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Simulink block diagram of the fractional delay Goodwin oscillator. Subsystem 1 containing a block of Transport Delay possesses the time lag effect, while subsystem 2 is a fractional integrator (with order of 0.9) which reflects the fading memory effect. The Step block is used for assigning the initial value in the beginning step, and the Outport block is used for collecting the simulation results.
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In dynamical systems theory, a system which can be described by differential equations is called a continuous dynamical system. In studies on genetic oscillation, most deterministic models at early stage are usually built on ordinary differential equations (ODE). Therefore, gene transcription which is a vital part in genetic oscillation is presuppo...
Citations
... It has been shown that transcription dynamics are slow and epigenetic dynamics such as DNA methylation dynamics are even slower [22,23]. Epigenetic mechanism by chromatin modification and/or DNA methylation may provide cellular biochemical memory by blocking or allowing transcription [24,25]. ...
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.
... It has been shown that transcription dynamics are slow and epigenetic dynamics such as DNA methylation dynamics are even slower [22,23]. Epigenetic mechanism by chromatin modification and/or DNA methylation may provide cellular biochemical memory by blocking or allowing transcription [24,25]. ...
Complex diseases such as cancer are caused by changes in the Gene Regulatory Networks. Systems that model the complex dynamics of these networks along with adapting to real gene expression data are closer to reality and can help understand the creation and treatment of cancer. In this paper, for the first time, modelling of gene regulatory networks is performed using delayed nonlinear variable order fractional systems in the state space by a new tool called GENAVOS. This tool uses gene expression time series data to identify and optimize system parameters. This software has several tools for analyzing system dynamics. The results show that the nonlinear variable order fractional systems have very good flexibility in adapting to real data. We found that regulatory networks in cancer cells actually have a larger delay parameter than in normal cells. It is also possible to create chaos, periodic and quasi-periodic oscillations by changing the delay, degradation and synthesis rates. Our findings indicate a profound effect of time-varying order on these networks, which may be related to a type of cellular memory due to epigenetic and environmental factors. We showed that by changing the delay parameter and the variable order function for a normal cell system, its behavior changes and becomes quite similar to the behavior of a cancer cell. This work also confirms the effective role of the miR-17-92 cluster in the cancer cell cycle. GENAVOS is available at https://github.com/hanif-y/GENAVOS with its user guide and MATLAB codes.
... In the last decades, dynamic systems have been intensively studied the fields of natural science and engineering technology. Especially the fractional-order dynamic systems described by the fractional-order derivative have received widespread concern because they are more accurate expressions of real systems with memory and inherited features where such characteristics are neglected or difficult to express with integer-order systems [1][2][3][4][5][6][7][8]. ...
This paper mainly focuses on the robust synchronization issue for drive-response fractional-order chaotic systems (FOCS) when they have unknown parameters and external disturbances. In order to achieve the goal, the sliding mode control scheme only using output information is designed, and at the same time, the structures of a sliding mode surface and a sliding mode controller are also constructed. A sufficient criterion is presented to ensure the robust synchronization of FOCS according to the stability theory of the fractional calculus and sliding mode control technique. In addition, the result can be applied to identical or non-identical chaotic systems with fractional-order. In the end, we build two practical examples to illustrate the feasibility of our theoretical results.
... Many real-world dynamical systems can be represented by fractional operators [8][9][10][11][12][13][14][15][16][17][18][19][20]. Fractional-order systems can be more suitable than integer-order ones in biological modeling due to the fact that, for instance, memory effects or nonlocal spatial correlations can be taken into account [21][22][23][24][25]. ...
This paper presents an alternative representation of a system of differential equations qualitatively showing the behavior of the biological rhythm of a crayfish during their transition from juvenile to adult stages. The model focuses on the interaction of four cellular oscillators coupled by diffusion of a hormone, a parameter (Formula presented.) is used to simulate the quality of communication among the oscillators, in biological terms, it measures developmental maturity of the crayfish. Since some quorum-sensing mechanism is assumed to be responsible for the synchronization of the biological oscillators, it is natural to investigate the possibility that the underlying diffusion process is not standard, i.e. it may be a so-called anomalous diffusion. In this case, it is well understood that diffusion equations with fractional derivatives describe these processes in a more realistic way. The alternative formulation of these equations contains fractional operators of Liouville–Caputo and Caputo–Fabrizio type. The numerical simulations of the equations reflect synchronization of ultradian rhythms leading to a circadian rhythm. The classical behavior is recovered when the order of the fractional derivative is (Formula presented.). We discuss possible biological implications.
... For example, a fractional derivative viscoelastic model has been proposed to describe the behavior of infant brain tissue under conditions consistent with the development of infant communicating hydrocephalus [25]. In the regulation of gene expression, the fractional-order model was introduced to describe the globally slow dynamics induced via discontinuous transcription [26]. In situations when the dynamics of individual nodes are modeled by FDEs, the networks under consideration become fractional-order complex networks (FCNs). ...
In this work, we propose a novel projective outer synchronization (POS) between unidirectionally coupled uncertain fractional-order complex networks through scalar transmitted signals. Based on the state observer theory, a control law is designed and some criteria are given in terms of linear matrix inequalities which guarantee global robust POS between such networks. Interestingly, in the POS regime, we show that different choices of scaling factor give rise to different outer synchrony, with various special cases including complete outer synchrony, anti-outer synchrony and even a state of amplitude death. Furthermore, it is demonstrated that although stability of POS is irrelevant to the inner-coupling strength, it will affect the convergence speed of POS. In particular, stronger inner synchronization can induce faster POS. The effectiveness of our method is revealed by numerical simulations on fractional-order complex networks with small-world communication topology.
... Note that even if an ODE describes an instantaneous process, the notion of instant depends on the time scale considered, whereas FDEs have the property of fading memory and depend on the range of α (0 < α < 1). Such memories can describe current events with the collective information from preceding events, while events in the far past can often be neglected compared to contributions from the near past [60]. Volterra defined the notion of fading memory as ''the principle of dissipation of hereditary action'' [61]. ...
This paper deals with pinning synchronization problem of fractional-order complex networks with Lipschitz-type nonlinear nodes and directed communication topology. We first reformulate the problem as a global asymptotic stability problem by describing network evolution in terms of error dynamics. Then, a novel frequency domain approach is developed by using Laplace transform, algebraic graph theory and generalized Gronwall inequality. We show that pinning synchronization can be ensured if the extended network topology contains a spanning tree and the coupling strength is large enough. Furthermore, we provide an easily testable criterion for global pinning synchronization depending on fractional-order, network topology, oscillator dynamics and state feedback. Numerical simulations are provided to illustrate the effectiveness of the theoretical analysis.
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