Mathematical model of the Drosophila circadian clock: loop regulation and transcriptional integration

Stable Expression of dmiR-283 in the Brain Promises Positive Effects in Endurance Exercise on Sleep-Wake Behavior in Aging Drosophila

Sleep-wake stability is imbalanced with natural aging, and microRNAs (miRNAs) play important roles in cell proliferation, apoptosis, and aging; however, the biological functions of miRNAs in regulating aging-related sleep-wake behavior remain unexplored. This study varied the expression pattern of dmiR-283 in Drosophila and the result showed that the aging decline in sleep-wake behavior was caused by the accumulation of brain dmiR-283 expression, whereas the core clock genes cwo and Notch signaling pathway might be suppressed, which regulate the aging process. In addition, to identify exercise intervention programs of Drosophila that promote healthy aging, mir-283^SP/+ and Pdf > mir-283^SP flies were driven to perform endurance exercise for a duration of 3 weeks starting at 10 and 30 days, respectively. The results showed that exercise starting in youth leads to an enhanced amplitude of sleep-wake rhythms, stable periods, increased activity frequency upon awakening, and the suppression of aging brain dmiR-283 expression in mir-283^SP/+ middle-aged flies. Conversely, exercise performed when the brain dmiR-283 reached a certain accumulation level showed ineffective or negative effects. In conclusion, the accumulation of dmiR-283 expression in the brain induced an age-dependent decline in sleep-wake behavior. Endurance exercise commencing in youth counteracts the increase in dmiR-283 in the aging brain, which ameliorates the deterioration of sleep-wake behavior during aging.

Long-range connections are crucial for synchronization transition in a computational model of Drosophila brain dynamics

The synchronization transition type has been the focus of attention in recent years because it is associated with many functional characteristics of the brain. In this paper, the synchronization transition in neural networks with sleep-related biological drives in Drosophila is investigated. An electrical synaptic neural network is established to research the difference between the synchronization transition of the network during sleep and wake, in which neurons regularly spike during sleep and chaotically spike during wake. The synchronization transition curves are calculated mainly using the global instantaneous order parameters S. The underlying mechanisms and types of synchronization transition during sleep are different from those during wake. During sleep, regardless of the network structure, a frustrated (discontinuous) transition can be observed. Moreover, the phenomenon of quasi periodic partial synchronization is observed in ring-shaped regular network with and without random long-range connections. As the network becomes dense, the synchronization of the network only needs to slightly increase the coupling strength g. While during wake, the synchronization transition of the neural network is very dependent on the network structure, and three mechanisms of synchronization transition have emerged: discontinuous synchronization (explosive synchronization and frustrated synchronization), and continuous synchronization. The random long-range connections is the main topological factor that plays an important role in the resulting synchronization transition. Furthermore, similarities and differences are found by comparing synchronization transition research for the Hodgkin-Huxley neural network in the beta-band and gammma-band, which can further improve the synchronization phase transition research of biologically motivated neural networks. A complete research framework can also be used to study coupled nervous systems, which can be extended to general coupled dynamic systems.

Mathematical Modeling in Circadian Rhythmicity

Circadian clocks are autonomous systems able to oscillate in a self-sustained manner in the absence of external cues, although such Zeitgebers are typically present. At the cellular level, the molecular clockwork consists of a complex network of interlocked feedback loops. This chapter discusses self-sustained circadian oscillators in the context of nonlinear dynamics theory. We suggest basic steps that can help in constructing a mathematical model and introduce how self-sustained generations can be modeled using ordinary differential equations. Moreover, we discuss how coupled oscillators synchronize among themselves or entrain to periodic signals. The development of mathematical models over the last years has helped to understand such complex network systems and to highlight the basic building blocks in which oscillating systems are built upon. We argue that, through theoretical predictions, the use of simple models can guide experimental research and is thus suitable to model biological systems qualitatively.

Collective Dynamics of Neural Networks With Sleep-Related Biological Drives in Drosophila

The collective electrophysiological dynamics of the brain as a result of sleep-related biological drives in Drosophila are investigated in this paper. Based on the Huber-Braun thermoreceptor model, the conductance-based neurons model is extended to a coupled neural network to analyze the local field potential (LFP). The LFP is calculated by using two different metrics: the mean value and the distance-dependent LFP. The distribution of neurons around the electrodes is assumed to have a circular or grid distribution on a two-dimensional plane. Regardless of which method is used, qualitatively similar results are obtained that are roughly consistent with the experimental data. During wake, the LFP has an irregular or a regular spike. However, the LFP becomes regular bursting during sleep. To further analyze the results, wavelet analysis and raster plots are used to examine how the LFP frequencies changed. The synchronization of neurons under different network structures is also studied. The results demonstrate that there are obvious oscillations at approximately 8 Hz during sleep that are absent during wake. Different time series of the LFP can be obtained under different network structures and the density of the network will also affect the magnitude of the potential. As the number of coupled neurons increases, the neural network becomes easier to synchronize, but the sleep and wake time described by the LFP spectrogram do not change. Moreover, the parameters that affect the durations of sleep and wake are analyzed.

Temperature robustness in Arabidopsis circadian clock models is facilitated by repressive interactions, autoregulation, and three-node feedbacks

The biological interactions underpinning the Arabidopsis circadian clock have been systematically uncovered and explored by biological experiments and mathematical models. This is captured by a series of published ordinary differential equation (ODE) models, which describe plant clock dynamics in response to light/dark conditions. However, understanding the role of temperature in resetting the clock (entrainment) and the mechanisms by which circadian rhythms maintain a near-24 h period over a range of temperatures (temperature compensation) is still unclear. Understanding entrainment and temperature compensation may elucidate the principles governing the structure of the circadian clock network. Here we explore the design principles of the Arabidopsis clock and its responses to changes in temperature. We analyse published clock models of Arabidopsis, spanning a range of complexity, and incorporate temperature-dependent dynamics into the parameters of translation rates in these models, to discern which regulatory patterns may best explain clock function and temperature compensation. We additionally construct three minimal clock models and explore what key features govern their rhythmicity and temperature robustness via a series of random parameterisations. Results show that the highly repressive interactions between the components of the plant clock, together with autoregulation patterns and three-node feedback loops, are associated with circadian function of the clock in general, and enhance its robustness to temperature variation in particular. However, because the networks governing clock function vary with time due to light and temperature conditions, we emphasise the importance of studying plant clock functionality in its entirety rather than as a set of discrete regulation patterns.

Thermosensitive alternative splicing senses and mediates temperature adaptation in Drosophila

Circadian rhythms are generated by the cyclic transcription, translation, and degradation of clock gene products, including timeless (tim), but how the circadian clock senses and adapts to temperature changes is not completely understood. Here, we show that temperature dramatically changes the splicing pattern of tim in Drosophila. We found that at 18°C, TIM levels are low because of the induction of two cold-specific isoforms: tim-cold and tim-short and cold. At 29°C, another isoform, tim-medium, is upregulated. Isoform switching regulates the levels and activity of TIM as each isoform has a specific function. We found that tim-short and cold encodes a protein that rescues the behavioral defects of tim⁰¹ mutants, and that flies in which tim-short and cold is abrogated have abnormal locomotor activity. In addition, miRNA-mediated control limits the expression of some of these isoforms. Finally, data that we obtained using minigenes suggest that tim alternative splicing might act as a thermometer for the circadian clock.

Heat the Clock: Entrainment and Compensation in Arabidopsis Circadian Rhythms

The circadian clock is a biological mechanism that permits some organisms to anticipate daily environmental variations. This clock generates biological rhythms, which can be reset by environmental cues such as cycles of light or temperature, a process known as entrainment. After entrainment, circadian rhythms typically persist with approximately 24 hours periodicity in free-running conditions, i.e. in the absence of environmental cues. Experimental evidence also shows that a free-running period close to 24 hours is maintained across a range of temperatures, a process known as temperature compensation. In the plant Arabidopsis, the effect of light on the circadian system has been widely studied and successfully modelled mathematically. However, the role of temperature in periodicity, and the relationship between entrainment and compensation, are not fully understood. Here we adapt recent models to incorporate temperature dependence by applying Arrhenius equations to the parameters of the models that characterize transcription, translation, and degradation rates. We show that the resulting models can exhibit thermal entrainment and temperature compensation, but that these phenomena emerge from physiologically different sets of processes. Further simulations combining thermal and photic forcing in more realistic scenarios clearly distinguish between the processes of entrainment and compensation, and reveal temperature compensation as an emergent property which can arise as a result of multiple temperature-dependent interactions. Our results consistently point to the thermal sensitivity of degradation rates as driving compensation and entrainment across a range of conditions.

Approximate kernel reconstruction for time-varying networks

Background

Most existing algorithms for modeling and analyzing molecular networks assume a static or time-invariant network topology. Such view, however, does not render the temporal evolution of the underlying biological process as molecular networks are typically "re-wired" over time in response to cellular development and environmental changes. In our previous work, we formulated the inference of time-varying or dynamic networks as a tracking problem, where the target state is the ensemble of edges in the network. We used the Kalman filter to track the network topology over time. Unfortunately, the output of the Kalman filter does not reflect known properties of molecular networks, such as sparsity.

Results

To address the problem of inferring sparse time-varying networks from a set of under-sampled measurements, we propose the Approximate Kernel RecONstruction (AKRON) Kalman filter. AKRON supersedes the Lasso regularization by starting from the Lasso-Kalman inferred network and judiciously searching the space for a sparser solution. We derive theoretical bounds for the optimality of AKRON. We evaluate our approach against the Lasso-Kalman filter on synthetic data. The results show that not only does AKRON-Kalman provide better reconstruction errors, but it is also better at identifying if edges exist within a network. Furthermore, we perform a real-world benchmark on the lifecycle (embryonic, larval, pupal, and adult stages) of the Drosophila Melanogaster.

Conclusions

We show that the networks inferred by the AKRON-Kalman filter are sparse and can detect more known gene-to-gene interactions for the Drosophila melanogaster than the Lasso-Kalman filter. Finally, all of the code reported in this contribution will be publicly available.

A systematic review of Drosophila short-term-memory genetics: Meta-analysis reveals robust reproducibility

Geneticists use olfactory conditioning in Drosophila to identify learning genes; however, little is known about how these genes are integrated into short-term memory (STM) pathways. Here, we investigated the hypothesis that the STM evidence base is weak. We performed systematic review and meta-analysis of the field. Using metrics to quantify variation between discovery articles and follow-up studies, we found that seven genes were both highly replicated, and highly reproducible. However, ∼80% of STM genes have never been replicated. While only a few studies investigated interactions, the reviewed genes could account for >1000% memory. This large summed effect size could indicate irreproducibility, many shared pathways, or that current assay protocols lack the specificity needed to identify core plasticity genes. Mechanistic theories of memory will require the convergence of evidence from system, circuit, cellular, molecular, and genetic experiments; systematic data synthesis is an essential tool for integrated neuroscience.

Entrainment of a Bacterial Synthetic Gene Oscillator through Proteolytic Queueing

Internal chemical oscillators (chemical clocks) direct the behavior of numerous biological systems, and maintenance of a given period and phase among many such oscillators may be important for their proper function. However, both environmental variability and fundamental molecular noise can cause biochemical oscillators to lose coherence. One solution to maintaining coherence is entrainment, where an external signal provides a cue that resets the phase of the oscillators. In this work, we study the entrainment of gene networks by a queueing interaction established by competition between proteins for a common proteolytic pathway. Principles of queueing entrainment are investigated for an established synthetic oscillator in Escherichia coli. We first explore this theoretically using a standard chemical reaction network model and a map-based model, both of which suggest that queueing entrainment can be achieved through pulsatile production of an additional protein competing for a common degradation pathway with the oscillator proteins. We then use a combination of microfluidics and fluorescence microscopy to verify that pulse trains modulating the production rate of a fluorescent protein targeted to the same protease (ClpXP) as the synthetic oscillator can entrain the oscillator.

Modelling with ANIMO: between fuzzy logic and differential equations

BACKGROUND

Computational support is essential in order to reason on the dynamics of biological systems. We have developed the software tool ANIMO (Analysis of Networks with Interactive MOdeling) to provide such computational support and allow insight into the complex networks of signaling events occurring in living cells. ANIMO makes use of timed automata as an underlying model, thereby enabling analysis techniques from computer science like model checking. Biology experts are able to use ANIMO via a user interface specifically tailored for biological applications. In this paper we compare the use of ANIMO with some established formalisms on two case studies.

RESULTS

ANIMO is a powerful and user-friendly tool that can compete with existing continuous and discrete paradigms. We show this by presenting ANIMO models for two case studies: Drosophila melanogaster circadian clock, and signal transduction events downstream of TNF α and EGF in HT-29 human colon carcinoma cells. The models were originally developed with ODEs and fuzzy logic, respectively.

CONCLUSIONS

Two biological case studies that have been modeled with respectively ODE and fuzzy logic models can be conveniently modeled using ANIMO. The ANIMO models require less parameters than ODEs and are more precise than fuzzy logic. For this reason we position the modelling paradigm of ANIMO between ODEs and fuzzy logic.

SMURC: High-Dimension Small-Sample Multivariate Regression With Covariance Estimation

We consider a high-dimension low sample-size multivariate regression problem that accounts for correlation of the response variables. The system is underdetermined as there are more parameters than samples. We show that the maximum likelihood approach with covariance estimation is senseless because the likelihood diverges. We subsequently propose a normalization of the likelihood function that guarantees convergence. We call this method small-sample multivariate regression with covariance (SMURC) estimation. We derive an optimization problem and its convex approximation to compute SMURC. Simulation results show that the proposed algorithm outperforms the regularized likelihood estimator with known covariance matrix and the sparse conditional Gaussian graphical model. We also apply SMURC to the inference of the wing-muscle gene network of the Drosophila melanogaster (fruit fly).

Clk post-transcriptional control denoises circadian transcription both temporally and spatially

The transcription factor CLOCK (CLK) is essential for the development and maintenance of circadian rhythms in Drosophila. However, little is known about how CLK levels are controlled. Here we show that Clk mRNA is strongly regulated post-transcriptionally through its 3' UTR. Flies expressing Clk transgenes without normal 3' UTR exhibit variable CLK-driven transcription and circadian behaviour as well as ectopic expression of CLK-target genes in the brain. In these flies, the number of the key circadian neurons differs stochastically between individuals and within the two hemispheres of the same brain. Moreover, flies carrying Clk transgenes with deletions in the binding sites for the miRNA bantam have stochastic number of pacemaker neurons, suggesting that this miRNA mediates the deterministic expression of CLK. Overall our results demonstrate a key role of Clk post-transcriptional control in stabilizing circadian transcription, which is essential for proper development and maintenance of circadian rhythms in Drosophila.

Tracking of time-varying genomic regulatory networks with a LASSO-Kalman smoother

It is widely accepted that cellular requirements and environmental conditions dictate the architecture of genetic regulatory networks. Nonetheless, the status quo in regulatory network modeling and analysis assumes an invariant network topology over time. In this paper, we refocus on a dynamic perspective of genetic networks, one that can uncover substantial topological changes in network structure during biological processes such as developmental growth. We propose a novel outlook on the inference of time-varying genetic networks, from a limited number of noisy observations, by formulating the network estimation as a target tracking problem. We overcome the limited number of observations (small n large p problem) by performing tracking in a compressed domain. Assuming linear dynamics, we derive the LASSO-Kalman smoother, which recursively computes the minimum mean-square sparse estimate of the network connectivity at each time point. The LASSO operator, motivated by the sparsity of the genetic regulatory networks, allows simultaneous signal recovery and compression, thereby reducing the amount of required observations. The smoothing improves the estimation by incorporating all observations. We track the time-varying networks during the life cycle of the Drosophila melanogaster. The recovered networks show that few genes are permanent, whereas most are transient, acting only during specific developmental phases of the organism.

Modeling circadian clocks: From equations to oscillations

Circadian rhythms are generated at the cellular level by a small but tightly regulated genetic network. In higher eukaryotes, interlocked transcriptional-translational feedback loops form the core of this network, which ensures the activation of the right genes (proteins) at the right time of the day. Understanding how such a complex molecular network can generate robust, self-sustained oscillations and accurately responds to signals from the environment (such as light and temperature) is greatly helped by mathematical modeling. In the present paper we review some mathematical models for circadian clocks, ranging from abstract, phenomenological models to the most detailed molecular models. We explain how the equations are derived, highlighting the challenges for the modelers, and how the models are analyzed. We show how to compute bifurcation diagrams, entrainment, and phase response curves. In the subsequent paper, we discuss, through a selection of examples, how modeling efforts have contributed to a better understanding of the dynamics of the circadian regulatory network.

Inverse perturbation for optimal intervention in gene regulatory networks

MOTIVATION

Analysis and intervention in the dynamics of gene regulatory networks is at the heart of emerging efforts in the development of modern treatment of numerous ailments including cancer. The ultimate goal is to develop methods to intervene in the function of living organisms in order to drive cells away from a malignant state into a benign form. A serious limitation of much of the previous work in cancer network analysis is the use of external control, which requires intervention at each time step, for an indefinite time interval. This is in sharp contrast to the proposed approach, which relies on the solution of an inverse perturbation problem to introduce a one-time intervention in the structure of regulatory networks. This isolated intervention transforms the steady-state distribution of the dynamic system to the desired steady-state distribution.

RESULTS

We formulate the optimal intervention problem in gene regulatory networks as a minimal perturbation of the network in order to force it to converge to a desired steady-state distribution of gene regulation. We cast optimal intervention in gene regulation as a convex optimization problem, thus providing a globally optimal solution which can be efficiently computed using standard toolboxes for convex optimization. The criteria adopted for optimality is chosen to minimize potential adverse effects as a consequence of the intervention strategy. We consider a perturbation that minimizes (i) the overall energy of change between the original and controlled networks and (ii) the time needed to reach the desired steady-state distribution of gene regulation. Furthermore, we show that there is an inherent trade-off between minimizing the energy of the perturbation and the convergence rate to the desired distribution. We apply the proposed control to the human melanoma gene regulatory network.

AVAILABILITY

The MATLAB code for optimal intervention in gene regulatory networks can be found online: http://syen.ualr.edu/nxbouaynaya/Bioinformatics2010.html.

Dynamics of the Drosophila circadian clock: theoretical anti-jitter network and controlled chaos

BACKGROUND

Electronic clocks exhibit undesirable jitter or time variations in periodic signals. The circadian clocks of humans, some animals, and plants consist of oscillating molecular networks with peak-to-peak time of approximately 24 hours. Clockwork orange (CWO) is a transcriptional repressor of Drosophila direct target genes.

METHODOLOGY/PRINCIPAL FINDINGS

Theory and data from a model of the Drosophila circadian clock support the idea that CWO controls anti-jitter negative circuits that stabilize peak-to-peak time in light-dark cycles (LD). The orbit is confined to chaotic attractors in both LD and dark cycles and is almost periodic in LD; furthermore, CWO diminishes the Euclidean dimension of the chaotic attractor in LD. Light resets the clock each day by restricting each molecular peak to the proximity of a prescribed time.

CONCLUSIONS/SIGNIFICANCE

The theoretical results suggest that chaos plays a central role in the dynamics of the Drosophila circadian clock and that a single molecule, CWO, may sense jitter and repress it by its negative loops.