Role of periodic forcing on the stochastic dynamics of a biomolecular clock

Biomolecular clocks produce sustained oscillations in mRNA/protein copy numbers that are subject to inherent copy-number fluctuations with important implications for proper cellular timekeeping. These random fluctuations embedded within periodic variations in copy numbers make the quantification of noise particularly challenging in stochastic gene oscillatory systems, unlike other non-oscillatory circuits. Motivated by diurnal cycles driving circadian clocks, we investigate the noise properties in the well-known Goodwin oscillator in the presence and absence of a periodic driving signal. We use two approaches to compute the noise as a function of time: (i) solving the moment dynamics derived from the linear noise approximation (LNA) assuming fluctuations are small relative to the mean and (ii) analyzing trajectories obtained from exact stochastic simulations of the Goodwin oscillator. Our results demonstrate that the LNA can predict the noise behavior quite accurately when the system shows damped oscillations or in the presence of external periodic forcing. However, the LNA could be misleading in the case of sustained oscillations without an external signal due to the propagation of large noise. Finally, we study the effect of random bursting of gene products on the clock stochastic dynamics. Our analysis reveals that the burst of mRNAs enhances the noise in the copy number regardless of the presence of external forcing, although the extent of fluctuations becomes less due to the forcing.

Role of intercellular coupling and delay on the synchronization of genetic oscillators

Living cells encode diverse biological clocks for circadian timekeeping and formation of rhythmic structures during embryonic development. A key open question is how these clocks synchronize across cells through intercellular coupling mechanisms. To address this question, we leverage the classical motif for genetic clocks the Goodwin oscillator where a gene product inhibits its own synthesis via time-delayed negative feedback. More specifically, we consider an interconnected system of two identical Goodwin oscillators (each operating in a single cell), where state information is conveyed between cells via a signaling pathway whose dynamics is modeled as a first-order system. In essence, the interaction between oscillators is characterized by an intercellular coupling strength and an intercellular time delay that represents the signaling response time. Systematic stability analysis characterizes the parameter regimes that lead to oscillatory dynamics, with high coupling strength found to destroy sustained oscillations. Within the oscillatory parameter regime we find both in-phase and anti-phase oscillations with the former more likely to occur for small intercellular time delays. Finally, we consider the stochastic formulation of the model with low-copy number fluctuations in biomolecular components. Interestingly, stochasticity leads to qualitatively different behaviors where in-phase oscillations are susceptible to the inherent fluctuations but not the anti-phase oscillations. In the context of the segmentation clock, such synchronized in-phase oscillations between cells are critical for the proper generation of repetitive segments during embryo development that eventually leads to the formation of the vertebral column.

Nonspecific protein binding can enhance the oscillatory regime of a biomolecular clock

Rhythms in gene regulatory networks are ubiquitous, from the circadian clock to the segmentation clock of vertebrates. There are many nonspecific protein binding sites (decoys) in a genome where regulatory proteins bind, and such decoys critically control the expression of a gene. The role decoys play on oscillatory regulatory networks is not well understood. Here, in the presence of decoy binding sites, we investigate the stability and the precision of the well-known Goodwin oscillator, a minimal model for regulatory oscillators. We derive the stability criterion in the presence of decoys and find that increasing decoy abundance can expand the parameter space where oscillating solutions exist. If the Goodwin system does not show any oscillation without decoy binding sites, a sustained oscillation is possible in their presence. Finally, we study precision in oscillations using stochastic simulations and find that decoy sites can make the oscillation more precise in terms of reducing noise in both the time period and the amplitude. Whereas, in an open-loop circuit, the gene expression can become noisier in the presence of decoy sites.

Parameter Identification and Adaptive Control of Uncertain Goodwin Oscillator Networks with Disturbances

This paper investigates the dynamic properties of a differential equation model of mammals’ circadian rhythms, including parameter identification, adaptive control, and outer synchronization. The circadian oscillator network is described by a Goodwin oscillator network, the couplings of which are from vasoactive intestinal polypeptides described by modified Van der Pol oscillators. We build up a drive-response system consisting of two networks with unknown parameters and disturbances. Then, we propose effective parameter updating laws to identify the unknown parameters and design adaptive control strategies to achieve outer synchronization in the drive-response system. As special cases, two succinct corollaries are presented for different instances. All the theoretical results are proved through strict mathematical deduction based on Lyapunov stability theory, and a numerical example is also carried out to illustrate the effectiveness.

Discrete-Time Mapping for an Impulsive Goodwin Oscillator with Three Delays

A popular biomathematics model of the Goodwin oscillator has been previously generalized to a more biologically plausible construct by introducing three time delays to portray the transport phenomena arising due to the spatial distribution of the model states. The present paper addresses a similar conversion of an impulsive version of the Goodwin oscillator that has found application in mathematical modeling, e.g. in endocrine systems with pulsatile hormone secretion. While the cascade structure of the linear continuous part pertinent to the Goodwin oscillator is preserved in the impulsive Goodwin oscillator, the static nonlinear feedback of the former is substituted with a pulse modulation mechanism thus resulting in hybrid dynamics of the closed-loop system. To facilitate the analysis of the mathematical model under investigation, a discrete mapping propagating the continuous state variables through the firing times of the impulsive feedback is derived. Due to the presence of multiple time delays in the considered model, previously developed mapping derivation approaches are not applicable here and a novel technique is proposed and applied. The mapping captures the dynamics of the original hybrid system and is instrumental in studying complex nonlinear phenomena arising in the impulsive Goodwin oscillator. A simulation example is presented to demonstrate the utility of the proposed approach in bifurcation analysis.