On the dephasing of genetic oscillators

Abstract

The digital nature of genes combined with the associated low copy numbers of proteins regulating them is a significant source of stochasticity, which affects the phase of biochemical oscillations. We show that unlike ordinary chemical oscillators, the dichotomic molecular noise of gene state switching in gene oscillators affects the stochastic dephasing in a way that may not always be captured by phenomenological limit cycle-based models. Through simulations of a realistic model of the NFκB/IκB network, we also illustrate the dephasing phenomena that are important for reconciling single-cell and population-based experiments on gene oscillators.

Keywords: gene networks; master equation; stochastic simulations.

Fig. 1.

A minimalist model of an NFκB/IκB genetic oscillator. Bold arrows indicate binding (k_on) and unbinding (k_off) of NFκB to the gene. Once bound, mRNA is produced, which initiates the synthesis (k_tl) of the IκB. The copies of mRNA also are constantly degraded. The IκB inhibits the NFκB by binding to it and preventing the activation of the gene. The IKK drives the irreversible degradation of IκB in the complex and prevents the full deactivation of NFκB.

Fig. 2.

Spectral solution of master equation for the NFκB/IκB gene oscillator. The axes are the decay rate , and the ratio of oscillation frequency to decay rate γ. Different points correspond to different eigenvalues. Slow modes are grouped around . Dominant rotational modes have a high ratio.

Fig. 3.

A mockup model of a genetic oscillator: a six-state directed cycle with an attached edge. The graph shows the dependence of the spectral gap on switching rate (G). (Inset) Dependence on G of the real part of the eigenvalue smallest in magnitude.

Fig. 4.

Normalized autocorrelation of steady-state fluctuations in an IκB population as a function of gene binding/unbinding rate coefficients.

Fig. 5.

Stationary probability distribution of noisy limit cycles of an NFκB/IκB oscillator as a function of system size Ω. The panels correspond to the cases of (A) nonadiabatic: slow gene binding/unbinding; (B) adiabatic: fast gene binding/unbinding; and (C) nondichotomic: absence of dichotomic gene binding/unbinding noise.

Fig. 6.

Normalized autocorrelation of steady-state fluctuations in an IκB population as a function of system size Ω.

Fig. 7.

Dependence of the dephasing time τ_ϕ on the rates of translation (k_tl) and gene-state switching (k_off).