Oscillations and temporal signalling in cells

Degradation-driven protein level oscillation in the yeast Saccharomyces cerevisiae

Generating robust, predictable perturbations in cellular protein levels will advance our understanding of protein function and enable the control of physiological outcomes in biotechnology applications. Timed periodic changes in protein levels play a critical role in the cell division cycle, cellular stress response, and development. Here we report the generation of robust protein level oscillations by controlling the protein degradation rate in the yeast Saccharomyces cerevisiae. Using a photo-sensitive degron and red fluorescent proteins as reporters, we show that under constitutive transcriptional induction, repeated triangular protein level oscillations as fast as 5-10 min-scale can be generated by modulating the protein degradation rate. Consistent with oscillations generated though transcriptional control, we observed a continuous decrease in the magnitude of oscillations as the input modulation frequency increased, indicating low-pass filtering of input perturbation. By using two red fluorescent proteins with distinct maturation times, we show that the oscillations in protein level is largely unaffected by delays originating from functional protein formation. Our study demonstrates the potential for repeated control of protein levels by controlling the protein degradation rate without altering the transcription rate.

A variable refractory period increases collective performance in noisy environments

In biological and artificial systems, synchronization is an important means of achieving coordination. During hunting, social spiders alternate their moving and stopping phases in unison as they move toward their prey. We combined fieldwork and modeling to investigate the behavioral rules that lead to the emergence of synchronized oscillations in hunting groups. We showed that an individual's decision to move depends on the relative intensity of vibrations emitted by the prey and the moving spiders. This rule allows the group to adapt quickly to any change in prey size or the number of spiders involved in the hunt. Such synchronization ensures that the spiders can locate their prey without being disturbed by signals from conspecifics and thus improves hunting performance.

Synchronized oscillations are found in all living systems, from cells to ecosystems and on varying time scales. A generic principle behind the production of oscillations involves a delay in the response of one entity to stimulations from the others in the system. Communication among entities is required for the emergence of synchronization, but its efficacy can be impaired by surrounding noise. In the social spider Anelosimus eximius, individuals coordinate their activity to catch large prey that are otherwise inaccessible to solitary hunters. When hunting in groups, dozens of spiders move rhythmically toward their prey by synchronizing moving and stopping phases. We proposed a mechanistic model implementing individual behavioral rules, all derived from field experiments, to elucidate the underlying principles of synchronization. We showed that the emergence of oscillations in spiders involves a refractory state, the duration of which depends on the relative intensity of prey versus conspecific signals. This flexible behavior allows individuals to rapidly adapt to variations in their vibrational landscapes. Exploring the model reveals that the benefits of synchronization resulting from improved accuracy in prey detection and reduced latency to capture prey more than offset the cost of the delay associated with immobility phases. Overall, our study shows that a refractory period whose duration is variable and dependent on information accessible to all entities in the system contributes to the emergence of self-organized oscillations in noisy environments. Our findings may inspire the design of artificial systems requiring fast and flexible synchronization between their components.

Oscillating the local milieu of polymersome interiors via single input-regulated bilayer crosslinking and permeability tuning

The unique permselectivity of cellular membranes is of crucial importance to maintain intracellular homeostasis while adapting to microenvironmental changes. Although liposomes and polymersomes have been widely engineered to mimic microstructures and functions of cells, it still remains a considerable challenge to synergize the stability and permeability of artificial cells and to imitate local milieu fluctuations. Herein, we report concurrent crosslinking and permeabilizing of pH-responsive polymersomes containing Schiff base moieties within bilayer membranes via enzyme-catalyzed acid production. Notably, this synergistic crosslinking and permeabilizing strategy allows tuning of the mesh sizes of the crosslinked bilayers with subnanometer precision, showing discriminative permeability toward maltooligosaccharides with molecular sizes of ~1.4-2.6 nm. The permselectivity of bilayer membranes enables intravesicular pH oscillation, fueled by a single input of glucose. This intravesicular pH oscillation can further drive the dissipative self-assembly of pH-sensitive dipeptides. Moreover, the permeabilization of polymersomes can be regulated by intracellular pH gradient as well, enabling the controlled release of encapsulated payloads.

Mathematical Modelling of p53 Signalling during DNA Damage Response: A Survey

No gene has garnered more interest than p53 since its discovery over 40 years ago. In the last two decades, thanks to seminal work from Uri Alon and Ghalit Lahav, p53 has defined a truly synergistic topic in the field of mathematical biology, with a rich body of research connecting mathematic endeavour with experimental design and data. In this review we survey and distill the extensive literature of mathematical models of p53. Specifically, we focus on models which seek to reproduce the oscillatory dynamics of p53 in response to DNA damage. We review the standard modelling approaches used in the field categorising them into three types: time delay models, spatial models and coupled negative-positive feedback models, providing sample model equations and simulation results which show clear oscillatory dynamics. We discuss the interplay between mathematics and biology and show how one informs the other; the deep connections between the two disciplines has helped to develop our understanding of this complex gene and paint a picture of its dynamical response. Although yet more is to be elucidated, we offer the current state-of-the-art understanding of p53 response to DNA damage.

Inference of Gene Regulatory Network Uncovers the Linkage between Circadian Clock and Crassulacean Acid Metabolism in Kalanchoë fedtschenkoi

The circadian clock drives time-specific gene expression, enabling biological processes to be temporally controlled. Plants that conduct crassulacean acid metabolism (CAM) photosynthesis represent an interesting case of circadian regulation of gene expression as stomatal movement is temporally inverted relative to stomatal movement in C3 plants. The mechanisms behind how the circadian clock enabled physiological differences at the molecular level is not well understood. Recently, the rescheduling of gene expression was reported as a mechanism to explain how CAM evolved from C3. Therefore, we investigated whether core circadian clock genes in CAM plants were re-phased during evolution, or whether networks of phase-specific genes were simply re-wired to different core clock genes. We identified candidate core clock genes based on gene expression features and then applied the Local Edge Machine (LEM) algorithm to infer regulatory relationships between this new set of core candidates and known core clock genes in Kalanchoë fedtschenkoi. We further inferred stomata-related gene targets for known and candidate core clock genes and constructed a gene regulatory network for core clock and stomata-related genes. Our results provide new insight into the mechanism of circadian control of CAM-related genes in K. fedtschenkoi, facilitating the engineering of CAM machinery into non-CAM plants for sustainable crop production in water-limited environments.

A tale of two rhythms: Locked clocks and chaos in biology

The fundamental mechanisms that control and regulate biological organisms exhibit a surprising level of complexity. Oscillators are perhaps the simplest motifs that produce time-varying dynamics and are ubiquitous in biological systems. It is also known that such biological oscillators interact with each other-for instance, circadian oscillators affect the cell cycle, and somitogenesis clock proteins in adjacent cells affect each other in developing embryos. Therefore, it is vital to understand the effects that can emerge from non-linear interaction between oscillations. Here, we show how oscillations typically arise in biology and take the reader on a tour through the great variety in dynamics that can emerge even from a single pair of coupled oscillators. We explain how chaotic dynamics can emerge and outline the methods of detecting this in experimental time traces. Finally, we discuss the potential role of such complex dynamical features in biological systems.

Kinetic Uncertainty Relations for the Control of Stochastic Reaction Networks

Nonequilibrium stochastic reaction networks are commonly found in both biological and nonbiological systems, but have remained hard to analyze because small differences in rate functions or topology can change the dynamics drastically. Here, we conjecture exact quantitative inequalities that relate the extent of fluctuations in connected components, for various network topologies. Specifically, we find that regardless of how two components affect each other's production rates, it is impossible to suppress fluctuations below the uncontrolled equivalents for both components: one must increase its fluctuations for the other to be suppressed. For systems in which components control each other in ringlike structures, it appears that fluctuations can only be suppressed in one component if all other components instead increase fluctuations, compared to the case without control. Even the general N-component system-with arbitrary connections and parameters-must have at least one component with increased fluctuations to reduce fluctuations in others. In connected reaction networks it thus appears impossible to reduce the statistical uncertainty in all components, regardless of the control mechanisms or energy dissipation.

Cell-cell bioelectrical interactions and local heterogeneities in genetic networks: a model for the stabilization of single-cell states and multicellular oscillations

Genetic networks operate in the presence of local heterogeneities in single-cell transcription and translation rates. Bioelectrical networks and spatio-temporal maps of cell electric potentials can influence multicellular ensembles. Could cell-cell bioelectrical interactions mediated by intercellular gap junctions contribute to the stabilization of multicellular states against local genetic heterogeneities? We theoretically analyze this question on the basis of two well-established experimental facts: (i) the membrane potential is a reliable read-out of the single-cell electrical state and (ii) when the cells are coupled together, their individual cell potentials can be influenced by ensemble-averaged electrical potentials. We propose a minimal biophysical model for the coupling between genetic and bioelectrical networks that associates the local changes occurring in the transcription and translation rates of an ion channel protein with abnormally low (depolarized) cell potentials. We then analyze the conditions under which the depolarization of a small region (patch) in a multicellular ensemble can be reverted by its bioelectrical coupling with the (normally polarized) neighboring cells. We show also that the coupling between genetic and bioelectric networks of non-excitable cells, modulated by average electric potentials at the multicellular ensemble level, can produce oscillatory phenomena. The simulations show the importance of single-cell potentials characteristic of polarized and depolarized states, the relative sizes of the abnormally polarized patch and the rest of the normally polarized ensemble, and intercellular coupling.

Genetically Encoded Fluorescent Biosensors Illuminate the Spatiotemporal Regulation of Signaling Networks

Cellular signaling networks are the foundation which determines the fate and function of cells as they respond to various cues and stimuli. The discovery of fluorescent proteins over 25 years ago enabled the development of a diverse array of genetically encodable fluorescent biosensors that are capable of measuring the spatiotemporal dynamics of signal transduction pathways in live cells. In an effort to encapsulate the breadth over which fluorescent biosensors have expanded, we endeavored to assemble a comprehensive list of published engineered biosensors, and we discuss many of the molecular designs utilized in their development. Then, we review how the high temporal and spatial resolution afforded by fluorescent biosensors has aided our understanding of the spatiotemporal regulation of signaling networks at the cellular and subcellular level. Finally, we highlight some emerging areas of research in both biosensor design and applications that are on the forefront of biosensor development.

Intercellular Connectivity and Multicellular Bioelectric Oscillations in Nonexcitable Cells: A Biophysical Model

Bioelectricity is emerging as a crucial mechanism for signal transmission and processing from the single-cell level to multicellular domains. We explore theoretically the oscillatory dynamics that result from the coupling between the genetic and bioelectric descriptions of nonexcitable cells in multicellular ensembles, connecting the genetic prepatterns defined over the ensemble with the resulting spatio-temporal map of cell potentials. These prepatterns assume the existence of a small patch in the ensemble with locally low values of the genetic rate constants that produce a specific ion channel protein whose conductance promotes the cell-polarized state (inward-rectifying channel). In this way, the short-range interactions of the cells within the patch favor the depolarized membrane potential state, whereas the long-range interaction of the patch with the rest of the ensemble promotes the polarized state. The coupling between the local and long-range bioelectric signals allows a binary control of the patch membrane potentials, and alternating cell polarization and depolarization states can be maintained for optimal windows of the number of cells and the intercellular connectivity in the patch. The oscillatory phenomena emerge when the feedback between the single-cell bioelectric and genetic dynamics is coupled at the multicellular level. In this way, the intercellular connectivity acts as a regulatory mechanism for the bioelectrical oscillations. The simulation results are qualitatively discussed in the context of recent experimental studies.

A minimal "push-pull" bistability model explains oscillations between quiescent and proliferative cell states

A minimal model for oscillating between quiescent and growth/proliferation states, dependent on the availability of a central metabolic resource, is presented. From the yeast metabolic cycles, metabolic oscillations in oxygen consumption are represented as transitions between quiescent and growth states. We consider metabolic resource availability, growth rates, and switching rates (between states) to model a relaxation oscillator explaining transitions between these states. This frustrated bistability model reveals a required communication between the metabolic resource that determines oscillations and the quiescent and growth state cells. Cells in each state reflect memory, or hysteresis of their current state, and “push–pull” cells from the other state. Finally, a parsimonious argument is made for a specific central metabolite as the controller of switching between quiescence and growth states. We discuss how an oscillator built around the availability of such a metabolic resource is sufficient to generally regulate oscillations between growth and quiescence through committed transitions.

Homogeneous Time Constants Promote Oscillations in Negative Feedback Loops

Biological oscillators are present in nearly all self-regulating systems, from individual cells to entire organisms. In any oscillator structure, a negative feedback loop is necessary, but not sufficient to guarantee the emergence of periodic behaviors. The likelihood of oscillations can be improved by careful tuning of the system time constants and by increasing the loop gain, yet it is unclear whether there is any general relationship between optimal time constants and loop gain. This issue is particularly relevant in genetic oscillators resulting from a chain of different subsequent biochemical events, each with distinct (and uncertain) kinetics. Using two families of genetic oscillators as model examples, we show that the loop gain required for oscillations is minimum when all elements in the loop have the same time constant. On the contrary, we show that homeostasis is ensured if a single element is considerably slower than the others.

Kinetics of lipid-nanoparticle-mediated intracellular mRNA delivery and function

mRNA delivery into cells forms the basis for one of the new and promising ways to treat various diseases. Among suitable carriers, lipid nanoparticles (LNPs) with a size of about 100 nm are now often employed. Despite high current interest in this area, the understanding of the basic details of LNP-mediated mRNA delivery and function is limited. To clarify the kinetics of mRNA release from LNPs, the author uses three generic models implying (i) exponential, (ii) diffusion-controlled, and (iii) detachment-controlled kinetic regimes, respectively. Despite the distinct differences in these kinetics, the associated transient kinetics of mRNA translation to the corresponding protein and its degradation are shown to be not too sensitive to the details of the mRNA delivery by LNPs (or other nanocarriers). In addition, the author illustrates how this protein may temporarily influence the expression of one gene or a few equivalent genes. The analysis includes positive or negative regulation of the gene transcription via the attachment of the protein without or with positive or negative feedback in the gene expression. Stable, bistable, and oscillatory schemes have been scrutinized in this context.

How likely are oscillations in a genetic feedback loop with delay?

Some genetic control networks display temporal oscillations as a result of delays in their homeostatic control. A relevant question about these systems is whether the oscillating regime is a rare feature, or it corresponds to a sizeable volume of the space of parameters. The answer is not trivial mainly due to the large number of parameters controlling the rate equations which describe the network. We have developed an efficient sampling scheme of the parameter space, based on a Monte Carlo algorithm, and applied it to a two-node system with delay, characterised by a 8-dimension parameter space. The result is that the volume fraction of the parameter space associated with oscillations is small but not negligible, and it is weakly dependent on the duration of the delay. The most critical parameter to control oscillations is the coupling production rates, which must have opposite sign, giving rise to a negative feedback loop. The oscillating regions are connected except along the equilibrium constants between the two species, not allowing neutral evolution along this parameter.

Transcription factor oscillations in neural stem cells: Implications for accurate control of gene expression

Naturally occurring oscillations in glucocorticoids induce a cyclic activation of the glucocorticoid receptor (GR), a well-characterized ligand-activated transcription factor. These cycles of GR activation/deactivation result in rapid GR exchange at genomic response elements and GR recycling through the chaperone machinery, ultimately generating pulses of GR-mediated transcriptional activity of target genes. In a recent article we have discussed the implications of circadian and high-frequency (ultradian) glucocorticoid oscillations for the dynamic control of gene expression in hippocampal neural stem/progenitor cells (NSPCs) (Fitzsimons et al., Front. Neuroendocrinol., 2016). Interestingly, this oscillatory transcriptional activity is common to other transcription factors, many of which regulate key biological functions in NSPCs, such as NF-kB, p53, Wnt and Notch. Here, we discuss the oscillatory behavior of these transcription factors, their role in a biologically accurate target regulation and the potential importance for a dynamic control of transcription activity and gene expression in NSPCs.

Elucidating the Functional Roles of Spatial Organization in Cross-Membrane Signal Transduction by a Hybrid Simulation Method

The ligand-binding of membrane receptors on cell surfaces initiates the dynamic process of cross-membrane signal transduction. It is an indispensable part of the signaling network for cells to communicate with external environments. Recent experiments revealed that molecular components in signal transduction are not randomly mixed, but spatially organized into distinctive patterns. These patterns, such as receptor clustering and ligand oligomerization, lead to very different gene expression profiles. However, little is understood about the molecular mechanisms and functional impacts of this spatial-temporal regulation in cross-membrane signal transduction. In order to tackle this problem, we developed a hybrid computational method that decomposes a model of signaling network into two simulation modules. The physical process of binding between receptors and ligands on cell surfaces are simulated by a diffusion-reaction algorithm, while the downstream biochemical reactions are modeled by stochastic simulation of Gillespie algorithm. These two processes are coupled together by a synchronization framework. Using this method, we tested the dynamics of a simple signaling network in which the ligand binding of cell surface receptors triggers the phosphorylation of protein kinases, and in turn regulates the expression of target genes. We found that spatial aggregation of membrane receptors at cellular interfaces is able to either amplify or inhibit downstream signaling outputs, depending on the details of clustering mechanism. Moreover, by providing higher binding avidity, the co-localization of ligands into multi-valence complex modulates signaling in very different ways that are closely related to the binding affinity between ligand and receptor. We also found that the temporal oscillation of the signaling pathway that is derived from genetic feedback loops can be modified by the spatial clustering of membrane receptors. In summary, our method demonstrates the functional importance of spatial organization in cross-membrane signal transduction. The method can be applied to any specific signaling pathway in cells.

Effects of acute tryptophan depletion on raphé functional connectivity in depression

Depression remains a great societal burden and a major treatment challenge. Most antidepressant medications target serotonergic raphé nuclei. Acute tryptophan depletion (ATD) modulates serotonin function. To better understand the raphé's role in mood networks, we studied raphé functional connectivity in depression. Fifteen depressed patients were treated with sertraline for 12 weeks and scanned during ATD and sham conditions. Based on our previous findings in a separate cohort, resting state MRI functional connectivity between raphé and other depression-related regions (ROIs) was analyzed in narrow frequency bands. ATD decreased raphé functional connectivity with the bilateral thalamus within 0.025-0.05 Hz, and also decreased raphé functional connectivity with the right pregenual anterior cingulate cortex within 0.05-0.1 Hz. Using the control broadband filter 0.01-0.1 Hz, no significant differences in raphé-ROI functional connectivity were observed. Post-hoc analysis by remission status suggested increased raphé functional connectivity with left pregenual anterior cingulate cortex in remitters (n=10) and decreased raphé functional connectivity with left thalamus in non-remitters (n=5), both within 0.025-0.05 Hz. Reducing serotonin function appears to alter coordination of these mood-related networks in specific, low frequency ranges. For examination of effects of reduced serotonin function on mood-related networks, specific low frequency BOLD fMRI signals can identify regions implicated in neural circuitry and may enable clinically-relevant interpretation of functional connectivity measures. The biological significance of these low frequency signals detected in the raphé merits further study.

Design principles for robust oscillatory behavior

Oscillatory responses are ubiquitous in regulatory networks of living organisms, a fact that has led to extensive efforts to study and replicate the circuits involved. However, to date, design principles that underlie the robustness of natural oscillators are not completely known. Here we study a three-component enzymatic network model in order to determine the topological requirements for robust oscillation. First, by simulating every possible topological arrangement and varying their parameter values, we demonstrate that robust oscillators can be obtained by augmenting the number of both negative feedback loops and positive autoregulations while maintaining an appropriate balance of positive and negative interactions. We then identify network motifs, whose presence in more complex topologies is a necessary condition for obtaining oscillatory responses. Finally, we pinpoint a series of simple architectural patterns that progressively render more robust oscillators. Together, these findings can help in the design of more reliable synthetic biomolecular networks and may also have implications in the understanding of other oscillatory systems.

Interplay between stochasticity and negative feedback leads to pulsed dynamics and distinct gene activity patterns

Gene expression is an inherently stochastic process that depends on the structure of the biochemical regulatory network in which the gene is embedded. Here we study the dynamical consequences of the interplay between stochastic gene switching and the widespread negative feedback regulatory loop in a simple model of a biochemical regulatory network. Using a simplified hybrid simulation approach, in which only the gene activation is modeled stochastically, we find that stochasticity in gene switching by itself can induce pulses in the system, providing also analytical insights into their origin. Furthermore, we find that this simple network is able to reproduce both exponential and peaked distributions of gene active and inactive times similar to those that have been observed experimentally. This simplified hybrid simulation approach also allows us to link these patterns to the dynamics of the system for each gene state.

Influence of parameter values on the oscillation sensitivities of two p53-Mdm2 models

Biomolecular networks that present oscillatory behavior are ubiquitous in nature. While some design principles for robust oscillations have been identified, it is not well understood how these oscillations are affected when the kinetic parameters are constantly changing or are not precisely known, as often occurs in cellular environments. Many models of diverse complexity level, for systems such as circadian rhythms, cell cycle or the p53 network, have been proposed. Here we assess the influence of hundreds of different parameter sets on the sensitivities of two configurations of a well-known oscillatory system, the p53 core network. We show that, for both models and all parameter sets, the parameter related to the p53 positive feedback, i.e. self-promotion, is the only one that presents sizeable sensitivities on extrema, periods and delay. Moreover, varying the parameter set values to change the dynamical characteristics of the response is more restricted in the simple model, whereas the complex model shows greater tunability. These results highlight the importance of the presence of specific network patterns, in addition to the role of parameter values, when we want to characterize oscillatory biochemical systems.

Inherent directionality explains the lack of feedback loops in empirical networks

We explore the hypothesis that the relative abundance of feedback loops in many empirical complex networks is severely reduced owing to the presence of an inherent global directionality. Aimed at quantifying this idea, we propose a simple probabilistic model in which a free parameter γ controls the degree of inherent directionality. Upon strengthening such directionality, the model predicts a drastic reduction in the fraction of loops which are also feedback loops. To test this prediction, we extensively enumerated loops and feedback loops in many empirical biological, ecological and socio-technological directed networks. We show that, in almost all cases, empirical networks have a much smaller fraction of feedback loops than network randomizations. Quite remarkably, this empirical finding is quantitatively reproduced, for all loop lengths, by our model by fitting its only parameter γ. Moreover, the fitted value of γ correlates quite well with another direct measurement of network directionality, performed by means of a novel algorithm. We conclude that the existence of an inherent network directionality provides a parsimonious quantitative explanation for the observed lack of feedback loops in empirical networks.

The pharmacodynamics of the p53-Mdm2 targeting drug Nutlin: the role of gene-switching noise

In this work we investigate, by means of a computational stochastic model, how tumor cells with wild-type p53 gene respond to the drug Nutlin, an agent that interferes with the Mdm2-mediated p53 regulation. In particular, we show how the stochastic gene-switching controlled by p53 can explain experimental dose-response curves, i.e., the observed inter-cell variability of the cell viability under Nutlin action. The proposed model describes in some detail the regulation network of p53, including the negative feedback loop mediated by Mdm2 and the positive loop mediated by PTEN, as well as the reversible inhibition of Mdm2 caused by Nutlin binding. The fate of the individual cell is assumed to be decided by the rising of nuclear-phosphorylated p53 over a certain threshold. We also performed in silico experiments to evaluate the dose-response curve after a single drug dose delivered in mice, or after its fractionated administration. Our results suggest that dose-splitting may be ineffective at low doses and effective at high doses. This complex behavior can be due to the interplay among the existence of a threshold on the p53 level for its cell activity, the nonlinearity of the relationship between the bolus dose and the peak of active p53, and the relatively fast elimination of the drug.

P53 is an antitumor gene regulating vital cellular functions such as repair of DNA damage, cellular suicide, and cell proliferation: in many tumors p53 is lowly expressed and/or mutated. Drugs targeting the biomolecular network of p53 are becoming important. The network includes the key proteins Mdm2 and PTEN, whose production is regulated by p53, and which, in turn, enact positive and negative feedbacks on p53. Drug Nutlin, inhibiting the p53 inhibitor Mdm2, might be important for tumors where p53 is underproduced but unmutated. We investigate the cellular mechanism of action of Nutlin. The basic concept of our mathematical model is that the experimentally observed cell-to-cell variability of Nutlin efficacy is caused by the randomness of gene activation/deactivation of Mdmd2 and PTEN. Indeed, the abundance/scarceness of p53 regulates the probability that the relative genes are active or inactive. The model reproduced the experimental cell-specific response to different doses of Nutlin (dose-response curves) in some types of tumor cells. Much clinical research focus on 'metronomic' drug delivery regimens, where instead of giving large doses with long intervals, smaller doses are frequently delivered. In our simulations, dose-splitting of Nutlin produced a response generally worse than the response to a single dose.

Stochastic oscillations induced by intrinsic fluctuations in a self-repressing gene

Biochemical reaction networks are subjected to large fluctuations attributable to small molecule numbers, yet underlie reliable biological functions. Thus, it is important to understand how regularity can emerge from noise. Here, we study the stochastic dynamics of a self-repressing gene with arbitrarily long or short response time. We find that when the mRNA and protein half-lives are approximately equal to the gene response time, fluctuations can induce relatively regular oscillations in the protein concentration. To gain insight into this phenomenon at the crossroads of determinism and stochasticity, we use an intermediate theoretical approach, based on a moment-closure approximation of the master equation, which allows us to take into account the binary character of gene activity. We thereby obtain differential equations that describe how nonlinearity can feed-back fluctuations into the mean-field equations to trigger oscillations. Finally, our results suggest that the self-repressing Hes1 gene circuit exploits this phenomenon to generate robust oscillations, inasmuch as its time constants satisfy precisely the conditions we have identified.

The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.

On the dephasing of genetic oscillators

The digital nature of genes combined with the associated low copy numbers of proteins regulating them is a significant source of stochasticity, which affects the phase of biochemical oscillations. We show that unlike ordinary chemical oscillators, the dichotomic molecular noise of gene state switching in gene oscillators affects the stochastic dephasing in a way that may not always be captured by phenomenological limit cycle-based models. Through simulations of a realistic model of the NFκB/IκB network, we also illustrate the dephasing phenomena that are important for reconciling single-cell and population-based experiments on gene oscillators.

A simple model of NF-κB dynamics reproduces experimental observations

The mathematical modeling of the NF-κB oscillations has attracted considerable attention in recent times, but there is a lack of simple models in the literature that can capture the main features of the dynamics of this important transcription factor. For this reason we propose a simple model that summarizes the key steps of the NF-κB pathway. We show that the resulting 5-dimensional dynamical system can reproduce different phenomena observed in experiments. Our model can display smooth and spiky oscillations in the amount of nuclear NF-κB and can reproduce the variety of dynamics observed when different stimulations such as TNF-α and LPS are used. Furthermore we show that the model can be easily extended to reproduce the expression of early, intermediate and late genes upon stimulation. As a final example we show that our simple model can mimic the different transcriptional outputs observed when cells are treated with two different drugs leading to nuclear localization of NF-κB: Leptomycin B and Cycloheximide.

The dynamics of genetic control in the cell: the good and bad of being late

The expression of genes in the cell is controlled by a complex interaction network involving proteins, RNA and DNA. The molecular events associated with the nodes of such a network take place on a variety of time scales, and thus cannot be regarded as instantaneous. In many cases, the cell is robust with respect to the delay in gene expression control, behaving as if it were instantaneous. However, there are specific cases in which delay gives rise to temporal oscillations. This is the case, for example, of the expression of tumour-suppressor protein p53, of protein Hes1, involved in the differentiation of stem cells, of NFkB and Wnt, in which case delay arises implicitly from the structure of the associated network. By means of delay rate equations, we study the kinetics of small regulatory networks, emphasizing the role of delay in an evolutionary context. These models suggest that oscillations are a typical outcome of the dynamics of regulatory networks, and evolution has to work to avoid them when not required (and not vice versa).

Generation of oscillating gene regulatory network motifs

Using an improved version of an evolutionary algorithm originally proposed by François and Hakim [Proc. Natl. Acad. Sci. USA 101, 580 (2004)], we generated small gene regulatory networks in which the concentration of a target protein oscillates in time. These networks may serve as candidates for oscillatory modules to be found in larger regulatory networks and protein interaction networks. The algorithm was run for 10(5) times to produce a large set of oscillating modules, which were systematically classified and analyzed. The robustness of the oscillations against variations of the kinetic rates was also determined, to filter out the least robust cases. Furthermore, we show that the set of evolved networks can serve as a database of models whose behavior can be compared to experimentally observed oscillations. The algorithm found three smallest (core) oscillators in which nonlinearities and number of components are minimal. Two of those are two-gene modules: the mixed feedback loop, already discussed in the literature, and an autorepressed gene coupled with a heterodimer. The third one is a single gene module which is competitively regulated by a monomer and a dimer. The evolutionary algorithm also generated larger oscillating networks, which are in part extensions of the three core modules and in part genuinely new modules. The latter includes oscillators which do not rely on feedback induced by transcription factors, but are purely of post-transcriptional type. Analysis of post-transcriptional mechanisms of oscillation may provide useful information for circadian clock research, as recent experiments showed that circadian rhythms are maintained even in the absence of transcription.

Finding novel pharmaceuticals in the systems biology era using multiple effective drug targets, phenotypic screening and knowledge of transporters: where drug discovery went wrong and how to fix it

Despite the sequencing of the human genome, the rate of innovative and successful drug discovery in the pharmaceutical industry has continued to decrease. Leaving aside regulatory matters, the fundamental and interlinked intellectual issues proposed to be largely responsible for this are: (a) the move from 'function-first' to 'target-first' methods of screening and drug discovery; (b) the belief that successful drugs should and do interact solely with single, individual targets, despite natural evolution's selection for biochemical networks that are robust to individual parameter changes; (c) an over-reliance on the rule-of-5 to constrain biophysical and chemical properties of drug libraries; (d) the general abandoning of natural products that do not obey the rule-of-5; (e) an incorrect belief that drugs diffuse passively into (and presumably out of) cells across the bilayers portions of membranes, according to their lipophilicity; (f) a widespread failure to recognize the overwhelmingly important role of proteinaceous transporters, as well as their expression profiles, in determining drug distribution in and between different tissues and individual patients; and (g) the general failure to use engineering principles to model biology in parallel with performing 'wet' experiments, such that 'what if?' experiments can be performed in silico to assess the likely success of any strategy. These facts/ideas are illustrated with a reasonably extensive literature review. Success in turning round drug discovery consequently requires: (a) decent systems biology models of human biochemical networks; (b) the use of these (iteratively with experiments) to model how drugs need to interact with multiple targets to have substantive effects on the phenotype; (c) the adoption of polypharmacology and/or cocktails of drugs as a desirable goal in itself; (d) the incorporation of drug transporters into systems biology models, en route to full and multiscale systems biology models that incorporate drug absorption, distribution, metabolism and excretion; (e) a return to 'function-first' or phenotypic screening; and (f) novel methods for inferring modes of action by measuring the properties on system variables at all levels of the 'omes. Such a strategy offers the opportunity of achieving a state where we can hope to predict biological processes and the effect of pharmaceutical agents upon them. Consequently, this should both lower attrition rates and raise the rates of discovery of effective drugs substantially.

A circadian clock-regulated toggle switch explains AtGRP7 and AtGRP8 oscillations in Arabidopsis thaliana

The circadian clock controls many physiological processes in higher plants and causes a large fraction of the genome to be expressed with a 24h rhythm. The transcripts encoding the RNA-binding proteins AtGRP7 (Arabidopsis thaliana Glycine Rich Protein 7) and AtGRP8 oscillate with evening peaks. The circadian clock components CCA1 and LHY negatively affect AtGRP7 expression at the level of transcription. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate post-transcriptionally: high protein levels promote the generation of an alternative splice form that is rapidly degraded. This clock-regulated feedback loop has been proposed to act as a molecular slave oscillator in clock output. While mathematical models describing the circadian core oscillator in Arabidopsis thaliana were introduced recently, we propose here the first model of a circadian slave oscillator. We define the slave oscillator in terms of ordinary differential equations and identify the model's parameters by an optimization procedure based on experimental results. The model successfully reproduces the pertinent experimental findings such as waveforms, phases, and half-lives of the time-dependent concentrations. Furthermore, we obtain insights into possible mechanisms underlying the observed experimental dynamics: the negative auto-regulation and reciprocal cross-regulation via alternative splicing could be responsible for the sharply peaking waveforms of the AtGRP7 and AtGRP8 mRNA. Moreover, our results suggest that the AtGRP8 transcript oscillations are subordinated to those of AtGRP7 due to a higher impact of AtGRP7 protein on alternative splicing of its own and of the AtGRP8 pre-mRNA compared to the impact of AtGRP8 protein. Importantly, a bifurcation analysis provides theoretical evidence that the slave oscillator could be a toggle switch, arising from the reciprocal cross-regulation at the post-transcriptional level. In view of this, transcriptional repression of AtGRP7 and AtGRP8 by LHY and CCA1 induces oscillations of the toggle switch, leading to the observed high-amplitude oscillations of AtGRP7 mRNA.

The circadian clock organizes the day in the life of a plant by causing 24h rhythms in gene expression. For example, the core clockwork of the model plant Arabidopsis thaliana causes the transcripts encoding the RNA-binding proteins AtGRP7 and AtGRP8 to undergo high amplitude oscillations with a peak at the end of the day. AtGRP7 and AtGRP8, in turn, negatively auto-regulate and reciprocally cross-regulate their own expression by causing alternative splicing of their pre-mRNAs, followed by rapid degradation of the alternatively spliced transcripts. This has led to the suggestion that they represent molecular slave oscillators downstream of the core clock. Using a mathematical model we obtain insights into possible mechanisms underlying the experimentally observed dynamics, e.g. a higher impact of AtGRP7 protein compared to the impact of AtGRP8 protein on the alternative splicing explains the experimentally observed phases of their transcript. Previously, components that reciprocally repress their own transcription (double negative loops) have been shown to potentially act as a toggle switch between two states. We provide theoretical evidence that the slave oscillator could be a bistable toggle switch as well, operating at the post-transcriptional level.

A semi-mechanistic integrated toxicokinetic-toxicodynamic (TK/TD) model for arsenic(III) in hepatocytes

BACKGROUND

A systems engineering approach is presented for describing the kinetics and dynamics that are elicited upon arsenic exposure of human hepatocytes. The mathematical model proposed here tracks the cellular reaction network of inorganic and organic arsenic compounds present in the hepatocyte and analyzes the production of toxicologically potent by-products and the signaling they induce in hepatocytes.

METHODS AND RESULTS

The present modeling effort integrates for the first time a cellular-level semi-mechanistic toxicokinetic (TK) model of arsenic in human hepatocytes with a cellular-level toxicodynamic (TD) model describing the arsenic-induced reactive oxygen species (ROS) burst, the antioxidant response, and the oxidative DNA damage repair process. The antioxidant response mechanism is described based on the Keap1-independent Nuclear Factor-erythroid 2-related factor 2 (Nrf2) signaling cascade and accounts for the upregulation of detoxifying enzymes. The ROS-induced DNA damage is simulated by coupling the TK/TD formulation with a model describing the multistep pathway of oxidative DNA repair. The predictions of the model are assessed against experimental data of arsenite-induced genotoxic damage to human hepatocytes; thereby capturing in silico the mode of the experimental dose-response curve.

CONCLUSIONS

The integrated cellular-level TK/TD model presented here provides significant insight into the underlying regulatory mechanism of Nrf2-regulated antioxidant response due to arsenic exposure. While computational simulations are in a fair good agreement with relevant experimental data, further analysis of the system unravels the role of a dynamic interplay among the feedback loops of the system in controlling the ROS upregulation and DNA damage response. This TK/TD framework that uses arsenic as an example can be further extended to other toxic or pharmaceutical agents.

Insm1a-mediated gene repression is essential for the formation and differentiation of Müller glia-derived progenitors in the injured retina

In zebrafish, retinal injury stimulates Müller glia (MG) reprograming; allowing them to generate multipotent progenitors that regenerate damaged cells and restore vision. Recent studies suggest transcriptional repression may underlie these events. To identify these repressors, we compared the transcriptomes of MG and MG-derived progenitors and identified insm1a, a transcriptional repressor exhibiting a biphasic pattern of expression that is essential for retina regeneration. Insm1a was found to suppress ascl1a and its own expression and link injury-dependent ascl1a induction with dickkopf (dkk) suppression, which is necessary for MG dedifferentiation. We also found that Insm1a was responsible for sculpting the zone of injury-responsive MG by suppressing hb-egf_a expression. Finally, we provide evidence that Insm1a stimulates progenitor cell cycle exit by suppressing a genetic program driving progenitor proliferation. Our studies identify Insm1a as a key regulator of retina regeneration and provide a mechanistic understanding of how it contributes to multiple phases of this process.

Dehydration stress-induced oscillations in LEA protein transcripts involves abscisic acid in the moss, Physcomitrella patens

• Physcomitrella patens is a bryophyte belonging to early diverging lineages of land plants following colonization of land in the Ordovician period. Mosses are typically found in refugial habitats and can experience rapidly fluctuating environmental conditions. The acquisition of dehydration tolerance by bryophytes is of fundamental importance as they lack water-conducting tissues and are generally one cell layer thick. • Here, we show that dehydration induced oscillations in the steady-state transcript abundances of two group 3 late embryogenesis abundant (LEA) protein genes in P. patens protonemata, and that the amplitudes of these oscillations are reflective of the severity of dehydration stress. • Dehydration stress also induced elevations in the concentrations of abscisic acid (ABA), and ABA alone can also induce dosage-dependent oscillatory increases in the steady-state abundance of LEA protein transcripts. Additionally, removal of ABA resulted in rapid attenuation of these oscillatory increases. • Our data demonstrate that dehydration stress-regulated expression of LEA protein genes is temporally dynamic and highlight the importance of oscillations as a robust mechanism for optimal responses. Our results suggest that dehydration stress-induced oscillations in the steady-state abundance of LEA protein transcripts may constitute an important cellular strategy for adaptation to life in a constantly changing environment.

Transcription factor oscillations induce differential gene expressions

Intracellular protein levels of diverse transcription factors (TFs) vary periodically with time. However, the effects of TF oscillations on gene expression, the primary role of TFs, are poorly understood. In this study, we determined these effects by comparing gene expression levels induced in the presence and in the absence of TF oscillations under same mean intracellular protein level of TF. For all the nonlinear TF transcription kinetics studied, an oscillatory TF is predicted to induce gene expression levels that are distinct from a nonoscillatory TF. The conditions dictating whether TF oscillations induce either higher or lower average gene expression levels were elucidated. Subsequently, the predicted effects from an oscillatory TF, which follows sigmoid transcription kinetics, were applied to demonstrate how oscillatory dynamics provide a mechanism for differential target gene transactivation. Generally, the mean TF concentration at which oscillations occur relative to the promoter binding affinity of a target gene determines whether the gene is up- or downregulated whereas the oscillation amplitude amplifies the magnitude of the differential regulation. Notably, the predicted trends of differential gene expressions induced by oscillatory NF-κB and glucocorticoid receptor match the reported experimental observations. Furthermore, the biological function of p53 oscillations is predicted to prime the cell for death upon DNA damage via differential upregulation of apoptotic genes. Lastly, given N target genes, an oscillatory TF can generate between (N-1) and (2N-1) distinct patterns of differential transactivation. This study provides insights into the mechanism for TF oscillations to induce differential gene expressions, and underscores the importance of TF oscillations in biological regulations.

Complex dynamics of transcription regulation

Transcription is a tightly regulated cellular function which can be triggered by endogenous (intrinsic) or exogenous (extrinsic) signals. The development of novel techniques to examine the dynamic behavior of transcription factors and the analysis of transcriptional activity at the single cell level with increased temporal resolution has revealed unexpected elements of stochasticity and dynamics of this process. Emerging research reveals a complex picture, wherein a wide range of time scales and temporal transcription patterns overlap to generate transcriptional programs. The challenge now is to develop a perspective that can guide us to common underlying mechanisms, and consolidate these findings. Here we review the recent literature on temporal dynamics and stochastic gene regulation patterns governed by intrinsic or extrinsic signals, utilizing the glucocorticoid receptor (GR)-mediated transcriptional model to illustrate commonality of these emerging concepts. This article is part of a Special Issue entitled: Chromatin in time and space.

Sic1 as a timer of Clb cyclin waves in the yeast cell cycle--design principle of not just an inhibitor

Cellular systems biology aims to uncover design principles that describe the properties of biological networks through interaction of their components in space and time. The cell cycle is a complex system regulated by molecules that are integrated into functional modules to ensure genome integrity and faithful cell division. In budding yeast, cyclin-dependent kinases (Cdk1/Clb) drive cell cycle progression, being activated and inactivated in a precise temporal sequence. In this module, which we refer to as the 'Clb module', different Cdk1/Clb complexes are regulated to generate waves of Clb activity, a functional property of cell cycle control. The inhibitor Sic1 plays a critical role in the Clb module by binding to and blocking Cdk1/Clb activity, ultimately setting the timing of DNA replication and mitosis. Fifteen years of research subsequent to the identification of Sic1 have lead to the development of an integrative approach that addresses its role in regulating the Clb module. Sic1 is an intrinsically disordered protein and achieves its inhibitory function by cooperative binding, where different structural regions stretch on the Cdk1/Clb surface. Moreover, Sic1 promotes S phase entry, facilitating Cdk1/Clb5 nuclear transport, and therefore revealing a double function of inhibitor/activator that rationalizes a mechanism to prevent precocious DNA replication. Interestingly, the investigation of Clb temporal dynamics by mathematical modelling and experimental validation provides evidence that Sic1 acts as a timer to coordinate oscillations of Clb cyclin waves. Here we review these findings, focusing on the design principle underlying the Clb module, which highlights the role of Sic1 in regulating phase-specific Cdk1/Clb activities.

Oscillabolastic model, a new model for oscillatory dynamics, applied to the analysis of Hes1 gene expression and Ehrlich ascites tumor growth

This paper introduces a new dynamical model, called the oscillabolastic model, to analyze the dynamical behavior of biomedical data when one observes oscillatory behavior. The proposed oscillabolastic model is sufficiently flexible to represent various types of oscillatory behavior. The oscillabolastic model is applied to two sets of data. The first data set deals with the oscillabolastic modeling of Ehrlich ascites tumor cells and the second one is the oscillabolastic modeling of the mean signal intensity of Hes1 gene expression in response to serum stimulation. A generalized oscillabolastic model is also suggested to accommodate cases in which predictor variables other than time are also involved.

EFFECT OF NON-CODING RNA ON BISTABILITY AND OSCILLATIONS IN THE mRNA-PROTEIN INTERPLAY

The feedbacks between the mRNA and protein synthesis may result in kinetic bistability and oscillations. Two generic models predicting bistability include, respectively, a gene with positive regulation of the mRNA production by protein and two genes with mutual suppression of the mRNA production due to negative regulation of the gene transcription by protein. The simplest model predicting oscillations describes a gene with negative regulation of the mRNA production by protein formed via mRNA translation and a few steps of conversion. We complement these models by the steps of non-coding RNA (ncRNA) formation and ncRNA-mRNA association and degradation. With this extension, the bistability can often be observed as well. Without and with ncRNA, the biochemistry behind the steady states may be different. In the latter case, for example, ncRNA may control the mRNA population in the situations when this population is relatively small, and one can observe a switch in the mRNA, protein and ncRNA populations. Our analysis of oscillatory kinetics of the mRNA-protein interplay shows that with ncRNA the oscillations may be observed in a wider range of parameters and the amplitude of oscillations may be larger.

Spatio-temporal modelling of the NF-κB intracellular signalling pathway: the roles of diffusion, active transport, and cell geometry

The nuclear factor kappa B (NF-κB) intracellular signalling pathway is central to many stressful, inflammatory, and innate immune responses. NF-κB proteins themselves are transcription factors for hundreds of genes. Experiments have shown that the NF-κB pathway can exhibit oscillatory dynamics-a negative feedback loop causes oscillatory nuclear-cytoplasmic translocation of NF-κB. Given that cell size and shape are known to influence intracellular signal transduction, we consider a spatio-temporal model of partial differential equations for the NF-κB pathway, where we model molecular movement by diffusion and, for several key species including NF-κB, by active transport as well. Through numerical simulations we find values for model parameters such that sustained oscillatory dynamics occur. Our spatial profiles and animations bear a striking resemblance to experimental images and movie clips employing fluorescent fusion proteins. We discover that oscillations in nuclear NF-κB may occur when active transport is across the nuclear membrane only, or when no species are subject to active transport. However, when active transport is across the nuclear membrane and NF-κB is additionally actively transported through the cytoplasm, oscillations are lost. Hence transport mechanisms in a cell will influence its response to activation of its NF-κB pathway. We also demonstrate that sustained oscillations in nuclear NF-κB are somewhat robust to changes in the shape of the cell, or the shape, location, and size of its nucleus, or the location of ribosomes. Yet if the cell is particularly flat or the nucleus sufficiently small, then oscillations are lost. Thus the geometry of a cell may partly determine its response to NF-κB activation. The NF-κB pathway is known to be constitutively active in several human cancers. Our spatially explicit modelling approach will allow us, in future work, to investigate targeted drug therapy of tumours.

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.

Modeling and analysis of the dynamic behavior of the XlnR regulon in Aspergillus niger

Background

In this paper the dynamics of the transcription-translation system for XlnR regulon in Aspergillus niger is modeled. The model is based on Hill regulation functions and uses ordinary differential equations. The network response to a trigger of D-xylose is considered and stability analysis is performed. The activating, repressive feedback, and the combined effect of the two feedbacks on the network behavior are analyzed.

Results

Simulation and systems analysis showed significant influence of activating and repressing feedback on metabolite expression profiles. The dynamics of the D-xylose input function has an important effect on the profiles of the individual metabolite concentrations. Variation of the time delay in the feedback loop has no significant effect on the pattern of the response. The stability and existence of oscillatory behavior depends on which proteins are involved in the feedback loop.

Conclusions

The dynamics in the regulation properties of the network are dictated mainly by the transcription and translation degradation rate parameters, and by the D-xylose consumption profile. This holds true with and without feedback in the network. Feedback was found to significantly influence the expression dynamics of genes and proteins. Feedback increases the metabolite abundance, changes the steady state values, alters the time trajectories and affects the response oscillatory behavior and stability conditions. The modeling approach provides insight into network behavioral dynamics particularly for small-sized networks. The analysis of the network dynamics has provided useful information for experimental design for future in vitro experimental work.

Global analysis of phase locking in gene expression during cell cycle: the potential in network modeling

Background

In nonlinear dynamic systems, synchrony through oscillation and frequency modulation is a general control strategy to coordinate multiple modules in response to external signals. Conversely, the synchrony information can be utilized to infer interaction. Increasing evidence suggests that frequency modulation is also common in transcription regulation.

Results

In this study, we investigate the potential of phase locking analysis, a technique to study the synchrony patterns, in the transcription network modeling of time course gene expression data. Using the yeast cell cycle data, we show that significant phase locking exists between transcription factors and their targets, between gene pairs with prior evidence of physical or genetic interactions, and among cell cycle genes. When compared with simple correlation we found that the phase locking metric can identify gene pairs that interact with each other more efficiently. In addition, it can automatically address issues of arbitrary time lags or different dynamic time scales in different genes, without the need for alignment. Interestingly, many of the phase locked gene pairs exhibit higher order than 1:1 locking, and significant phase lags with respect to each other. Based on these findings we propose a new phase locking metric for network reconstruction using time course gene expression data. We show that it is efficient at identifying network modules of focused biological themes that are important to cell cycle regulation.

Conclusions

Our result demonstrates the potential of phase locking analysis in transcription network modeling. It also suggests the importance of understanding the dynamics underlying the gene expression patterns.