Design principles of biochemical oscillators

Hybrid modeling and simulation of stochastic effects on progression through the eukaryotic cell cycle

The eukaryotic cell cycle is regulated by a complicated chemical reaction network. Although many deterministic models have been proposed, stochastic models are desired to capture noise in the cell resulting from low numbers of critical species. However, converting a deterministic model into one that accurately captures stochastic effects can result in a complex model that is hard to build and expensive to simulate. In this paper, we first apply a hybrid (mixed deterministic and stochastic) simulation method to such a stochastic model. With proper partitioning of reactions between deterministic and stochastic simulation methods, the hybrid method generates the same primary characteristics and the same level of noise as Gillespie's stochastic simulation algorithm, but with better efficiency. By studying the results generated by various partitionings of reactions, we developed a new strategy for hybrid stochastic modeling of the cell cycle. The new approach is not limited to using mass-action rate laws. Numerical experiments demonstrate that our approach is consistent with characteristics of noisy cell cycle progression, and yields cell cycle statistics in accord with experimental observations.

Dose-response aligned circuits in signaling systems

Cells use biological signal transduction pathways to respond to environmental stimuli and the behavior of many cell types depends on precise sensing and transmission of external information. A notable property of signal transduction that was characterized in the Saccharomyces cerevisiae yeast cell and many mammalian cells is the alignment of dose-response curves. It was found that the dose response of the receptor matches closely the dose responses of the downstream. This dose-response alignment (DoRA) renders equal sensitivities and concordant responses in different parts of signaling system and guarantees a faithful information transmission. The experimental observations raise interesting questions about the nature of the information transmission through DoRA signaling networks and design principles of signaling systems with this function. Here, we performed an exhaustive computational analysis on network architectures that underlie the DoRA function in simple regulatory networks composed of two and three enzymes. The minimal circuits capable of DoRA were examined with Michaelis-Menten kinetics. Several motifs that are essential for the dynamical function of DoRA were identified. Systematic analysis of the topology space of robust DoRA circuits revealed that, rather than fine-tuning the network's parameters, the function is primarily realized by enzymatic regulations on the controlled node that are constrained in limiting regions of saturation or linearity.

Mechanisms generating bistability and oscillations in microRNA-mediated motifs

The importance of post-transcriptional regulation by microRNAs (miRNAs) has recently been recognized in almost all cellular processes. When participating in cellular processes, miRNAs mainly mediate mRNA degradation or translational repression. Recently computational and experimental studies have identified an abundance of motifs involving miRNAs and transcriptional factors (TFs). The simplest motif is a two-node miRNA-mediated feedback loop (MFL) in which a TF regulates an miRNA and the TF itself is negatively regulated by the miRNA. In this paper we present a general computational model for the MFL based on biochemical regulations and explore its dynamics by using bifurcation analysis. Our results show that the MFL can behave either as switches or as oscillators, depending on the TF as a repressor or an activator. These functional features are consistent with the widespread appearance of miRNAs in fate decisions such as proliferation, differentiation, and apoptosis during development. We found that under the interplay of a TF and an miRNA, the MFL model can behave as switches for wide ranges of parameters even without cooperative binding of the TF. In addition, oscillations induced by the miRNA in the MFL model require neither an additional positive feedback loop, nor self-activation of the gene, nor cooperative binding of the TF, nor saturated degradation. Therefore, the MFL may provide a general network structure to induce bistability or oscillations. It is hoped that the results presented here will provide a new view on how gene expression is regulated by miRNAs and further guidance for experiments. Moreover, the insight gained from this study is also expected to provide a basis for the investigation of more complex networks assembled by simple building blocks.

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.

Patterning embryos with oscillations: structure, function and dynamics of the vertebrate segmentation clock

The segmentation clock is an oscillating genetic network thought to govern the rhythmic and sequential subdivision of the elongating body axis of the vertebrate embryo into somites: the precursors of the segmented vertebral column. Understanding how the rhythmic signal arises, how it achieves precision and how it patterns the embryo remain challenging issues. Recent work has provided evidence of how the period of the segmentation clock is regulated and how this affects the anatomy of the embryo. The ongoing development of real-time clock reporters and mathematical models promise novel insight into the dynamic behavior of the clock.

Bigger, better, faster: principles and models of AKAP anchoring protein signaling

A kinase anchoring proteins (AKAPs) bind multiple signaling proteins and have subcellular targeting domains that allow them to greatly impact cellular signaling. AKAPs localize, specify, amplify, and accelerate signal transduction within the cell by bringing signaling proteins together in space and time. AKAPs also organize higher-order network motifs such as feed forward and feedback loops that may create complex network responses, including adaptation, oscillation, and ultrasensitivity. Computational models have begun to provide an insight into how AKAPs regulate signaling dynamics and cardiovascular pathophysiology. Models of mitogen-activated protein kinase and epidermal growth factor receptor scaffolds have revealed additional design principles and new methods for representing signaling scaffolds mathematically. Coupling computational modeling with quantitative experimental approaches will be increasingly necessary for dissecting the diverse information processing functions performed by AKAP signaling complexes.

Regulation of oscillation dynamics in biochemical systems with dual negative feedback loops

Feedback controls are central to cellular regulation. Negative-feedback mechanisms are well known to underline oscillatory dynamics. However, the presence of multiple negative-feedback mechanisms is common in oscillatory cellular systems, raising intriguing questions of how they cooperate to regulate oscillations. In this work, we studied the dynamical properties of a set of general biochemical motifs with dual, nested negative-feedback structures. We showed analytically and then confirmed numerically that, in these motifs, each negative-feedback loop exhibits distinctly different oscillation-controlling functions. The longer, outer feedback loop was found to promote oscillations, whereas the short, inner loop suppresses and can even eliminate oscillations. We found that the position of the inner loop within the coupled motifs affects its repression strength towards oscillatory dynamics. Bifurcation analysis indicated that emergence of oscillations may be a strict parametric requirement and thus evolutionarily tricky. Investigation of the quantitative features of oscillations (i.e. frequency, amplitude and mean value) revealed that coupling negative feedback provides robust tuning of the oscillation dynamics. Finally, we demonstrated that the mitogen-activated protein kinase (MAPK) cascades also display properties seen in the general nested feedback motifs. The findings and implications in this study provide novel understanding of biochemical negative-feedback regulation in a mixed wiring context.

A symmetric dual feedback system provides a robust and entrainable oscillator

Many organisms have evolved molecular clocks to anticipate daily changes in their environment. The molecular mechanisms by which the circadian clock network produces sustained cycles have extensively been studied and transcriptional-translational feedback loops are common structures to many organisms. Although a simple or single feedback loop is sufficient for sustained oscillations, circadian clocks implement multiple, complicated feedback loops. In general, different types of feedback loops are suggested to affect the robustness and entrainment of circadian rhythms.

To reveal the mechanism by which such a complex feedback system evolves, we quantify the robustness and light entrainment of four competing models: the single, semi-dual, dual, and redundant feedback models. To extract the global properties of those models, all plausible kinetic parameter sets that generate circadian oscillations are searched to characterize their oscillatory features. To efficiently perform such analyses, we used the two-phase search (TPS) method as a fast and non-biased search method and quasi-multiparameter sensitivity (QMPS) as a fast and exact measure of robustness to uncertainty of all kinetic parameters.

So far the redundant feedback model has been regarded as the most robust oscillator, but our extensive analysis corrects or overcomes this hypothesis. The dual feedback model, which is employed in biology, provides the most robust oscillator to multiple parameter perturbations within a cell and most readily entrains to a wide range of light-dark cycles. The kinetic symmetry between the dual loops and their coupling via a protein complex are found critically responsible for robust and entrainable oscillations. We first demonstrate how the dual feedback architecture with kinetic symmetry evolves out of many competing feedback systems.

Inferring Biological Mechanisms by Data-Based Mathematical Modelling: Compartment-Specific Gene Activation during Sporulation in Bacillus subtilis as a Test Case

Biological functionality arises from the complex interactions of simple components. Emerging behaviour is difficult to recognize with verbal models alone, and mathematical approaches are important. Even few interacting components can give rise to a wide range of different responses, that is, sustained, transient, oscillatory, switch-like responses, depending on the values of the model parameters. A quantitative comparison of model predictions and experiments is therefore important to distinguish between competing hypotheses and to judge whether a certain regulatory behaviour is at all possible and plausible given the observed type and strengths of interactions and the speed of reactions. Here I will review a detailed model for the transcription factor σ(F), a regulator of cell differentiation during sporulation in Bacillus subtilis. I will focus in particular on the type of conclusions that can be drawn from detailed, carefully validated models of biological signaling networks. For most systems, such detailed experimental information is currently not available, but accumulating biochemical data through technical advances are likely to enable the detailed modelling of an increasing number of pathways. A major challenge will be the linking of such detailed models and their integration into a multiscale framework to enable their analysis in a larger biological context.

Quantifying the dynamics of coupled networks of switches and oscillators

Complex network dynamics have been analyzed with models of systems of coupled switches or systems of coupled oscillators. However, many complex systems are composed of components with diverse dynamics whose interactions drive the system's evolution. We, therefore, introduce a new modeling framework that describes the dynamics of networks composed of both oscillators and switches. Both oscillator synchronization and switch stability are preserved in these heterogeneous, coupled networks. Furthermore, this model recapitulates the qualitative dynamics for the yeast cell cycle consistent with the hypothesized dynamics resulting from decomposition of the regulatory network into dynamic motifs. Introducing feedback into the cell-cycle network induces qualitative dynamics analogous to limitless replicative potential that is a hallmark of cancer. As a result, the proposed model of switch and oscillator coupling provides the ability to incorporate mechanisms that underlie the synchronized stimulus response ubiquitous in biochemical systems.

Predictive models for cellular signaling networks

This chapter provides an introduction to the formulation and analysis of differential-equation-based models for biological regulatory networks. In the first part, we discuss basic reaction types and the use of mass action kinetics and of simplifying approximations in the development of models for biological signaling. In the second part we introduce phase plane and linear stability analysis to evaluate the time evolution and identify the long-term attractors of dynamic systems. We then discuss the use of bifurcation diagrams to evaluate the parameter dependency of qualitative network behaviors (i.e., the emergence of oscillations or switches), and we give measures for the sensitivity and robustness of the signaling output.

Model-based investigation of the transcriptional activity of p53 and its feedback loop regulation via 14-3-3σ

Experiments have recently shown that p53 expression can display oscillations in response to certain stress signals. In this work, mathematical modelling and bifurcation analysis are combined to investigate under which conditions the oscillation of p53 could propagate to its direct downstream transcription targets. The authors' analysis suggests that oscillations of p53 will propagate only to proteins with medium-fast mRNA and protein turnover rates. The authors retrieved data concerning the half-life of mRNA and protein for a number of p53-promoted genes and found that, according to their model, most of them are not able to inherit the oscillation of p53 because of their slow turnover rates. However, their analysis indicates that p53 oscillation may actually fine-tune the expression pattern of a protein when it is integrated with a second oscillatory signal. The authors also consider the case of additional regulatory loops affecting p53 oscillations and involving proteins transcriptionally induced by p53. Their results for 14-3-3σ, a protein that targets the p53 inhibitor MDM2 for degradation, suggest that the addition of feedback-loop regulation may modulate basic properties of p53 oscillation and induce quick cessation of them under certain physiological conditions. Moreover, the interplay between DNA damage and 14-3-3σ may induce bistability in the oscillation of p53.

Information routing driven by background chatter in a signaling network

Living systems are capable of processing multiple sources of information simultaneously. This is true even at the cellular level, where not only coexisting signals stimulate the cell, but also the presence of fluctuating conditions is significant. When information is received by a cell signaling network via one specific input, the existence of other stimuli can provide a background activity –or chatter– that may affect signal transmission through the network and, therefore, the response of the cell. Here we study the modulation of information processing by chatter in the signaling network of a human cell, specifically, in a Boolean model of the signal transduction network of a fibroblast. We observe that the level of external chatter shapes the response of the system to information carrying signals in a nontrivial manner, modulates the activity levels of the network outputs, and effectively determines the paths of information flow. Our results show that the interactions and node dynamics, far from being random, confer versatility to the signaling network and allow transitions between different information-processing scenarios.

Far from being silent and static, the habitat of a cell is usually composed by multiple and simultaneous signals. We can consider nutrients, hormones, temperature, light, and other stimuli as elements building a default environment in which cells grow, divide and die. This environment, which has an intrinsically fluctuating nature, is the setting in which cells process all incoming stimuli. Here we examine the role that this background activity –or signaling chatter– plays in the transmission of information in a typical human cell. We address this question using a cellular model of signal transduction that we simulate using both random and periodic stimuli. We find that the level of background chatter determines the response of the whole signaling network to external stimuli. Different areas of the network are activated by specific levels of background activity, routing the information through chatter-dependent paths. In this way, different levels of chatter allow the network to select between different responses, given the same stimulus. These features depend on the architecture and functional connectivity of a truly biological network, since we find that randomized versions of the model are incapable of showing this behavior.

A whole-body model for glycogen regulation reveals a critical role for substrate cycling in maintaining blood glucose homeostasis

Timely, and sometimes rapid, metabolic adaptation to changes in food supply is critical for survival as an organism moves from the fasted to the fed state, and vice versa. These transitions necessitate major metabolic changes to maintain energy homeostasis as the source of blood glucose moves away from ingested carbohydrates, through hepatic glycogen stores, towards gluconeogenesis. The integration of hepatic glycogen regulation with extra-hepatic energetics is a key aspect of these adaptive mechanisms. Here we use computational modeling to explore hepatic glycogen regulation under fed and fasting conditions in the context of a whole-body model. The model was validated against previous experimental results concerning glycogen phosphorylase a (active) and glycogen synthase a dynamics. The model qualitatively reproduced physiological changes that occur during transition from the fed to the fasted state. Analysis of the model reveals a critical role for the inhibition of glycogen synthase phosphatase by glycogen phosphorylase a. This negative regulation leads to high levels of glycogen synthase activity during fasting conditions, which in turn increases substrate (futile) cycling, priming the system for a rapid response once an external source of glucose is restored. This work demonstrates that a mechanistic understanding of the design principles used by metabolic control circuits to maintain homeostasis can benefit from the incorporation of mathematical descriptions of these networks into “whole-body” contextual models that mimic in vivo conditions.

Homeostasis of blood glucose concentrations during circadian shifts in survival-related activities, sleep and food availability is crucial for the survival of mammals. This process depends upon glucose intake, short-term storage as glycogen, and gluconeogenesis. The integration of hepatic glycogen anabolic and catabolic dynamics with whole body energetics is critical for survival. In this paper we use computational modeling to investigate the potential survival advantage of substrate (futile) cycling of glycogen and glycogen precursors. Our simulations, combined with published experimental results of other researchers, indicate that as the body enters a state of fasting, the activity of enzymes involved in the synthesis of glycogen increases leading to increased substrate cycling. This increase in substrate cycling allows the system to respond more rapidly once new external sources of glucose become available. The whole-body computational model developed for this work allows the metabolic control circuitry to be studied under simulated in vivo conditions, providing functional insights that are not evident when individual modules of glycogen regulatory circuitry are examined in isolation.

A simple negative interaction in the positive transcriptional feedback of a single gene is sufficient to produce reliable oscillations

Negative and positive transcriptional feedback loops are present in natural and synthetic genetic oscillators. A single gene with negative transcriptional feedback needs a time delay and sufficiently strong nonlinearity in the transmission of the feedback signal in order to produce biochemical rhythms. A single gene with only positive transcriptional feedback does not produce oscillations. Here, we demonstrate that this single-gene network in conjunction with a simple negative interaction can also easily produce rhythms. We examine a model comprised of two well-differentiated parts. The first is a positive feedback created by a protein that binds to the promoter of its own gene and activates the transcription. The second is a negative interaction in which a repressor molecule prevents this protein from binding to its promoter. A stochastic study shows that the system is robust to noise. A deterministic study identifies that the dynamics of the oscillator are mainly driven by two types of biomolecules: the protein, and the complex formed by the repressor and this protein. The main conclusion of this paper is that a simple and usual negative interaction, such as degradation, sequestration or inhibition, acting on the positive transcriptional feedback of a single gene is a sufficient condition to produce reliable oscillations. One gene is enough and the positive transcriptional feedback signal does not need to activate a second repressor gene. This means that at the genetic level an explicit negative feedback loop is not necessary. The model needs neither cooperative binding reactions nor the formation of protein multimers. Therefore, our findings could help to clarify the design principles of cellular clocks and constitute a new efficient tool for engineering synthetic genetic oscillators.

Plasticity of TGF-β signaling

Background

The family of TGF-β ligands is large and its members are involved in many different signaling processes. These signaling processes strongly differ in type with TGF-β ligands eliciting both sustained or transient responses. Members of the TGF-β family can also act as morphogen and cellular responses would then be expected to provide a direct read-out of the extracellular ligand concentration. A number of different models have been proposed to reconcile these different behaviours. We were interested to define the set of minimal modifications that are required to change the type of signal processing in the TGF-β signaling network.

Results

To define the key aspects for signaling plasticity we focused on the core of the TGF-β signaling network. With the help of a parameter screen we identified ranges of kinetic parameters and protein concentrations that give rise to transient, sustained, or oscillatory responses to constant stimuli, as well as those parameter ranges that enable a proportional response to time-varying ligand concentrations (as expected in the read-out of morphogens). A combination of a strong negative feedback and fast shuttling to the nucleus biases signaling to a transient rather than a sustained response, while oscillations were obtained if ligand binding to the receptor is weak and the turn-over of the I-Smad is fast. A proportional read-out required inefficient receptor activation in addition to a low affinity of receptor-ligand binding. We find that targeted modification of single parameters suffices to alter the response type. The intensity of a constant signal (i.e. the ligand concentration), on the other hand, affected only the strength but not the type of the response.

Conclusions

The architecture of the TGF-β pathway enables the observed signaling plasticity. The observed range of signaling outputs to TGF-β ligand in different cell types and under different conditions can be explained with differences in cellular protein concentrations and with changes in effective rate constants due to cross-talk with other signaling pathways. It will be interesting to uncover the exact cellular differences as well as the details of the cross-talks in future work.

MicroRNA-mediated positive feedback loop and optimized bistable switch in a cancer network Involving miR-17-92

MicroRNAs (miRNAs) are small, noncoding RNAs that play an important role in many key biological processes, including development, cell differentiation, the cell cycle and apoptosis, as central post-transcriptional regulators of gene expression. Recent studies have shown that miRNAs can act as oncogenes and tumor suppressors depending on the context. The present work focuses on the physiological significance of miRNAs and their role in regulating the switching behavior. We illustrate an abstract model of the Myc/E2F/miR-17-92 network presented by Aguda et al. (2008), which is composed of coupling between the E2F/Myc positive feedback loops and the E2F/Myc/miR-17-92 negative feedback loop. By systematically analyzing the network in close association with plausible experimental parameters, we show that, in the presence of miRNAs, the system bistability emerges from the system, with a bistable switch and a one-way switch presented by Aguda et al. instead of a single one-way switch. Moreover, the miRNAs can optimize the switching process. The model produces a diverse array of response-signal behaviors in response to various potential regulating scenarios. The model predicts that this transition exists, one from cell death or the cancerous phenotype directly to cell quiescence, due to the existence of miRNAs. It was also found that the network involving miR-17-92 exhibits high noise sensitivity due to a positive feedback loop and also maintains resistance to noise from a negative feedback loop.

Modeling circadian clocks: Roles, advantages, and limitations

Circadian rhythms are endogenous oscillations characterized by a period of about 24h. They constitute the biological rhythms with the longest period known to be generated at the molecular level. The abundance of genetic information and the complexity of the molecular circuitry make circadian clocks a system of choice for theoretical studies. Many mathematical models have been proposed to understand the molecular regulatory mechanisms that underly these circadian oscillations and to account for their dynamic properties (temperature compensation, entrainment by light dark cycles, phase shifts by light pulses, rhythm splitting, robustness to molecular noise, intercellular synchronization). The roles and advantages of modeling are discussed and illustrated using a variety of selected examples. This survey will lead to the proposal of an integrated view of the circadian system in which various aspects (interlocked feedback loops, inter-cellular coupling, and stochasticity) should be considered together to understand the design and the dynamics of circadian clocks. Some limitations of these models are commented and challenges for the future identified.

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.

Efficient characterization of high-dimensional parameter spaces for systems biology

Background

A biological system's robustness to mutations and its evolution are influenced by the structure of its viable space, the region of its space of biochemical parameters where it can exert its function. In systems with a large number of biochemical parameters, viable regions with potentially complex geometries fill a tiny fraction of the whole parameter space. This hampers explorations of the viable space based on "brute force" or Gaussian sampling.

Results

We here propose a novel algorithm to characterize viable spaces efficiently. The algorithm combines global and local explorations of a parameter space. The global exploration involves an out-of-equilibrium adaptive Metropolis Monte Carlo method aimed at identifying poorly connected viable regions. The local exploration then samples these regions in detail by a method we call multiple ellipsoid-based sampling. Our algorithm explores efficiently nonconvex and poorly connected viable regions of different test-problems. Most importantly, its computational effort scales linearly with the number of dimensions, in contrast to "brute force" sampling that shows an exponential dependence on the number of dimensions. We also apply this algorithm to a simplified model of a biochemical oscillator with positive and negative feedback loops. A detailed characterization of the model's viable space captures well known structural properties of circadian oscillators. Concretely, we find that model topologies with an essential negative feedback loop and a nonessential positive feedback loop provide the most robust fixed period oscillations. Moreover, the connectedness of the model's viable space suggests that biochemical oscillators with varying topologies can evolve from one another.

Conclusions

Our algorithm permits an efficient analysis of high-dimensional, nonconvex, and poorly connected viable spaces characteristic of complex biological circuitry. It allows a systematic use of robustness as a tool for model discrimination.

Molecular level dynamics of genetic oscillator--the effect of protein-protein interaction

Uncovering how interactions of a set of molecular components influence the system's dynamic behavior is important for understanding intracellular processes and elucidating design principles, but unfortunately, there are limited efforts for studying this issue. Here, we study the effect of distinct post-translational dynamics controlled by protein dimerization on oscillations in the repressilator. For this, we propose three biologically motivated model scenarios of the repressilator with monomer or dimer being the active form of repressor, and with protein-protein interactions. It is found that the dimer dissociation constant can tune oscillatory regions, frequency and amplitude. Introducing a modified linear noise approximation to evaluate fluctuations of amplitude and period in the oscillatory systems, we show that different dimerization leads to a different effect on period and amplitude in reducing noise. The manipulation of the circuit's biochemical properties provides a practical strategy for designing a robust and tunable oscillator.

Does the potential for chaos constrain the embryonic cell-cycle oscillator?

Although many of the core components of the embryonic cell-cycle network have been elucidated, the question of how embryos achieve robust, synchronous cellular divisions post-fertilization remains unexplored. What are the different schemes that could be implemented by the embryo to achieve synchronization? By extending a cell-cycle model previously developed for embryos of the frog Xenopus laevis to include the spatial dimensions of the embryo, we establish a novel role for the rapid, fertilization-initiated calcium wave that triggers cell-cycle oscillations. Specifically, in our simulations a fast calcium wave results in synchronized cell cycles, while a slow wave results in full-blown spatio-temporal chaos. We show that such chaos would ultimately lead to an unpredictable patchwork of cell divisions across the embryo. Given this potential for chaos, our results indicate a novel design principle whereby the fast calcium-wave trigger following embryo fertilization synchronizes cell divisions.

Cell divisions across an embryo occur in rapid synchrony - like clockwork - starting within minutes of fertilization. How does an embryo achieve this remarkable uniformity? Simple diffusion is too slow: typical proteins diffuse with a rate of 10 µm²/s, requiring nearly 14 hours to traverse a 1 mm embryo. An exciting idea is that the embryo is an active medium, much like the heart where pulses of electrical activity result in organized contractions. However, just as the heart can have arrhythmias, our model predicts that oscillations in the embryo can become chaotic. What would be the biological consequences of this behavior? How do embryos avoid chaos? Our work provides potential answers to these questions: Chaos would lead to an unpredictable patchwork of cell divisions across the embryo - clearly a fatal defect in development. To avoid chaos then, we predict that cell-cycle oscillations need to be triggered throughout the embryo at almost precisely the same time. The threat that chaos will mar development therefore explains the mystery of why embryos universally employ a fast calcium wave to trigger cell-cycle oscillations. In this way, developing organisms get the synchronizing benefits of an active medium without suffering the destructive consequences of chaotic arrhythmias.

Robust entrainment of circadian oscillators requires specific phase response curves

The circadian clocks keeping time in many living organisms rely on self-sustained biochemical oscillations entrained by external cues, such as light, to the 24-h cycle induced by Earth's rotation. However, environmental cues are unreliable due to the variability of habitats, weather conditions, or cue-sensing mechanisms among individuals. A tempting hypothesis is that circadian clocks have evolved so as to be robust to fluctuations in the signal that entrains them. To support this hypothesis, we analyze the synchronization behavior of weakly and periodically forced oscillators in terms of their phase response curve (PRC), which measures phase changes induced by a perturbation applied at different times of the cycle. We establish a general relationship between the robustness of key entrainment properties, such as stability and oscillator phase, on the one hand, and the shape of the PRC as characterized by a specific curvature or the existence of a dead zone, on the other hand. The criteria obtained are applied to computational models of circadian clocks and account for the disparate robustness properties of various forcing schemes. Finally, the analysis of PRCs measured experimentally in several organisms strongly suggests a case of convergent evolution toward an optimal strategy for maintaining a clock that is accurate and robust to environmental fluctuations.

Modeling the cell cycle: why do certain circuits oscillate?

Computational modeling and the theory of nonlinear dynamical systems allow one to not simply describe the events of the cell cycle, but also to understand why these events occur, just as the theory of gravitation allows one to understand why cannonballs fly in parabolic arcs. The simplest examples of the eukaryotic cell cycle operate like autonomous oscillators. Here, we present the basic theory of oscillatory biochemical circuits in the context of the Xenopus embryonic cell cycle. We examine Boolean models, delay differential equation models, and especially ordinary differential equation (ODE) models. For ODE models, we explore what it takes to get oscillations out of two simple types of circuits (negative feedback loops and coupled positive and negative feedback loops). Finally, we review the procedures of linear stability analysis, which allow one to determine whether a given ODE model and a particular set of kinetic parameters will produce oscillations.

Optimizing periodicity and polymodality in noise-induced genetic oscillators

Many cellular functions are based on the rhythmic organization of biological processes into self-repeating cascades of events. Some of these periodic processes, such as the cell cycles of several species, exhibit conspicuous irregularities in the form of period skippings, which lead to polymodal distributions of cycle lengths. A recently proposed mechanism that accounts for this quantized behavior is the stabilization of a Hopf-unstable state by molecular noise. Here we investigate the effect of varying noise in a model system, namely an excitable activator-repressor genetic circuit, that displays this noise-induced stabilization effect. Our results show that an optimal noise level enhances the regularity (coherence) of the cycles, in a form of coherence resonance. Similar noise levels also optimize the multimodal nature of the cycle lengths. Together, these results illustrate how molecular noise within a minimal gene regulatory motif confers robust generation of polymodal patterns of periodicity.

Programming an in vitro DNA oscillator using a molecular networking strategy

Living organisms perform and control complex behaviours by using webs of chemical reactions organized in precise networks. This powerful system concept, which is at the very core of biology, has recently become a new foundation for bioengineering. Remarkably, however, it is still extremely difficult to rationally create such network architectures in artificial, non-living and well-controlled settings. We introduce here a method for such a purpose, on the basis of standard DNA biochemistry. This approach is demonstrated by assembling de novo an efficient chemical oscillator: we encode the wiring of the corresponding network in the sequence of small DNA templates and obtain the predicted dynamics. Our results show that the rational cascading of standard elements opens the possibility to implement complex behaviours in vitro. Because of the simple and well-controlled environment, the corresponding chemical network is easily amenable to quantitative mathematical analysis. These synthetic systems may thus accelerate our understanding of the underlying principles of biological dynamic modules.

Min protein patterns emerge from rapid rebinding and membrane interaction of MinE

In Escherichia coli, the pole-to-pole oscillation of the Min proteins directs septum formation to midcell, which is required for symmetric cell division. In vitro, protein waves emerge from the self-organization of MinD, a membrane-binding ATPase, and its activator MinE. For wave propagation, the proteins need to cycle through states of collective membrane binding and unbinding. Although MinD presumably undergoes cooperative membrane attachment, it is unclear how synchronous detachment is coordinated. We used confocal and single-molecule microscopy to elucidate the order of events during Min wave propagation. We propose that protein detachment at the rear of the wave, and the formation of the E-ring, are accomplished by two complementary processes: first, local accumulation of MinE due to rapid rebinding, leading to dynamic instability; and second, a structural change induced by membrane-interaction of MinE in an equimolar MinD-MinE (MinDE) complex, which supports the robustness of pattern formation.

Bistability and oscillations in gene regulation mediated by small noncoding RNAs

The interplay of small noncoding RNAs (sRNAs), mRNAs, and proteins has been shown to play crucial roles in almost all cellular processes. As key post-transcriptional regulators of gene expression, the mechanisms and roles of sRNAs in various cellular processes still need to be fully understood. When participating in cellular processes, sRNAs mainly mediate mRNA degradation or translational repression. Here, we show how the dynamics of two minimal architectures is drastically affected by these two mechanisms. A comparison is also given to reveal the implication of the fundamental differences. This study may help us to analyze complex networks assembled by simple modules more easily. A better knowledge of the sRNA-mediated motifs is also of interest for bio-engineering and artificial control.

Signal processing in cellular clocks

Many biochemical events within a cell need to be timed properly to occur at specific times of day, after other events have happened within the cell or in response to environmental signals. The cellular biochemical feedback loops that time these events have already received much recent attention in the experimental and modeling communities. Here, we show how ideas from signal processing can be applied to understand the function of these clocks. Consider two signals from the network s(t) and r(t), either two variables of a model or two experimentally measured time courses. We show how s(t) can be decomposed into two parts, the first being a function of r(t), and the second the derivative of a function of r(t). Geometric principles are then derived that can be used to understand when oscillations appear in biochemical feedback loops, the period of these oscillations, and their time course. Specific examples of this theory are provided that show how certain networks are prone or not prone to oscillate, how individual biochemical processes affect the period, and how oscillations in one chemical species can be deduced from oscillations in other parts of the network.

Computational modeling of mammalian signaling networks

One of the most exciting developments in signal transduction research has been the proliferation of studies in which a biological discovery was initiated by computational modeling. In this study, we review the major efforts that enable such studies. First, we describe the experimental technologies that are generally used to identify the molecular components and interactions in, and dynamic behavior exhibited by, a network of interest. Next, we review the mathematical approaches that are used to model signaling network behavior. Finally, we focus on three specific instances of 'model-driven discovery': cases in which computational modeling of a signaling network has led to new insights that have been verified experimentally.

Delay in feedback repression by cryptochrome 1 is required for circadian clock function

Direct evidence for the requirement of delay in feedback repression in the mammalian circadian clock has been elusive. Cryptochrome 1 (Cry1), an essential clock component, displays evening-time expression and serves as a strong repressor at morning-time elements (E box/E' box). In this study, we reveal that a combination of day-time elements (D box) within the Cry1-proximal promoter and night-time elements (RREs) within its intronic enhancer gives rise to evening-time expression. A synthetic composite promoter produced evening-time expression, which was further recapitulated by a simple phase-vector model. Of note, coordination of day-time with night-time elements can modulate the extent of phase delay. A genetic complementation assay in Cry1(-/-):Cry2(-/-) cells revealed that substantial delay of Cry1 expression is required to restore circadian rhythmicity, and its prolonged delay slows circadian oscillation. Taken together, our data suggest that phase delay in Cry1 transcription is required for mammalian clock function.

Automated analysis of biological oscillator models using mode decomposition

MOTIVATION

Oscillating signals produced by biological systems have shapes, described by their Fourier spectra, that can potentially reveal the mechanisms that generate them. Extracting this information from measured signals is interesting for the validation of theoretical models, discovery and classification of interaction types, and for optimal experiment design.

RESULTS

An automated workflow is described for the analysis of oscillating signals. A software package is developed to match signal shapes to hundreds of a priori viable model structures defined by a class of first-order differential equations. The package computes parameter values for each model by exploiting the mode decomposition of oscillating signals and formulating the matching problem in terms of systems of simultaneous polynomial equations. On the basis of the computed parameter values, the software returns a list of models consistent with the data. In validation tests with synthetic datasets, it not only shortlists those model structures used to generate the data but also shows that excellent fits can sometimes be achieved with alternative equations. The listing of all consistent equations is indicative of how further invalidation might be achieved with additional information. When applied to data from a microarray experiment on mice, the procedure finds several candidate model structures to describe interactions related to the circadian rhythm. This shows that experimental data on oscillators is indeed rich in information about gene regulation mechanisms.

AVAILABILITY

The software package is available at http://babylone.ulb.ac.be/autoosc/.

Saccharomyces cerevisiae as a model organism: a comparative study

BACKGROUND

Model organisms are used for research because they provide a framework on which to develop and optimize methods that facilitate and standardize analysis. Such organisms should be representative of the living beings for which they are to serve as proxy. However, in practice, a model organism is often selected ad hoc, and without considering its representativeness, because a systematic and rational method to include this consideration in the selection process is still lacking.

METHODOLOGY/PRINCIPAL FINDINGS

In this work we propose such a method and apply it in a pilot study of strengths and limitations of Saccharomyces cerevisiae as a model organism. The method relies on the functional classification of proteins into different biological pathways and processes and on full proteome comparisons between the putative model organism and other organisms for which we would like to extrapolate results. Here we compare S. cerevisiae to 704 other organisms from various phyla. For each organism, our results identify the pathways and processes for which S. cerevisiae is predicted to be a good model to extrapolate from. We find that animals in general and Homo sapiens in particular are some of the non-fungal organisms for which S. cerevisiae is likely to be a good model in which to study a significant fraction of common biological processes. We validate our approach by correctly predicting which organisms are phenotypically more distant from S. cerevisiae with respect to several different biological processes.

CONCLUSIONS/SIGNIFICANCE

The method we propose could be used to choose appropriate substitute model organisms for the study of biological processes in other species that are harder to study. For example, one could identify appropriate models to study either pathologies in humans or specific biological processes in species with a long development time, such as plants.

Synthetic in vitro transcriptional oscillators

The construction of synthetic biochemical circuits from simple components illuminates how complex behaviors can arise in chemistry and builds a foundation for future biological technologies. A simplified analog of genetic regulatory networks, in vitro transcriptional circuits, provides a modular platform for the systematic construction of arbitrary circuits and requires only two essential enzymes, bacteriophage T7 RNA polymerase and Escherichia coli ribonuclease H, to produce and degrade RNA signals. In this study, we design and experimentally demonstrate three transcriptional oscillators in vitro. First, a negative feedback oscillator comprising two switches, regulated by excitatory and inhibitory RNA signals, showed up to five complete cycles. To demonstrate modularity and to explore the design space further, a positive-feedback loop was added that modulates and extends the oscillatory regime. Finally, a three-switch ring oscillator was constructed and analyzed. Mathematical modeling guided the design process, identified experimental conditions likely to yield oscillations, and explained the system's robust response to interference by short degradation products. Synthetic transcriptional oscillators could prove valuable for systematic exploration of biochemical circuit design principles and for controlling nanoscale devices and orchestrating processes within artificial cells.

Minimum protein oscillator based on multisite phosphorylation∕dephosphorylation

The authors propose a novel minimum oscillator whereby a protein with multiple phosphorylation sites directly embedded in a negative feedback loop can exhibit oscillation. They demonstrate that if the fully phosphorylated substrate inhibits the first phosphorylation step in a cooperative manner, multisite substrates can exhibit oscillatory behaviour at the presence of a kinase and phosphatase. With a fixed number of sites, the non-linearity of the negative feedback and the substrate∕enzyme ratio must be above certain threshold values to generate undamped oscillation. There is an inverse relationship between the number of phosphorylation sites and the minimum non-linearity of the negative feedback required for oscillation; that is, the ultrasensitivity and time delay rooted in multisite phosphorylation compensate for the explicit non-linearity in the negative feedback. The period and amplitude of oscillation are mainly determined by the number of phosphorylation sites and the substrate∕enzyme ratio. The authors' results suggest that a multisite protein can be exploited for the construction of a synthetic protein oscillator featuring simplicity, robustness and tunability.

A comprehensive molecular interaction map of the budding yeast cell cycle

With the accumulation of data on complex molecular machineries coordinating cell-cycle dynamics, coupled with its central function in disease patho-physiologies, it is becoming increasingly important to collate the disparate knowledge sources into a comprehensive molecular network amenable to systems-level analyses. In this work, we present a comprehensive map of the budding yeast cell-cycle, curating reactions from ∼600 original papers. Toward leveraging the map as a framework to explore the underlying network architecture, we abstract the molecular components into three planes—signaling, cell-cycle core and structural planes. The planar view together with topological analyses facilitates network-centric identification of functions and control mechanisms. Further, we perform a comparative motif analysis to identify around 194 motifs including feed-forward, mutual inhibitory and feedback mechanisms contributing to cell-cycle robustness. We envisage the open access, comprehensive cell-cycle map to open roads toward community-based deeper understanding of cell-cycle dynamics.

Models at the single cell level

Many cellular behaviors cannot be completely captured or appropriately described at the cell population level. Noise induced by stochastic chemical reactions, spatially polarized signaling networks, and heterogeneous cell-cell communication are among the many phenomena that require fine-grained analysis. Accordingly, the mathematical models used to describe such systems must be capable of single cell or subcellular resolution. Here, we review techniques for modeling single cells, including models of stochastic chemical kinetics, spatially heterogeneous intracellular signaling, and spatial stochastic systems. We also briefly discuss applications of each type of model.

Physical oncology: a bench-to-bedside quantitative and predictive approach

Cancer models relating basic science to clinical care in oncology may fail to address the nuances of tumor behavior and therapy, as in the case, discussed herein, of the complex multiscale dynamics leading to the often-observed enhanced invasiveness, paradoxically induced by the very antiangiogenic therapy designed to destroy the tumor. Studies would benefit from approaches that quantitatively link the multiple physical and temporal scales from molecule to tissue in order to offer outcome predictions for individual patients. Physical oncology is an approach that applies fundamental principles from the physical and biological sciences to explain certain cancer behaviors as observable characteristics arising from the underlying physical and biochemical events. For example, the transport of oxygen molecules through tissue affects phenotypic characteristics such as cell proliferation, apoptosis, and adhesion, which in turn underlie the patient-scale tumor growth and invasiveness. Our review of physical oncology illustrates how tumor behavior and treatment response may be a quantifiable function of marginally stable molecular and/or cellular conditions modulated by inhomogeneity. By incorporating patient-specific genomic, proteomic, metabolomic, and cellular data into multiscale physical models, physical oncology could complement current clinical practice through enhanced understanding of cancer behavior, thus potentially improving patient survival.

Spatial and temporal information coding and noise in the NF-κB system

NF-κB (nuclear factor κB) regulates cellular stress and the immune responses to infection. Its activation results in oscillations in nuclear NF-κB abundance. We treated cells with repeated short pulses of TNFα (tumour necrosis factor α) at various intervals to mimic pulsatile inflammatory signals. At all pulse intervals analysed, we observed synchronous cycles of NF-κB nuclear translocation. Lower frequency stimulations gave repeated full-amplitude translocations, whereas higher frequency pulses gave translocations with reduced amplitude, indicating that the system failed to reset completely. Deterministic and stochastic mathematical models predicted how negative feedback loops might regulate both system resetting and cellular heterogeneity. Altering the stimulation interval gave different patterns of NF-κB-dependent gene expression, supporting a functional role for oscillation frequency. The causes of cell-to-cell variation and the possible functions of these processes in cells and tissues are discussed. The NF-κB system is just one of a number of known biological oscillators that include calcium signalling, transcription cycles, p53, the segmentation clock, the circadian clock, the cell cycle and seasonal rhythms. The way such cycles are integrated could be part of the answer as to how organisms achieve complexity while retaining the robustness of cellular decision-making processes.

Systems-level feedback in cell-cycle control

Alternation of chromosome replication and segregation is essential for successful completion of the cell cycle and it requires an oscillation of Cdk1 (cyclin-dependent kinase 1)-CycB (cyclin B) activity. In the present review, we illustrate the essential features of checkpoint controlled and uncontrolled cell-cycle oscillations by using mechanical metaphors. Despite variations in the molecular details of the oscillatory mechanism, the underlying network motifs responsible for the oscillations are always well-conserved. The checkpoint-controlled cell cycles are always driven by a negative-feedback loop amplified by double-negative feedbacks (antagonism).

Time-series clustering of gene expression in irradiated and bystander fibroblasts: an application of FBPA clustering

Background

The radiation bystander effect is an important component of the overall biological response of tissues and organisms to ionizing radiation, but the signaling mechanisms between irradiated and non-irradiated bystander cells are not fully understood. In this study, we measured a time-series of gene expression after α-particle irradiation and applied the Feature Based Partitioning around medoids Algorithm (FBPA), a new clustering method suitable for sparse time series, to identify signaling modules that act in concert in the response to direct irradiation and bystander signaling. We compared our results with those of an alternate clustering method, Short Time series Expression Miner (STEM).

Results

While computational evaluations of both clustering results were similar, FBPA provided more biological insight. After irradiation, gene clusters were enriched for signal transduction, cell cycle/cell death and inflammation/immunity processes; but only FBPA separated clusters by function. In bystanders, gene clusters were enriched for cell communication/motility, signal transduction and inflammation processes; but biological functions did not separate as clearly with either clustering method as they did in irradiated samples. Network analysis confirmed p53 and NF-κB transcription factor-regulated gene clusters in irradiated and bystander cells and suggested novel regulators, such as KDM5B/JARID1B (lysine (K)-specific demethylase 5B) and HDACs (histone deacetylases), which could epigenetically coordinate gene expression after irradiation.

Conclusions

In this study, we have shown that a new time series clustering method, FBPA, can provide new leads to the mechanisms regulating the dynamic cellular response to radiation. The findings implicate epigenetic control of gene expression in addition to transcription factor networks.

From Physics to Pharmacology?

Over the last fifty years there has been an explosion of biological data, leading to the realization that to fully explain biological mechanisms it is necessary to interpret them as complex dynamical systems. The first stage of this interpretation is to determine which components (proteins, genes or metabolites) of the system interact. This is usually represented by a graph, or network. The behavior of this network can then be investigated using mathematical modeling. In vivo these biological networks show several remarkable (and seemingly paradoxical) properties including robustness, plasticity and sensitivity. Erroneous behavior of these networks is often associated with disease. Hence understanding the system-level properties can have important implications for the treatment of disease. Systems biology is an organized approach to quantitatively describe and elucidate the behavior of these complex networks. This review focuses on the progress and future challenges of a systems approach to biology.