Intercellular Delay Regulates the Collective Period of Repressively Coupled Gene Regulatory Oscillator Networks

Abstract

Most biological rhythms are generated by a population of cellular oscillators coupled through intercellular signaling. Recent experimental evidence shows that the collective period may differ significantly from the autonomous period in the presence of intercellular delays. The phenomenon has been investigated using delay-coupled phase oscillators, but the proposed phase model contains no direct biological mechanism, which may weaken the model's reliability in unraveling biophysical principles. Based on a published gene regulatory oscillator model, we analyze the collective period of delay-coupled biological oscillators using the multivariable harmonic balance technique. We prove that, in contradiction to the common intuition that the collective period increases linearly with the coupling delay, the collective period turns out to be a periodic function of the intercellular delay. More surprisingly, the collective period may even decrease with the intercellular delay when the delay resides in certain regions. The collective period is given in a closed-form in terms of biochemical reaction constants and thus provides biological insights as well as guidance in synthetic-biological-oscillator design. Simulation results are given based on a segmentation clock model to confirm the theoretical predictions.

Travelling waves in somitogenesis: collective cellular properties emerge from time-delayed juxtacrine oscillation coupling

“…In each region, the collective period increases monotonically (Figure 5A) and the collective amplitude describes a parabola with a local maximum (Figure 5B). A similar pattern has been reported partially in the context of the zebrafish PSM oscillator, Hes/her oscillations in neural differentiation, and a variety of synchronization phenomena across the natural sciences (Wang et al 2014; Morelli et al, 2009; Herrgen et al, 2010; Momiji & Monk, 2009; Sadeghi & Valizadeh, 2014; Pavlides et al, 2015; Vanag et al, 2016; Wetzel et al, 2017). Nevertheless, in contrast to other mathematical descriptions, we do not observe oscillation death, a stable non-oscillating state, which has been hypothesized to occur between the synchronization and anti-synchronization phase space regions (Figure 5B, Shimojo et al, 2016).…”

“…3, 4). A similar model has been used for analytical purposes (Wang et al, 2014), but we incorporate only two time delays, instead of three. Our model corresponds to the her7;hes6 double mutant.…”

“…Theoretical models show that coupling via Notch signalling can lead to either synchronization or salt-and- pepper pattern of oscillation phases, depending on the time delay involved in Notch signalling (Lewis, 2003; Tiedemann et al, 2007; Ay et al, 2013). Mathematical formulations show that the collective period of coupled oscillators might differ from the period of uncoupled oscillators, depending in a non-trivial way on both, the coupling strength and the coupling delay (Niebur et al, 1991; Morelli et al, 2009; Herrgen et al, 2010; Wang et al, 2014). In the PSM, the collective period is a local property and related measurements are discussed in section 1.3.…”

“…Lewis’ representation combines both, the her1 and the her7 oscillator and couples neighbouring cells via deltaC (Lewis, 2003). In a similar fashion, we couple only the her1 oscillators, similarly as previously done in analytical studies (Wang et al, 2014), but our model incorporates only two time delays, instead of three. Using this reductive approach, we can recapitulate the principles that govern the travelling wave.…”

Delay decomposition approach to $$\mathcal {H}_{\infty }$$ H ∞ filtering analysis of genetic oscillator networks with time-varying delays

“…However, in genetic networks, time delay plays a crucial role, which is regarded as a source of instability and oscillations (Lakshmanan et al 2014;Mao 2013;Mathiyalagan et al 2012) due to the slow process of transcription and translation associated with mRNA and protein respectively. It has been observed that in gene regulator oscillator model (Wang et al 2014) intercellular delay regulates the collective period of coupled cellular oscillators. By using Lyapunov stability theory and matrix inequality approach, the synchronization criteria for coupled genetic oscillators with delayed coupling has been investigated in Li and Lam (2011).…”