Many biochemically dynamical systems are controlled by intrinsic rhythms generated by specialized cellular clocks within the organism itself. These rhythm generators are composed of thousands of clock cells that are intrinsically diverse, but nevertheless manage to function in a coherent oscillatory state [1–6]. However, the synchronization mechanisms by which this collective behavior arises remains to be understood, even if individual clock cells are known to operate through biochemical networks comprising of multiple regulatory feedback loops [7–11]. The complexity of these cellular systems has hindered a complete understanding of natural genetic oscillators and their synchronization [12–16].
Recently designed synthetic genetic oscillators can offer an alternative approach, and provide a relatively well-controlled test bed in which the function and behavior of natural genetic oscillators can be isolated and characterized in detail [1, 3, 7, 8, 17]. As an example, a synthetic biological oscillator, termed the “repressilator,” was developed in Escherichia coli from a network of three transcriptional repressors that inhibit one another in a cyclic way [18–21]. Spontaneous oscillations were observed in individual cells within a growing culture, although substantial variability and noise were present among the different cells. Another synthetic oscillator was designed and built that exhibited damped oscillatory responses to perturbations in culture . Recently, several mechanisms of intercell coupling of synthetic genetic oscillators have been discussed, to enhance the oscillating response of the synthetic biological system . In general, coupling among oscillators is not sufficient to achieve synchronization, and many ensembles of coupled oscillators exhibit phase dispersion rather than a synchronized state, because the oscillators may actively resist oscillation or because the coupling is too small. Therefore, the synchronization of a population of nonlinear stochastic coupled oscillators must be analyzed carefully.
In previous studies [1, 12–16, 23–26], the collective behavior of synthetic genetic oscillators was discussed on the basis of cell-to-cell communication through quorum sensing, which may lead to synchronization in an ensemble of identical genetic oscillators. In general, intercellular communication is accomplished by transmitting individual cell reactions via intercellular signals to neighboring cells, which can generate a global cellular synchronization at the level of molecules, tissues, organs, or the body . The ability to communicate among cells is an absolute requisite to ensure appropriate and robust synchronization at all levels in organisms living in uncertain environments. Synchronization of coupled networks has been investigated intensively in past decades because of its biological implications and potential applications [3, 4, 20]. The global synchronization mechanism of oscillators via direct coupling has been derived based on the Lyapunov method and linearization technique [27–29]. Different synchronization mechanisms in a population of nonlinear stochastic genetic oscillators with noise and impulse control inputs have also been previously discussed [7–9].
Generally, biological systems or organisms are subject to time-varying uncertainties, assumed to be in the form of both internal noise resulting from the birth and death of biochemical molecules, and external noise resulting from environmental perturbations [15, 19, 30–34]. It has been shown that environmental molecular noise plays an important role in the stochastic behavioral phenomena of biological systems at various levels. In particular, gene regulation is an inherently noisy process [35–37], from transcriptional control, alternative splicing, translation, and diffusion to biological modification of transcription factors. Such stochastic noise cannot only significantly affect the dynamics of biological systems, but also may be exploited by living organisms to actively facilitate certain cellular functions, such as cellular communication and synchronization. In this study, the time-varying parameter fluctuations in gene regulation processes are modeled as stochastic intrinsic kinetic noise, whereas environmental molecular noise of the host cells is modeled as an extrinsic disturbance of synthetic genetic oscillators. A population of synthetic oscillators coupled by quorum sensing is modeled by a set of coupled nonlinear stochastic equations with intrinsic and extrinsic noise. Our main purpose is to discuss the robust synchronization mechanism of a population of nonlinear stochastic coupled synthetic genetic oscillators under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise on the host cells.
Based on nonlinear stochastic equations of coupled synthetic genetic oscillators distributed in different host cells, the robust synchronization mechanism is discussed from the H
noise-filtering perspective. The robust ability to tolerate stochastic kinetic parameter fluctuations and the filtering ability to attenuate extrinsic environmental molecular noise to maintain synchronization of nonlinear stochastic coupled synthetic genetic oscillations was measured from the nonlinear stochastic system theory point of view. In the case where robust synchronization cannot be maintained or is corrupted by intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise, some robust synchronization design methods are discussed to enhance synchronization. Both the physical insight into the robust synchronization mechanisms and the designs to improve these mechanisms require solving nonlinear HJIs, which cannot be easily achieved by any analytical or numerical method at present. In order to simplify this analysis and design, the Takagi-Sugeno (T-S) fuzzy model [38–40] is employed to interpolate several local linear stochastic genetic oscillators to approximate the nonlinear stochastic genetic oscillators. Consequently, the analysis and design of robust synchronization can be achieved by solving of a set of LMIs , via the help of the LMI toolbox in MATLAB.
If the robust synchronization of nonlinear stochastic coupled synthetic genetic oscillators could not be achieved spontaneously, an external control input was developed to synchronize the coupled synthetic genetic oscillators. External stimulation inputs are known to play an important role in the synchronization of biological rhythms. For instance, many organisms display a circadian rhythm of 24-hours periodicity entrained to the light–dark cycle . Other examples include physiological rhythms stimulated by regular or periodic inputs occurring in the context of medical devices, synchronization of electronic genetic networks by an external forcing, i.e., external voltage , and a wide variety of regular and irregular rhythms induced by periodic stimulation of squid giant axons . In general, physiological oscillations can be synchronized by appropriate external or internal stimuli. Recently, Wang et al. constructed an impulse control system to model the process of periodical injection of coupling substances with constant or random impulse control into a common extracellular medium and studied its effect on the dynamics of collective rhythms . Synchronization with time delays via direct coupling and linearization methods has also been discussed [29, 43]. In our study, a control input design method was developed to guarantee the robust synchronization of synthetic genetic oscillators under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. In order to avoid solving the complicated HJI, the fuzzy interpolation method was also employed to simplify the control design procedure by only solving a set of simple LMIs .
The contributions of this paper are fourfold: (1) A nonlinear stochastic system is introduced to model a population of nonlinear stochastic coupled synthetic genetic oscillators under random intrinsic kinetic parameter fluctuations and extrinsic molecule noise in vivo; (2) The conditions of robust synchronization are developed from the nonlinear stochastic dynamical system point of view, i.e. if the synchronization robustness ≥ intrinsic robustness + extrinsic robustness, then the intrinsic parameter fluctuation can be tolerated and the extrinsic noise can be buffered so that the robust synchronization of coupled oscillators can be guaranteed. Therefore, we could obtain better insight into synchronization mechanisms of coupled synthetic molecular systems distributed in host cells, and to provide further systematic analysis and control design to improve the synchronization robustness. Further, if the robust synchronization conditions cannot be guaranteed, some synchronization control schemes are also developed to improve the synchronization robustness of coupled synthetic gene networks; (3) The fuzzy interpolation method is introduced to simplify the analysis and design procedure of robust synchronization of coupled nonlinear stochastic synthetic gene networks; and (4) An external inducer input control design method is also developed to guarantee robust synchronization of the nonlinear stochastic synthetic genetic oscillators, when spontaneous synchronization cannot be achieved under intrinsic kinetic parameter fluctuations and extrinsic environmental molecular noise. Finally, a design example is provided in silico to illustrate the design procedure and to confirm the performance of the proposed design methods.