Oscillatory phosphorylation of yeast Fus3 MAP kinase controls periodic gene expression and morphogenesis

Signal-transduction networks can display complex dynamic behavior such as oscillations in the activity of key components [1-6], but it is often unclear whether such dynamic complexity is actually important for the network's regulatory functions [7, 8]. Here, we found that the mitogen-activated protein kinase (MAPK) Fus3, a key regulator of the yeast mating-pheromone response, undergoes sustained oscillations in its phosphorylation and activation state during continuous pheromone exposure. These MAPK activity oscillations led to corresponding oscillations in mating-gene expression. Oscillations in MAPK activity and gene expression required the negative regulator of G protein signaling Sst2 and partially required the MAPK phosphatase Msg5. Peaks in Fus3 activation correlated with periodic rounds of cell morphogenesis, with each peak preceding the formation of an additional mating projection. Preventing projection formation did not eliminate MAPK oscillation, but preventing MAPK oscillation blocked the formation of additional projections. A mathematical model was developed that reproduced several features of the observed oscillatory dynamics. These observations demonstrate a role for MAPK activity oscillation in driving a periodic downstream response and explain how the pheromone signaling pathway, previously thought to desensitize after 1-3 hr, controls morphology changes that continue for a much longer time.     

Delay model of circadian gene expression with two negative feedback loops

In this theoretical paper we propose a quantitative minimal model for circadian gene expression based on two negative feedback loops. We perform numerical simulations to analyse its dynamics and parameter sensitivities in free-running conditions, and verify the entrainability by a single periodic driver. We furthermore apply two simultaneously acting external drivers, leading to aperiodic oscillations in the case of a single-loop system. These can be turned into regular periodic oscillations by introduction of a second loop. Our studies confirm the increasing evidence that multiple feedback loops increase the robustness of regulatory systems, and stress the particular situation of systems that are close to transition from free-running oscillation to steady-state behaviour. We discuss possible molecular realisations of the featured feedback loops and suggest the application of complex patterns of external stimulation as a generally useful approach to assess the functionality of models of circadian systems.     

Modelling periodic oscillation in gene regulatory networks by cyclic feedback systems

In this paper, we develop a new methodology to analyze and design periodic oscillators of biological networks, in particular gene regulatory networks with multiple genes, proteins and time delays, by using negative cyclic feedback systems. We show that negative cyclic feedback networks have no stable equilibria but stable periodic orbits when certain conditions are satisfied. Specifically, we first prove the basic properties of the biological networks composed of cyclic feedback loops, and then extend our results to general cyclic feedback network with less restriction, thereby making our theoretical analysis and design of oscillators easy to implement, even for large-scale systems. Finally, we use one circadian network formed by a period protein (PER) and per mRNA, and one biologically plausible synthetic gene network, to demonstrate the theoretical results. Since there is less restriction on the network structure, the results of this paper can be expected to apply to a wide variety of areas on modelling, analyzing and designing of biological systems.     

Sources of Variability in a Synthetic Gene Oscillator

Synthetic gene oscillators are small, engineered genetic circuits that produce periodic variations in target protein expression. Like other gene circuits, synthetic gene oscillators are noisy and exhibit fluctuations in amplitude and period. Understanding the origins of such variability is key to building predictive models that can guide the rational design of synthetic circuits. Here, we developed a method for determining the impact of different sources of noise in genetic oscillators by measuring the variability in oscillation amplitude and correlations between sister cells. We first used a combination of microfluidic devices and time-lapse fluorescence microscopy to track oscillations in cell lineages across many generations. We found that oscillation amplitude exhibited high cell-to-cell variability, while sister cells remained strongly correlated for many minutes after cell division. To understand how such variability arises, we constructed a computational model that identified the impact of various noise sources across the lineage of an initial cell. When each source of noise was appropriately tuned the model reproduced the experimentally observed amplitude variability and correlations, and accurately predicted outcomes under novel experimental conditions. Our combination of computational modeling and time-lapse data analysis provides a general way to examine the sources of variability in dynamic gene circuits.     

Cellular oscillators in animal segmentation

Kinetic modeling of developmental dynamics requires detailed knowledge about genetic and metabolic networks that underlie developmental processes. However, such knowledge is not available for a vast majority of developmental processes. Here, we present an coarse-grained, phenomenological model of periodic pattern formation in multicellular organisms based on cellular oscillators (CO) that can be applied to systems for which little or no molecular data is available. An oscillatory process within cells serves as a developmental clock whose period is tightly regulated by cell-autonomous and non-autonomous mechanisms. A spatial pattern is generated as a result of an initial temporal ordering of the cell oscillators freezing into spatial order as the clocks slow down and stop at different times or phases in their cycles. When applied to vertebrate somitogenesis, the CO model can reproduce the dynamics of periodic gene expression patterns observed in the presomitic mesoderm. Different somite lengths can be generated by altering the period of the oscillation. There is evidence that a CO-type mechanism might also underlie segment formation in certain invertebrates, such as annelids and short germ insects. This suggests that the dynamical principles of sequential segmentation might be equivalent throughout the animal kingdom although most of the genes involved in segment determination differ between distant phyla.     

Frequency Dependent Topological Patterns of Resting-State Brain Networks

The topological organization underlying brain networks has been extensively investigated using resting-state fMRI, focusing on the low frequency band from 0.01 to 0.1 Hz. However, the frequency specificities regarding the corresponding brain networks remain largely unclear. In the current study, a data-driven method named complementary ensemble empirical mode decomposition (CEEMD) was introduced to separate the time series of each voxel into several intrinsic oscillation rhythms with distinct frequency bands. Our data indicated that the whole brain BOLD signals could be automatically divided into five specific frequency bands. After applying the CEEMD method, the topological patterns of these five temporally correlated networks were analyzed. The results showed that global topological properties, including the network weighted degree, network efficiency, mean characteristic path length and clustering coefficient, were observed to be most prominent in the ultra-low frequency bands from 0 to 0.015 Hz. Moreover, the saliency of small-world architecture demonstrated frequency-density dependency. Compared to the empirical mode decomposition method (EMD), CEEMD could effectively eliminate the mode-mixing effects. Additionally, the robustness of CEEMD was validated by the similar results derived from a split-half analysis and a conventional frequency division method using the rectangular window band-pass filter. Our findings suggest that CEEMD is a more effective method for extracting the intrinsic oscillation rhythms embedded in the BOLD signals than EMD. The application of CEEMD in fMRI data analysis will provide in-depth insight in investigations of frequency specific topological patterns of the dynamic brain networks.     

Detecting periodic patterns in unevenly spaced gene expression time series using Lomb-Scargle periodograms

MOTIVATION: Periodic patterns in time series resulting from biological experiments are of great interest. The commonly used Fast Fourier Transform (FFT) algorithm is applicable only when data are evenly spaced and when no values are missing, which is not always the case in high-throughput measurements. The choice of statistic to evaluate the significance of the periodic patterns for unevenly spaced gene expression time series has not been well substantiated. METHODS: The Lomb-Scargle periodogram approach is used to search time series of gene expression to quantify the periodic behavior of every gene represented on the DNA array. The Lomb-Scargle periodogram analysis provides a direct method to treat missing values and unevenly spaced time points. We propose the combination of a Lomb-Scargle test statistic for periodicity and a multiple hypothesis testing procedure with controlled false discovery rate to detect significant periodic gene expression patterns. RESULTS: We analyzed the Plasmodium falciparum gene expression dataset. In the Quality Control Dataset of 5080 expression patterns, we found 4112 periodic probes. In addition, we identified 243 probes with periodic expression in the Complete Dataset, which could not be examined in the original study by the FFT analysis due to an excessive number of missing values. While most periodic genes had a period of 48 h, some had a period close to 24 h. Our approach should be applicable for detection and quantification of periodic patterns in any unevenly spaced gene expression time-series data.     

Frequency domain analysis reveals external periodic fluctuations can generate sustained p53 oscillation

p53 is a well-known tumor suppressor protein that regulates many pathways, such as ones involved in cell cycle and apoptosis. The p53 levels are known to oscillate without damping after DNA damage, which has been a focus of many recent studies. A negative feedback loop involving p53 and MDM2 has been reported to be responsible for this oscillatory behavior, but questions remain as how the dynamics of this loop alter in order to initiate and maintain the sustained or undamped p53 oscillation. Our frequency domain analysis suggests that the sustained p53 oscillation is not completely dictated by the negative feedback loop; instead, it is likely to be also modulated by periodic DNA repair-related fluctuations that are triggered by DNA damage. According to our analysis, the p53-MDM2 feedback mechanism exhibits adaptability in different cellular contexts. It normally filters noise and fluctuations exerted on p53, but upon DNA damage, it stops performing the filtering function so that DNA repair-related oscillatory signals can modulate the p53 oscillation. Furthermore, it is shown that the p53-MDM2 feedback loop increases its damping ratio allowing p53 to oscillate at a frequency more synchronized with the other cellular efforts to repair the damaged DNA, while suppressing its inherent oscillation-generating capability. Our analysis suggests that the overexpression of MDM2, observed in many types of cancer, can disrupt the operation of this adaptive mechanism by making it less responsive to the modulating signals after DNA damage occurs.     

Formulas for intrinsic noise evaluation in oscillatory genetic networks

The linear noise approximation is a useful method for stochastic noise evaluations in genetic regulatory networks, where the covariance equation described as a Lyapunov equation plays a central role. We discuss the linear noise approximation method for evaluations of an intrinsic noise in autonomously oscillatory genetic networks; in such oscillatory networks, the covariance equation becomes a periodic differential equation that provides generally an unbounded covariance matrix, so that the standard method of noise evaluation based on the covariance matrix cannot be adopted directly. In this paper, we develop a new method of noise evaluation in oscillatory genetic networks; first, we investigate structural properties, e.g., orbital stability and periodicity, of the solutions to the covariance equation given as a periodic Lyapunov differential equation by using the Floquet-Lyapunov theory, and propose a global measure for evaluating stochastic amplitude fluctuations on the periodic trajectory; we also derive an evaluation formula for the period fluctuation. Finally, we apply our method to a model of circadian oscillations based on negative auto-regulation of gene expression, and show validity of our method by comparing the evaluation results with stochastic simulations.     

Inter-Network Interactions: Impact of Connections between Oscillatory Neuronal Networks on Oscillation Frequency and Pattern

Oscillations in electrical activity are a characteristic feature of many brain networks and display a wide variety of temporal patterns. A network may express a single oscillation frequency, alternate between two or more distinct frequencies, or continually express multiple frequencies. In addition, oscillation amplitude may fluctuate over time. The origin of this complex repertoire of activity remains unclear. Different cortical layers often produce distinct oscillation frequencies. To investigate whether interactions between different networks could contribute to the variety of oscillation patterns, we created two model networks, one generating on its own a relatively slow frequency (20 Hz; slow network) and one generating a fast frequency (32 Hz; fast network). Taking either the slow or the fast network as source network projecting connections to the other, or target, network, we systematically investigated how type and strength of inter-network connections affected target network activity. For high inter-network connection strengths, we found that the slow network was more effective at completely imposing its rhythm on the fast network than the other way around. The strongest entrainment occurred when excitatory cells of the slow network projected to excitatory or inhibitory cells of the fast network. The fast network most strongly imposed its rhythm on the slow network when its excitatory cells projected to excitatory cells of the slow network. Interestingly, for lower inter-network connection strengths, multiple frequencies coexisted in the target network. Just as observed in rat prefrontal cortex, the target network could express multiple frequencies at the same time, alternate between two distinct oscillation frequencies, or express a single frequency with alternating episodes of high and low amplitude. Together, our results suggest that input from other oscillating networks may markedly alter a network''s frequency spectrum and may partly be responsible for the rich repertoire of temporal oscillation patterns observed in the brain.