Internal noise-sustained circadian rhythms in a Drosophila model

Noise-induced bistability in a simple mutual inhibition system

In this study, we study noise-induced bistability in a simple bivariate mutual inhibition system with slow fluctuating responses to external signals. We give a general condition that the marginal stationary probability density of one of the two variables experiences a transition from a unimodal shape to a bimodal one. We show that the transition occurs even when the stationary probability density of the response to external signals is monotone. The mechanism for the transition is investigated in terms of the calculation of the mean first passage time. We also discuss the genericity of the transition mechanism.

Restoring circadian gene profiles in clock networks using synthetic feedback control

The circadian system-an organism's built-in biological clock-is responsible for orchestrating biological processes to adapt to diurnal and seasonal variations. Perturbations to the circadian system (e.g., pathogen attack, sudden environmental change) often result in pathophysiological responses (e.g., jetlag in humans, stunted growth in plants, etc.) In view of this, synthetic biologists are progressively adapting the idea of employing synthetic feedback control circuits to alleviate the effects of perturbations on circadian systems. To facilitate the design of such controllers, suitable models are required. Here, we extend our recently developed model for the plant circadian clock-termed the extended S-System model-to model circadian systems across different kingdoms of life. We then use this modeling strategy to develop a design framework, based on an antithetic integral feedback (AIF) controller, to restore a gene's circadian profile when it is subject to loss-of-function due to external perturbations. The use of the AIF controller is motivated by its recent successful experimental implementation. Our findings provide circadian biologists with a systematic and general modeling and design approach for implementing synthetic feedback control of circadian systems.

Common noise induced synchronous circadian oscillations in uncoupled non-identical systems

The effect of common noise on the collective behavior of circadian oscillation systems was studied in an elementary circadian clock model. It is shown that common noise could induce synchronous oscillations in two uncoupled non-identical systems in the deterministic stable steady state region. The synchronicity of common noise induced oscillations is suppressed by the internal noise, but is not remarkably decreased within a wide range of internal noise intensity. This demonstrates that the common noise induced synchronous oscillations are rather robust to internal fluctuations.

Intrinsic noise alters the frequency spectrum of mesoscopic oscillatory chemical reaction systems

Mesoscopic oscillatory reaction systems, for example in cell biology, can exhibit stochastic oscillations in the form of cyclic random walks even if the corresponding macroscopic system does not oscillate. We study how the intrinsic noise from molecular discreteness influences the frequency spectrum of mesoscopic oscillators using as a model system a cascade of coupled Brusselators away from the Hopf bifurcation. The results show that the spectrum of an oscillator depends on the level of noise. In particular, the peak frequency of the oscillator is reduced by increasing noise, and the bandwidth increased. Along a cascade of coupled oscillators, the peak frequency is further reduced with every stage and also the bandwidth is reduced. These effects can help understand the role of noise in chemical oscillators and provide fingerprints for more reliable parameter identification and volume measurement from experimental spectra.

Robustness of circadian rhythms in the presence of molecular fluctuations: an investigation based on a mechanistic, statistical theory and a simulation algorithm

After a very brief introduction to a mechanistic and statistical theory of molecular fluctuations in chemical reactions developed by Joel Keizer, we explore the robustness of a circadian rhythm model by using the theory and the exact stochastic simulation (ESS). The comparative study shows that the theory reflects the effects of the dynamics of the model on the robustness more than ESS does. Even though the theory is a macroscopic one, the robustness of the model compares well with that computed from the ESS when the system size is larger than 50. The robustness increases nonlinearly with the system size and it reaches an asymptotic value at higher system sizes. As we can expect from the dynamics of the system, the robustness is minimum near the bifurcation point and as the most sensitive parameter increases away from the bifurcation point the robustness according to the theory as well as the ESS increases and then reaches to a steady value.

Internal noise enhanced oscillation in a delayed circadian pacemaker

The effect of internal noise in a delayed circadian oscillator is studied by using both chemical Langevin equations and stochastic normal form theory. It is found that internal noise can induce circadian oscillation even if the delay time τ is below the deterministic Hopf bifurcation τ(h). We use signal-to-noise ratio (SNR) to quantitatively characterize the performance of such noise induced oscillations and a threshold value of SNR is introduced to define the so-called effective oscillation. Interestingly, the τ-range for effective stochastic oscillation, denoted as Δτ(EO), shows a bell-shaped dependence on the intensity of internal noise which is inversely proportional to the system size. We have also investigated how the rates of synthesis and degradation of the clock protein influence the SNR and thus Δτ(EO). The decay rate K(d) could significantly affect Δτ(EO), while varying the gene expression rate K(e) has no obvious effect if K(e) is not too small. Stochastic normal form analysis and numerical simulations are in good consistency with each other. This work provides us comprehensive understandings of how internal noise and time delay work cooperatively to influence the dynamics of circadian oscillations.

Multiscale complexity in the mammalian circadian clock

The field of systems biology studies how the interactions among individual components (e.g. genes and proteins) yield interesting and complex behavior. The circadian (daily) timekeeping system in mammals is an ideal system to study complexity because of its many biological scales (from genes to animal behavior). A wealth of data at each of these scales has recently been discovered. Within each scale, modeling can advance our understanding of challenging problems that arise in studying mammalian timekeeping. However, future work must focus on bridging the multiple spatial and temporal scales in the modeling of SCN network. Here we review recent advances, and then delve into a few areas that are promising research directions. We also discuss the flavor of modeling needed (simple or detailed) as well as new techniques that are needed to meet the challenges in modeling data across scales.

Stochastic approaches in systems biology

The discrete and random occurrence of chemical reactions far from thermodynamic equilibrium, and low copy numbers of chemical species, in systems biology necessitate stochastic approaches. This review is an effort to give the reader a flavor of the most important stochastic approaches relevant to systems biology. Notions of biochemical reaction systems and the relevant concepts of probability theory are introduced side by side. This leads to an intuitive and easy-to-follow presentation of a stochastic framework for modeling subcellular biochemical systems. In particular, we make an effort to show how the notion of propensity, the chemical master equation (CME), and the stochastic simulation algorithm arise as consequences of the Markov property. Most stochastic modeling reviews focus on stochastic simulation approaches--the exact stochastic simulation algorithm and its various improvements and approximations. We complement this with an outline of an analytical approximation. The most common formulation of stochastic models for biochemical networks is the CME. Although stochastic simulations are a practical way to realize the CME, analytical approximations offer more insight into the influence of randomness on system's behavior. Toward that end, we cover the chemical Langevin equation and the related Fokker-Planck equation and the two-moment approximation (2MA). Throughout the text, two pedagogical examples are used to key illustrate ideas. With extensive references to the literature, our goal is to clarify key concepts and thereby prepare the reader for more advanced texts.

A new class of highly efficient exact stochastic simulation algorithms for chemical reaction networks

We introduce an alternative formulation of the exact stochastic simulation algorithm (SSA) for sampling trajectories of the chemical master equation for a well-stirred system of coupled chemical reactions. Our formulation is based on factored-out, partial reaction propensities. This novel exact SSA, called the partial-propensity direct method (PDM), is highly efficient and has a computational cost that scales at most linearly with the number of chemical species, irrespective of the degree of coupling of the reaction network. In addition, we propose a sorting variant, SPDM, which is especially efficient for multiscale reaction networks.

Stochastic simulation of delay-induced circadian rhythms in Drosophila

Circadian rhythms are ubiquitous in all eukaryotes and some prokaryotes. Several computational models with or without time delays have been developed for circadian rhythms. Exact stochastic simulations have been carried out for several models without time delays, but no exact stochastic simulation has been done for models with delays. In this paper, we proposed a detailed and a reduced stochastic model with delays for circadian rhythms in Drosophila based on two deterministic models of Smolen et al. and employed exact stochastic simulation to simulate circadian oscillations. Our simulations showed that both models can produce sustained oscillations and that the oscillation is robust to noise in the sense that there is very little variability in oscillation period although there are significant random fluctuations in oscillation peaks. Moreover, although average time delays are essential to simulation of oscillation, random changes in time delays within certain range around fixed average time delay cause little variability in the oscillation period. Our simulation results also showed that both models are robust to parameter variations and that oscillation can be entrained by light/dark circles. Our simulations further demonstrated that within a reasonable range around the experimental result, the rates that dclock and per promoters switch back and forth between activated and repressed sites have little impact on oscillation period.

Noise-induced coherence in multicellular circadian clocks

In higher organisms, circadian rhythms are generated by a multicellular genetic clock that is entrained very efficiently to the 24-h light-dark cycle. Most studies done so far of these circadian oscillators have considered a perfectly periodic driving by light, in the form of either a square wave or a sinusoidal modulation. However, in natural conditions, organisms are subject to nonnegligible fluctuations in the light level all through the daily cycle. In this article, we investigate how the interplay between light fluctuations and intercellular coupling affects the dynamics of the collective rhythm in a large ensemble of nonidentical, globally coupled cellular clocks modeled as Goodwin oscillators. On the basis of experimental considerations, we assume an inverse dependence of the cell-cell coupling strength on the light intensity, in such a way that the larger the light intensity, the weaker the coupling. Our results show a noise-induced rhythm generation for constant light intensities at which the clock is arrhythmic in the noise-free case. Importantly, the rhythm shows a resonancelike phenomenon as a function of the noise intensity. Such improved coherence can be only observed at the level of the overt rhythm and not at the level of the individual oscillators, thus suggesting a cooperative effect of noise, coupling, and the emerging synchronization between the oscillators.

Optimal spatial synchronization on scale-free networks via noisy chemical synapses

We show that the spatial synchronization of noise-induced excitations on scale-free networks, mediated through nonlinear chemical coupling, depends vitally on the intensity of additive noise and the coupling strength. In particular, a twofold optimization is needed for achieving maximal spatial synchrony, thus indicating the existence of an optimal noise intensity as well as an optimal coupling strength. On the other hand, the traditional linear coupling via gap junctions, while still requiring a fine-tuning of the noise intensity, does not postulate the existence of an optimal coupling strength since the synchronization increases monotonously with the increasing coupling strength. Presented results reveal inherent differences in optimal spatial synchronization evoked by chemical and electrical coupling, and could hence help to pinpoint their specific roles in networked systems.