Molecular noise induces concentration oscillations in chemical systems with stable node steady states

Abstract

It is well known that internal or molecular noise induces concentration oscillations in chemical systems whose deterministic models exhibit damped oscillations. In this article we show, using the linear-noise approximation of the chemical master equation, that noise can also induce oscillations in systems whose deterministic descriptions admit no damped oscillations, i.e., systems with a stable node. This non-intuitive phenomenon is remarkable since, unlike noise-induced oscillations in systems with damped deterministic oscillations, it cannot be explained by noise excitation of the deterministic resonant frequency of the system. We here prove the following general properties of stable-node noise-induced oscillations for systems with two species: (i) the upper bound of their frequency is given by the geometric mean of the real eigenvalues of the Jacobian of the system, (ii) the upper bound of the Q-factor of the oscillations is inversely proportional to the distance between the real eigenvalues of the Jacobian, and (iii) these oscillations are not necessarily exhibited by all interacting chemical species in the system. The existence and properties of stable-node oscillations are verified by stochastic simulations of the Brusselator, a cascade Brusselator reaction system, and two other simple chemical systems involving auto-catalysis and trimerization. It is also shown how external noise induces stable node oscillations with different properties than those stimulated by internal noise.

Effects of bursty protein production on the noisy oscillatory properties of downstream pathways

“…We consider a general two variable Fokker-Planck equation with linear drift and diffusion coefficients of the form: One can use this equation to derive an equation for the power spectrum of the fluctuations in the number of molecules of species X i () and this is found to be15: where , α 2 is the same as α 1 but with 1 and 2 interchanged, β i = D ii , p = [ Det ( J )] 2 and q = [ Tr ( J )] 2 − 2 Det ( J ). Here Tr and Det refer to the matrix trace and determinant respectively.…”

“…The sharpness of the peak indicates the quality of the oscillation; see Ref. 15 for a detailed discussion of quality measures for noisy oscillators.…”

“…3 (b) and (d) we show the quality factor of the noise-induced oscillations as a function of the mean burst size μ 1 . The quality factor is defined as , where is the frequency at which maximum power is obtained and Δ ω 99% is the difference of the frequencies at which the power takes 99% of its maximum value; this measure was introduced in15 and shown to be highly reflective of the rhythmicity visible in a time series of noise-induced oscillations. For these far-from Hopf oscillations the maximum possible value of Q 99% is ≈ 5 (Ref 15).…”

“…Preliminary investigations show that in non-bursty systems if the LNA predicts a peak in the power spectrum of fluctuations for systems far from the Hopf bifurcation then the spectrum calculated from stochastic simulations shows a peak even if the molecule numbers are very small (see Fig. 6 in15); the quality of the oscillations may, however, be lower than that predicted by the LNA. Hence we expect the consideration of terms of higher order than the LNA to have little or no effect on the necessary conditions derived in this paper since these are specifically for the existence or non-existence of a peak in the power spectrum.…”

“…The power spectrum estimate (the periodogram) is then obtained by a discrete Fourier transform of this time series (See Appendix C of Ref. 15 for further details). The choices of sampling parameters for the spectra in Fig.…”