surrounding these assumptions by studying the temporal dynamics of microstates and its clustering space using tools from dynamical systems analysis, fractal, and chaos theory to highlight the shortcomings in microstate analysis. The results show evidence of complex and chaotic EEG dynamics outside the GFP peaks, which is being missed by microstate analysis. Furthermore, the winner-takes-all approach of only one microstate being active at a time is found to be inadequate since the dynamic EEG scalp topology does not always resemble that of the assigned microstate, and there is competition among the different microstate classes. Finally, clustering space analysis shows that the four microstates do not cluster into four distinct and separable clusters. Taken collectively, these results show that the discontinuous description of EEG microstates is inadequate when looking at nonstationary short-scale EEG dynamics.

Multi-fidelity modeling in sequential design for stability identification in dynamic time-delay systems

Time delay arises in a variety of real-world complex systems. A high-fidelity simulation generally renders high accuracy to simulate the dynamic evolution of such complex systems and appraise quantity of interest for process design and response optimization. Identification of limit states exemplifies such applications, which outlines the boundary that separates distinct regions (e.g., stability region) in parameter space. While design of experiments is the common procedure to evaluate decision functions to sketch the boundary, it crucially relies on the quantity and quality of sampling points. This has made it infeasible to explore a large parameter design space with expensive-to-evaluate high-fidelity simulations. Furthermore, the complex contour of stability region in time-delay systems nullifies most existing sequential design paradigms, including adaptive classification approaches. On the other hand, low-fidelity surrogate modeling efficiently emulates a high-fidelity simulation, albeit at the expense of accuracy, not ideal to inspect the system behavior near the critical boundary. In this study, we investigate a multifidelity approach to delineate the stability region in a sequential fashion: sampling points are first evaluated by the low-fidelity surrogate modeling, and only those selected according to the exploration-exploitation trade-off principle are then assessed by a high-fidelity simulation to approximate the stability boundary. The application in a numerical case study of the delayed Mathieu equation as well as a real-world machining process corroborates the proposed approach.

On developing a framework for detection of oscillations in data

Oscillation is a phenomenon very commonly observed in systems, ranging from simple ones to complex distributed network. Several techniques have been proposed in the literature for detecting oscillations to study their importance in domains ranging from physiology to climate studies. However, there is a lack of a common framework accommodative of important features of data such as non-stationarity, intermittent oscillations, measurement noise, multimodal oscillations, and the like. In this article, we outline a framework that addresses these challenges, the results of which can then be analyzed along with appropriate knowledge about the underlying system. We present results of an extensive simulation study that establishes the robustness and reliability of the proposed technique and demonstrate its applicability to real datasets in climate and in industrial datasets.

Bifurcation delay in a network of locally coupled slow-fast systems

We study the evolution of bifurcation delay in a network of locally coupled slow-fast systems. Our study reveals that a tiny perturbation even in a single node causes asymmetry in bifurcation delay. We investigate the evolution of bifurcation delay as a function of various parameters, such as feedback coupling strength, amplitude of external force, frequency of external force, and delay coupling strength. We show that a traveling wave is generated as the result of introducing local parameter mismatch, and the bifurcation delay shows a dip in the spatial profile. We believe that these spatiotemporal patterns in bifurcation delay shed light on the dynamics of neuronal networks.

Bistable emergence of oscillations in growing Bacillus subtilis biofilms

Biofilm communities of Bacillus subtilis bacteria have recently been shown to exhibit collective growth-rate oscillations mediated by electrochemical signaling to cope with nutrient starvation. These oscillations emerge once the colony reaches a large enough number of cells. However, it remains unclear whether the amplitude of the oscillations, and thus their effectiveness, builds up over time gradually or if they can emerge instantly with a nonzero amplitude. Here we address this question by combining microfluidics-based time-lapse microscopy experiments with a minimal theoretical description of the system in the form of a delay-differential equation model. Analytical and numerical methods reveal that oscillations arise through a subcritical Hopf bifurcation, which enables instant high-amplitude oscillations. Consequently, the model predicts a bistable regime where an oscillating and a nonoscillating attractor coexist in phase space. We experimentally validate this prediction by showing that oscillations can be triggered by perturbing the media conditions, provided the biofilm size lies within an appropriate range. The model also predicts that the minimum size at which oscillations start decreases with stress, a fact that we also verify experimentally. Taken together, our results show that collective oscillations in cell populations can emerge suddenly with nonzero amplitude via a discontinuous transition.

Critical dynamics of Hopf bifurcations in the corticothalamic system: Transitions from normal arousal states to epileptic seizures

A physiologically based corticothalamic model of large-scale brain activity is used to analyze critical dynamics of transitions from normal arousal states to epileptic seizures, which correspond to Hopf bifurcations. This relates an abstract normal form quantitatively to underlying physiology that includes neural dynamics, axonal propagation, and time delays. Thus, a bridge is constructed that enables normal forms to be used to interpret quantitative data. The normal form of the Hopf bifurcations with delays is derived using Hale's theory, the center manifold theorem, and normal form analysis, and it is found to be explicitly expressed in terms of transfer functions and the sensitivity matrix of a reduced open-loop system. It can be applied to understand the effect of each physiological parameter on the critical dynamics and determine whether the Hopf bifurcation is supercritical or subcritical in instabilities that lead to absence and tonic-clonic seizures. Furthermore, the effects of thalamic and cortical nonlinearities on the bifurcation type are investigated, with implications for the roles of underlying physiology. The theoretical predictions about the bifurcation type and the onset dynamics are confirmed by numerical simulations and provide physiologically based criteria for determining bifurcation types from first principles. The results are consistent with experimental data from previous studies, imply that new regimes of seizure transitions may exist in clinical settings, and provide a simplified basis for control-systems interventions. Using the normal form, and the full equations from which it is derived, more complex dynamics, such as quasiperiodic cycles and saddle cycles, are discovered near the critical points of the subcritical Hopf bifurcations.

Chaos and Hyperchaos in a Model of Ribosome Autocatalytic Synthesis

Any vital activities of the cell are based on the ribosomes, which not only provide the basic machinery for the synthesis of all proteins necessary for cell functioning during growth and division, but for biogenesis itself. From this point of view, ribosomes are self-replicating and autocatalytic structures. In current work we present an elementary model in which the autocatalytic synthesis of ribosomal RNA and proteins, as well as enzymes ensuring their degradation are described with two monotonically increasing functions. For certain parameter values, the model, consisting of one differential equation with delayed argument, demonstrates both stationary and oscillatory dynamics of the ribosomal protein synthesis, which can be chaotic and hyperchaotic dependent on the value of the delayed argument. The biological interpretation of the modeling results and parameter estimation suggest the feasibility of chaotic dynamics in molecular genetic systems of eukaryotes, which depends only on the internal characteristics of functioning of the translation system.

Chaos and hyperchaos in simple gene network with negative feedback and time delays

Today there are examples that prove the existence of chaotic dynamics at all levels of organization of living systems, except intracellular, although such a possibility has been theoretically predicted. The lack of experimental evidence of chaos generation at the intracellular level in vivo may indicate that during evolution the cell got rid of chaos. This work allows the hypothesis that one of the possible mechanisms for avoiding chaos in gene networks can be a negative evolutionary selection, which prevents fixation or realization of regulatory circuits, creating too mild, from the biological point of view, conditions for the emergence of chaos. It has been shown that one of such circuits may be a combination of negative autoregulation of expression of transcription factors at the level of their synthesis and degradation. The presence of such a circuit results in formation of multiple branches of chaotic solutions as well as formation of hyperchaos with equal and sufficiently low values of the delayed argument that can be implemented not only in eukaryotic, but in prokaryotic cells.

Impact of time delays on oscillatory dynamics of interlinked positive and negative feedback loops

Interlinking a positive feedback loop (PFL) with a negative feedback loop (NFL) constitutes a typical motif in genetic networks, performing various functions in cell signaling. How time delay in feedback regulation affects the dynamics of such systems still remains unclear. Here, we investigate three systems of interlinked PFL and NFL with time delays: a synthetic genetic oscillator, a three-node circuit, and a simplified single-node model. The stability of steady states and the routes to oscillation in the single-node model are analyzed in detail. The amplitude and period of oscillations vary with a pointwise periodicity over a range of time delay. Larger-amplitude oscillations can be induced when the PFL has an appropriately long delay, in comparison with the PFL with no delay or short delay; this conclusion holds true for all the three systems. We unravel the underlying mechanism for the above effects via analytical derivation under a limiting condition. We also develop a stochastic algorithm for simulating a single reaction with two delays and show that robust oscillations can be maintained by the PFL with a properly long delay in the single-node system. This work presents an effective method for constructing robust large-amplitude oscillators and interprets why similar circuit architectures are engaged in timekeeping systems such as circadian clocks.