Periodic perturbation of genetic oscillations

Abstract

Kinetics of gene expression may be oscillatory due to the feedbacks between the mRNA, non-coding RNA and protein synthesis. In complex genetic networks, the kinetic oscillations generated in one subnetwork may influence oscillations in another subnetwork. To clarify what may happen in such situations, we have performed a mean-field analysis and Monte Carlo simulations of periodic perturbation of the oscillatory kinetics of the simplest genetic network including a gene with negative regulation of the mRNA production by protein, obtained via mRNA translation and two steps of conversion. Our analysis shows universal and specific features of the kinetics under consideration. Our simulations indicate that due to fluctuations only some of these features can really be observed. Specifically, the main frequencies obtained by the Fourier expansion of the mean-field and Monte Carlo kinetics are found to be often similar except that the Monte Carlo distribution of frequencies near the main frequencies is somewhat wider.

Introduction

Periodic perturbation or, in other words, forcing of oscillatory kinetics is a very basic interdisciplinary subject attracting attention already a few decades (see, e.g., reviews [1], [2], [3], recent articles [4], [5], [6], [7], and references therein). Customarily, the analysis of this effect is based on the mean-field (MF) kinetic equations. The corresponding kinetics exhibit many interesting universal and specific features. To show and classify such kinetics, one usually uses the perturbation frequency and amplitude, ω and α, as governing parameters. This parameter plane can be divided up into regions where the dynamic behavior is qualitatively the same. Boundaries of the regions correspond to bifurcations. Typically, one can observe stable periodic oscillations (limit cycle) and quasi-periodic oscillations associated usually with a doubly periodic attractor (2-torus). Quasi-periodic oscillations usually occur provided that the perturbation is not too strong (i.e., α is not too large). In the sea of such oscillations, there are a multitude of narrow regions (Arnol’d tongues) corresponding to entrainment by frequencies kω/l (k and l are integers). With decreasing α, these regions touch the ω/ω_∘ axis at ω = kω_∘/l, where ω_∘ is the natural frequency. In practice, one can observe only the main tongues. If α is large, oscillations occur, as a rule, with the imposed frequency, ω. In addition, one can sometimes find relatively narrow regions corresponding to fully chaotic oscillations associated with strange attractors. The difference between quasi-periodic and fully chaotic oscillations can be seen in the distribution of the frequencies obtained by expanding the kinetics into Fourier series. Quasi-periodic oscillations exhibit primarily the natural frequency, imposed frequency, and multiple frequencies. Chaos associated with strange attractors is characterized by a broader distribution of frequencies. With decreasing reactant population, due to fluctuations, the natural oscillations exhibit stochastic features and eventually may disappear. In this limit, the perturbation-related kinetic features, predicted by the MF theory, may be partly or completely smeared.

To motivate our present study in this area, we recall that the gene transcription in cells is usually governed by regulatory proteins. Due to the abundance of the protein-mediated feedbacks, the kinetics of the formation of RNAs (mRNAs and non-coding RNAs) and proteins in cells are often complex, e.g., bistable or oscillatory even in the simplest genetic networks (reviewed in Refs. [8], [9], [10], [11], [12]). The kinetic models of oscillatory genetic networks are now numerous (see, e.g., reviews [10], [11], [12] and articles [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48]). Many of them are generic and focused on the simplest schemes. There are also more complex and specific models scrutinizing, e.g., the cell cycle [16], [19], [37]. The design of relatively complex schemes of genetic oscillators is discussed by Kobayashi et al. [44], [45]. Oscillations in complex distributed genetic networks containing many genes are analyzed in Refs. [47], [48]. The bulk of these models is based on the MF kinetic equations ignoring fluctuations of the RNA and protein populations.

In cells, the numbers of copies of each RNA and protein are often relatively low [49], [50] (lower or often appreciably lower than a few hundreds and thousands, respectively) and accordingly the gene expression frequently exhibits stochastic features (reviewed in Refs. [8], [9], [12], [51], [52], [53]). The corresponding theoretical studies are now numerous and focused primarily on the non-oscillatory kinetics. The stochastic oscillatory kinetics of gene expression were analyzed, e.g., in Refs. [17], [18], [20], [26], [27], [38].

The effect of periodic perturbations on bistable kinetics of gene expression has recently been scrutinized in Refs. [54], [55], [56], [57]. In this study, we analyze the situation when a gene forming the simplest oscillatory network is regulated by protein which itself belongs to another oscillatory network (the latter oscillations may be spontaneous and include only RNAs and proteins or induced, e.g., by the external carbon source [58]). In other words, this means that the former oscillations are periodically perturbed. Such perturbations are of considerable intrinsic interest. The potential biological significance of this effect for entraining and/or amplifying genetic oscillations has just been recognized [59]. In fact, this area is now open for studies and debate. To our knowledge, the first corresponding theoretical results were obtained by Hasty et al. [14]. They used a two-variable MF oscillatory model (it was later employed also in Ref. [60]) with mutually-regulated synthesis and degradation of proteins. The periodic perturbation, αsin (ωt), was added to the protein-regulated rate of formation of one of the proteins, i.e., the regulation and perturbation were assumed to be independent. For this model, Hasty et al. have constructed a course-grained kinetic phase diagram and illustrated the ability of amplification of perturbations. Battogtokh et al. [61], [62] have analyzed periodically perturbed genetic oscillations with emphasis on the eukaryotic cell cycle. The perturbation was added to the MF equations for the growth of the cell mass and the cell-cycle phase [61] or to the rate of synthesis of one of the enzymes [62]. Tsumoto et al. [63] have presented a MF model showing how light adaptation may result in the entrainment of circadian oscillations. Herzel et al. [64], [65] have recently used simple MF amplitude-phase oscillator models to illustrate how coupling of oscillators may influence the entrainment of circadian clocks. In particular, Herzel et al. [64] have explained their experimental finding that the lung clocks entrain to extreme zeitgeber cycles, whereas the suprachiasmatic nucleus clocks do not. Concerning the experiments, we refer to a recent interesting study by Mondragón-Palomino et al. [66] where the phases of hundreds of synthetic genetic oscillators have been tracked relative to a common external stimulus (the transcriptional inducer, arabinose) to map the entrainment regions predicted by a detailed model of the clock. Synthetic oscillators were observed to be frequency-locked in wide intervals of the external period and showed higher-order resonance.

As already noticed, the kinetic models of oscillatory kinetics of gene expression are now numerous. Many simplest generic models include the delay between gene transcription and formation of regulatory protein(s) (for the classification of the mechanisms of genetic oscillations, see, e.g., Refs. [11], [38]). In our work, we study the effect of periodic perturbations on one of the simplest generic oscillators including a gene with negative regulation of the mRNA production by protein formed with a delay [this model belongs to Goodwin’s class (reviewed in Ref. [11])]. Using the MF kinetic equations, we first show how the oscillatory kinetics can be perturbed in this case (Section 2). Complementing the MF treatment, we present Monte Carlo (MC) simulations illustrating the role of fluctuations in such kinetics (Section 3).

Compared to previous studies (Refs. [14], [60], [61], [62], [63], [64], [65]), our treatment has four novel and/or complementary ingredients. First, the model we adopt is different. Secondly, we employ a different scheme of perturbation (compared to Refs. [14], [60], [61], [62]). Specifically, the perturbation is assumed to modulate the mRNA synthesis rate, and the effects of the inherent regulation and perturbation are factorized (in Refs. [14], [60], [61], [62], these effects are additive). Thirdly, we exhibit typical MF kinetics in much more detail. In addition, we present MC simulations illustrating smearing of some of the MF kinetic features (in Refs. [14], [60], [61], [62], [63], [64], [65], the stochastic effects are not discussed). In particular, we illustrate evolution of the kinetics with decreasing mRNA and protein populations. Our MF and MC results are expected to be of interest for general readership.

In our model, the delay between gene transcription and formation of the regulatory protein is related to protein conversion steps [see steps (3) in the scheme described below in Section 2.1]. For a simple negative loop, the delay is a necessary condition for oscillations [11]. It can be introduced by using ordinary differential equations (as in this work) or, slightly more formally, by employing delay differential equations [9], [11]. In the deterministic case in the absence of periodic perturbation, these two approaches are nearly equivalent [11]. In the deterministic case with periodic perturbation, the results are also expected to be similar although it should be proved. In the stochastic regime, the type of fluctuations depends on the mechanism of the delay. For example, the type of fluctuations related to protein conversion is different compared to that related to the delay occurring due to mRNA translation. Under such circumstances, one cannot exclude tiny differences in the results predicted by the two approaches. With this reservation, we believe that our general conclusions are valid irrespective of the mechanism of the delay.