Design principles of biochemical oscillators

Modelling emergence of oscillations in communicating bacteria: a structured approach from one to many cells

Population-level measurements of phenotypic behaviour in biological systems may not necessarily reflect individual cell behaviour. To assess qualitative changes in the behaviour of a single cell, when alone and when part of a community, we developed an agent-based model describing the metabolic states of a population of quorum-coupled cells. The modelling is motivated by published experimental work of a synthetic genetic regulatory network (GRN) used in Escherichia coli cells that exhibit oscillatory behaviour across the population. To decipher the mechanisms underlying oscillations in the system, we investigate the behaviour of the model via numerical simulation and bifurcation analysis. In particular, we study the effect of an increase in population size as well as the spatio-temporal behaviour of the model. Our results demonstrate that oscillations are possible only in the presence of a high concentration of the coupling chemical and are due to a time scale separation in key regulatory components of the system. The model suggests that the population establishes oscillatory behaviour as the system's preferred stable state. This is achieved via an effective increase in coupling across the population. We conclude that population effects in GRN design need to be taken into consideration and be part of the design process. This is important in planning intervention strategies or designing specific cell behaviours.

Cell cycle control across the eukaryotic kingdom

Almost two billion years of evolution have generated a vast and amazing variety of eukaryotic life with approximately 8.7 million extant species. Growth and reproduction of all of these organisms depend on faithful duplication and distribution of their chromosomes to the newly forming daughter cells in a process called the cell cycle. However, most of what is known today about cell cycle control comes from a few model species that belong to the unikonts; that is, to only one of five 'supergroups' that comprise the eukaryotic kingdom. Recently, analyzing species from distantly related clades is providing insights into general principles of cell cycle regulation and shedding light on its evolution. Here, referring to animal and fungal as opposed to non-unikont systems, especially flowering plants from the archaeplastid supergroup, we compare the conservation of central cell cycle regulator functions, the structure of network topologies, and the evolutionary dynamics of substrates of core cell cycle kinases.

Enhanced entrainability of genetic oscillators by period mismatch

Biological oscillators coordinate individual cellular components so that they function coherently and collectively. They are typically composed of multiple feedback loops, and period mismatch is unavoidable in biological implementations. We investigated the advantageous effect of this period mismatch in terms of a synchronization response to external stimuli. Specifically, we considered two fundamental models of genetic circuits: smooth and relaxation oscillators. Using phase reduction and Floquet multipliers, we numerically analysed their entrainability under different coupling strengths and period ratios. We found that a period mismatch induces better entrainment in both types of oscillator; the enhancement occurs in the vicinity of the bifurcation on their limit cycles. In the smooth oscillator, the optimal period ratio for the enhancement coincides with the experimentally observed ratio, which suggests biological exploitation of the period mismatch. Although the origin of multiple feedback loops is often explained as a passive mechanism to ensure robustness against perturbation, we study the active benefits of the period mismatch, which include increasing the efficiency of the genetic oscillators. Our findings show a qualitatively different perspective for both the inherent advantages of multiple loops and their essentiality.

Phase responses of oscillating components in a signaling pathway

Signal transduction pathways control various events in mammalian cells such as growth, proliferation, differentiation, apoptosis, or migration in response to environmental stimuli. Because of their importance, the activity of signaling pathways is controlled by multiple modes of positive and negative feedback regulation. Although negative feedback regulation primarily functions to stabilize a system, it also becomes a source of emerging oscillations. For example, the oscillatory behavior of mitogen-activated protein kinase (MAPK) activity has been theoretically proposed earlier and experimentally verified recently. However, the physiological function of such oscillatory behavior in biological systems remains unclear. To understand the functional aspects of this behavior, one should analyze the oscillation dynamics from a mathematical point of view. In this study, we applied the phase reduction method to two simple, structurally similar phosphorylation-dephosphorylation cycle models with negative feedback loops (Models A and B) and a MAPK cascade model, whose dynamics all show oscillation. We found that all three models we tested have a Type II phase response. In addition, we found that when a pair of each models A and B coupled through a weak diffusion interaction, they could synchronize with a zero phase difference. A pair of MAPK cascade models also showed synchronous oscillation, however, when a time delay was introduced into the coupling, it showed an asynchronous response. These results imply that structurally similar or even identical biological oscillators can produce differentiated dynamics in response to external perturbations when the cellular environment is altered. Synchronous or asynchronous oscillation may add strength to or dampen the efficiency of signal propagation, depending on subcellular distances and cell density. Phase response analysis allows prediction of dynamics changes in oscillations in multiple cellular environments.

Design principles of regulatory networks: searching for the molecular algorithms of the cell

A challenge in biology is to understand how complex molecular networks in the cell execute sophisticated regulatory functions. Here we explore the idea that there are common and general principles that link network structures to biological functions, principles that constrain the design solutions that evolution can converge upon for accomplishing a given cellular task. We describe approaches for classifying networks based on abstract architectures and functions, rather than on the specific molecular components of the networks. For any common regulatory task, can we define the space of all possible molecular solutions? Such inverse approaches might ultimately allow the assembly of a design table of core molecular algorithms that could serve as a guide for building synthetic networks and modulating disease networks.

Delay-managed tradeoff in the molecular dynamics of the segmentation clock

The molecular segmentation clock is a complex regulatory network that governs the periodic somite segmentation in vertebrate embryos. Underlying the rhythm of the segmentation clock is a single-cell level pace-making circuit, where inevitable molecular noise and time delay impose normal operating constraints to the pace-making. However, how the molecular mechanisms of the core circuit of the segmentation clock coordinate the operating constraints and maintain the rhythmic nature of the developmental process remains poorly understood. To address this question, we construct two biologically-motivated mathematical models with multiple clock protein transcription binding sites, with differential or rate-limited decay rates for protein monomers and dimers. We demonstrate that the rate-limited decay significantly enlarges the parameter space of noise-induced and delay-induced oscillations. Interestingly, focusing on the stochastic characters of noise-induced and delay-induced oscillations in terms of phase coherence and phase averaged amplitude noise in the polar coordinate, we find that there is a delay-managed tradeoff between phase coherence and phase averaged amplitude noise. In particular, the model with both multiple binding sites and rate-limited decay can show regular tunability as the delay increases. Our results indicate that transcriptional and post-translational mechanisms constrain the combined effects of noise and delay on the molecular dynamics of the segmentation clock.

Minimal models for cell-cycle control based on competitive inhibition and multisite phosphorylations of Cdk substrates

The eukaryotic cell cycle is characterized by alternating oscillations in the activities of cyclin-dependent kinase (Cdk) and the anaphase-promoting complex (APC). Successful completion of the cell cycle is dependent on the precise, temporally ordered appearance of these activities. A modest level of Cdk activity is sufficient to initiate DNA replication, but mitosis and APC activation require an elevated Cdk activity. In present-day eukaryotes, this temporal order is provided by a complex network of regulatory proteins that control both Cdk and APC activities via sharp thresholds, bistability, and time delays. Using simple computational models, we show here that these dynamical features of cell-cycle organization could emerge in a control system driven by a single Cdk/cyclin complex and APC wired in a negative-feedback loop. We show that ordered phosphorylation of cellular proteins could be explained by multisite phosphorylation/dephosphorylation and competition of substrates for interconverting kinase (Cdk) and phosphatase. In addition, the competition of APC substrates for ubiquitylation can create and maintain sustained oscillations in cyclin levels. We propose a sequence of models that gets closer and closer to a realistic model of cell-cycle control in yeast. Since these models lack the elaborate control mechanisms characteristic of modern eukaryotes, they suggest that bistability and time delay may have characterized eukaryotic cell divisions before the current cell-cycle control network evolved in all its complexity.

Transcriptional bursting diversifies the behaviour of a toggle switch: hybrid simulation of stochastic gene expression

Hybrid models for gene expression combine stochastic and deterministic representations of the underlying biophysical mechanisms. According to one of the simplest hybrid formalisms, protein molecules are produced in randomly occurring bursts of a randomly distributed size while they are degraded deterministically. Here, we use this particular formalism to study two key regulatory motifs-the autoregulation loop and the toggle switch. The distribution of burst times is determined and used as a basis for the development of exact simulation algorithms for gene expression dynamics. For the autoregulation loop, the simulations are compared to an analytic solution of a master equation. Simulations of the toggle switch reveal a number of qualitatively distinct scenarios with implications for the modelling of cell-fate selection.

pH wave-front propagation in the urea-urease reaction

The urease-catalyzed hydrolysis of urea displays feedback that results in a switch from acid (pH ~3) to base (pH ~9) after a controllable period of time (from 10 to >5000 s). Here we show that the spatially distributed reaction can support pH wave fronts propagating with a speed of the order of 0.1-1 mm min(-1). The experimental results were reproduced qualitatively in reaction-diffusion simulations including a Michaelis-Menten expression for the urease reaction with a bell-shaped rate-pH dependence. However, this model fails to predict that at lower enzyme concentrations, the unstirred reaction does not always support fronts when the well-stirred reaction still rapidly switches to high pH.

Delayed protein synthesis reduces the correlation between mRNA and protein fluctuations

Recent experimental results indicate that, in single Escherichia coli cells, the fluctuations in mRNA level are uncorrelated with those of protein. However, a basic two-stage model for prokaryotic gene expression suggests that there ought to be a degree of correlation between the two. Therefore, it is important to investigate realistic modifications of the basic model that have the potential to reduce the theoretical level of the correlation. In this work, we focus on translational and reporter maturation delay, reporting that its introduction into the two-stage model reduces the cross correlation between instantaneous mRNA and protein levels. Our results indicate that the experimentally observed sample correlation coefficient between mRNA and protein levels may increase if the protein measurements are shifted back in time by the value of the delay.

Mathematical modeling of an oscillating gene circuit to unravel the circadian clock network of Arabidopsis thaliana

The Arabidopsis thaliana circadian clock is an interconnected network highly tractable to systems approaches. Most elements in the transcriptional-translational oscillator were identified by genetic means and the expression of clock genes in various mutants led to the founding hypothesis of a positive-negative feedback loop being the core clock. The identification of additional clock genes beyond those defined in the core led to the use of systems approaches to decipher this angiosperm oscillator circuit. Kinetic modeling was first used to explain periodicity effects of various circadian mutants. This conformed in a flexible way to experimental details. Such observations allowed a recursive use of hypothesis generating from modeling, followed by experimental corroboration. More recently, the biochemical finding of new description of a DNA-binding activity for one class of clock components directed improvements in feature generation, one of which revealed that the core of the oscillator is a negative-negative feedback loop. The recursive use of modeling and experimental validation has thus revealed many essential transcriptional components that drive negative arms in the circadian oscillator. What awaits is to more fully describe the positive arms and an understanding of how additional pathways converge on the clock.

Mathematical modeling of fission yeast Schizosaccharomyces pombe cell cycle: exploring the role of multiple phosphatases

Cell cycle is the central process that regulates growth and division in all eukaryotes. Based on the environmental condition sensed, the cell lies in a resting phase G0 or proceeds through the cyclic cell division process (G1→S→G2→M). These series of events and phase transitions are governed mainly by the highly conserved Cyclin dependent kinases (Cdks) and its positive and negative regulators. The cell cycle regulation of fission yeast Schizosaccharomyces pombe is modeled in this study. The study exploits a detailed molecular interaction map compiled based on the published model and experimental data. There are accumulating evidences about the prominent regulatory role of specific phosphatases in cell cycle regulations. The current study emphasizes the possible role of multiple phosphatases that governs the cell cycle regulation in fission yeast S. pombe. The ability of the model to reproduce the reported regulatory profile for the wild-type and various mutants was verified though simulations.

ELECTRONIC SUPPLEMENTARY MATERIAL

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Translational applications of evaluating physiologic variability in human endotoxemia

Dysregulation of the inflammatory response is a critical component of many clinically challenging disorders such as sepsis. Inflammation is a biological process designed to lead to healing and recovery, ultimately restoring homeostasis; however, the failure to fully achieve those beneficial results can leave a patient in a dangerous persistent inflammatory state. One of the primary challenges in developing novel therapies in this area is that inflammation is comprised of a complex network of interacting pathways. Here, we discuss our approaches towards addressing this problem through computational systems biology, with a particular focus on how the presence of biological rhythms and the disruption of these rhythms in inflammation may be applied in a translational context. By leveraging the information content embedded in physiologic variability, ranging in scale from oscillations in autonomic activity driving short-term heart rate variability to circadian rhythms in immunomodulatory hormones, there is significant potential to gain insight into the underlying physiology.

Phase resetting reveals network dynamics underlying a bacterial cell cycle

Genomic and proteomic methods yield networks of biological regulatory interactions but do not provide direct insight into how those interactions are organized into functional modules, or how information flows from one module to another. In this work we introduce an approach that provides this complementary information and apply it to the bacterium Caulobacter crescentus, a paradigm for cell-cycle control. Operationally, we use an inducible promoter to express the essential transcriptional regulatory gene ctrA in a periodic, pulsed fashion. This chemical perturbation causes the population of cells to divide synchronously, and we use the resulting advance or delay of the division times of single cells to construct a phase resetting curve. We find that delay is strongly favored over advance. This finding is surprising since it does not follow from the temporal expression profile of CtrA and, in turn, simulations of existing network models. We propose a phenomenological model that suggests that the cell-cycle network comprises two distinct functional modules that oscillate autonomously and couple in a highly asymmetric fashion. These features collectively provide a new mechanism for tight temporal control of the cell cycle in C. crescentus. We discuss how the procedure can serve as the basis for a general approach for probing network dynamics, which we term chemical perturbation spectroscopy (CPS).

During the cell cycle, the cell progresses through a series of stages that are associated with various cell cycle events such as replication of genetic materials. Genetic and molecular dissections have revealed that the cell cycle is regulated by a network of interacting molecules that produces oscillatory dynamics. The major cell cycle regulators have been identified previously in different species and the activity of these regulators oscillates. However, the question of how cell cycle regulators coordinate different cell cycle events during the cell cycle remains controversial. Here, we investigate this question in a model bacterial system for cell cycle, Caulobacter crescentus. We perturb the expression of the master cell cycle regulator ctrA in a pulsatile fashion and quantify the response of the cell cycle to such perturbations. The measured response is contradictory to the existing mechanism of Caulobacter cell cycle control, which views the cell cycle progression as a sequential activation/inhibition process. We propose a new model that involves coupling of multiple oscillators and show the quantitative agreement between this new model and our measurements. We expect this procedure to be generalized and applied to a broad range of systems to obtain information that complements that obtained from other methods.

Computing with competition in biochemical networks

Cells rely on limited resources such as enzymes or transcription factors to process signals and make decisions. However, independent cellular pathways often compete for a common molecular resource. Competition is difficult to analyze because of its nonlinear global nature, and its role remains unclear. Here we show how decision pathways such as transcription networks may exploit competition to process information. Competition for one resource leads to the recognition of convex sets of patterns, whereas competition for several resources (overlapping or cascaded regulons) allows even more general pattern recognition. Competition also generates surprising couplings, such as correlating species that share no resource but a common competitor. The mechanism we propose relies on three primitives that are ubiquitous in cells: multiinput motifs, competition for a resource, and positive feedback loops.

A dynamic and intricate regulatory network determines Pseudomonas aeruginosa virulence

Pseudomonas aeruginosa is a metabolically versatile bacterium that is found in a wide range of biotic and abiotic habitats. It is a major human opportunistic pathogen causing numerous acute and chronic infections. The critical traits contributing to the pathogenic potential of P. aeruginosa are the production of a myriad of virulence factors, formation of biofilms and antibiotic resistance. Expression of these traits is under stringent regulation, and it responds to largely unidentified environmental signals. This review is focused on providing a global picture of virulence gene regulation in P. aeruginosa. In addition to key regulatory pathways that control the transition from acute to chronic infection phenotypes, some regulators have been identified that modulate multiple virulence mechanisms. Despite of a propensity for chaotic behaviour, no chaotic motifs were readily observed in the P. aeruginosa virulence regulatory network. Having a ‘birds-eye’ view of the regulatory cascades provides the forum opportunities to pose questions, formulate hypotheses and evaluate theories in elucidating P. aeruginosa pathogenesis. Understanding the mechanisms involved in making P. aeruginosa a successful pathogen is essential in helping devise control strategies.

A design principle for a posttranslational biochemical oscillator

Multisite phosphorylation plays an important role in biological oscillators such as the circadian clock. Its general role, however, has been elusive. In this theoretical study, we show that a simple substrate with two modification sites acted upon by two opposing enzymes (e.g., a kinase and a phosphatase) can show oscillations in its modification state. An unbiased computational analysis of this oscillator reveals two common characteristics: a unidirectional modification cycle and sequestering of an enzyme by a specific modification state. These two motifs cause a substrate to act as a coupled system in which a unidirectional cycle generates single-molecule oscillators, whereas sequestration synchronizes the population by limiting the available enzyme under conditions in which substrate is in excess. We also demonstrate the conditions under which the oscillation period is temperature compensated, an important feature of the circadian clock. This theoretical model will provide a framework for analyzing and synthesizing posttranslational oscillators.

Inphase and antiphase synchronization in a delay-coupled system with applications to a delay-coupled FitzHugh-Nagumo system

A time delay is inevitable in the coupled system and is an essential property of the coupling, which cannot be neglected in many realistic coupled systems. In this paper, we first study the existence of a Hopf bifurcation induced by coupling time delay and then investigate the influence of coupling time delay on the patterns of Hopf-bifurcating periodic oscillations. How the coupling time delay leads to complex scenarios of synchronized inphase or antiphase oscillations is analytically investigated. As an example, we study the delay-coupled FitzHugh-Nagumo system. We find conditional stability, absolute stability, and stability switches of the steady state provoked by the coupling time delay. Then we investigate the inphase and antiphase synchronized periodic solutions induced by delay, and determine the direction and stability of these bifurcating periodic orbits by employing the center manifold reduction and normal form theory. We find that in the region where stability switches occur, there exist synchronization transitions, i.e., synchronized dynamics can be switched from inphase (antiphase) to antiphase (inphase) and back to inphase (antiphase) and so on just by progressive increase of the coupling time delay.

The cell cycle switch computes approximate majority

Both computational and biological systems have to make decisions about switching from one state to another. The ‘Approximate Majority’ computational algorithm provides the asymptotically fastest way to reach a common decision by all members of a population between two possible outcomes, where the decision approximately matches the initial relative majority. The network that regulates the mitotic entry of the cell-cycle in eukaryotes also makes a decision before it induces early mitotic processes. Here we show that the switch from inactive to active forms of the mitosis promoting Cyclin Dependent Kinases is driven by a system that is related to both the structure and the dynamics of the Approximate Majority computation. We investigate the behavior of these two switches by deterministic, stochastic and probabilistic methods and show that the steady states and temporal dynamics of the two systems are similar and they are exchangeable as components of oscillatory networks.

A spatial physiological model for p53 intracellular dynamics

In this paper we design and analyse a physiologically based model representing the accumulation of protein p53 in the nucleus after triggering of ATM by DNA damage. The p53 protein is known to have a central role in the response of the cell to cytotoxic or radiotoxic insults resulting in DNA damage. A reasonable requirement for a model describing intracellular signalling pathways is taking into account the basic feature of eukaryotic cells: the distinction between nucleus and cytoplasm. Our aim is to show, on a simple reaction network describing p53 dynamics, how this basic distinction provides a framework which is able to yield expected oscillatory dynamics without introducing either positive feedbacks or delays in the reactions. Furthermore we prove that oscillations appear only if some spatial constraints are respected, e.g. if the diffusion coefficients correspond to known biological values. Finally we analyse how the spatial features of a cell influence the dynamic response of the p53 network to DNA damage, pointing out that the protein oscillatory dynamics is indeed a response that is robust towards changes with respect to cellular environments. Even if we change the cell shape or its volume or better its ribosomal distribution, we observe that DNA damage yields sustained oscillations of p53.

Cytokine combination therapy prediction for bone remodeling in tissue engineering based on the intracellular signaling pathway

The long-term performance of tissue-engineered bone grafts is determined by a dynamic balance between bone regeneration and resorption. We proposed using embedded cytokine slow-releasing hydrogels to tune this balance toward a desirable final bone density. In this study we established a systems biology model, and quantitatively explored the combinatorial effects of delivered cytokines from hydrogels on final bone density. We hypothesized that: 1) bone regeneration was driven by transcription factors Runx2 and Osterix, which responded to released cytokines, such as Wnt, BMP2, and TGFβ, drove the development of osteoblast lineage, and contributed to bone mass generation; and 2) the osteoclast lineage, on the other hand, governed the bone resorption, and communications between these two lineages determined the dynamics of bone remodeling. In our model, Intracellular signaling pathways were represented by ordinary differential equations, while the intercellular communications and cellular population dynamics were modeled by stochastic differential equations. Effects of synergistic cytokine combinations were evaluated by Loewe index and Bliss index. Simulation results revealed that the Wnt/BMP2 combinations released from hydrogels showed best control of bone regeneration and synergistic effects, and suggested optimal dose ratios of given cytokine combinations released from hydrogels to most efficiently control the long-term bone remodeling. We revealed the characteristics of cytokine combinations of Wnt/BMP2 which could be used to guide the design of in vivo bone scaffolds and the clinical treatment of some diseases such as osteoporosis.

Tuning genetic clocks employing DNA binding sites

Periodic oscillations play a key role in cell physiology from the cell cycle to circadian clocks. The interplay of positive and negative feedback loops among genes and proteins is ubiquitous in these networks. Often, delays in a negative feedback loop and/or degradation rates are a crucial mechanism to obtain sustained oscillations. How does nature control delays and kinetic rates in feedback networks? Known mechanisms include proper selection of the number of steps composing a feedback loop and alteration of protease activity, respectively. Here, we show that a remarkably simple means to control both delays and effective kinetic rates is the employment of DNA binding sites. We illustrate this design principle on a widely studied activator-repressor clock motif, which is ubiquitous in natural systems. By suitably employing DNA target sites for the activator and/or the repressor, one can switch the clock “on” and “off” and precisely tune its period to a desired value. Our study reveals a design principle to engineer dynamic behavior in biomolecular networks, which may be largely exploited by natural systems and employed for the rational design of synthetic circuits.

Adjoint method provides phase response functions for delay-induced oscillations

Limit-cycle oscillations induced by time delay are widely observed in various systems, but a systematic phase-reduction theory for them has yet to be developed. Here we present a practical theoretical framework to calculate the phase response function Z(θ), a fundamental quantity for the theory, of delay-induced limit cycles with infinite-dimensional phase space. We show that Z(θ) can be obtained as a zero eigenfunction of the adjoint equation associated with an appropriate bilinear form for the delay differential equations. We confirm the validity of the proposed framework for two biological oscillators and demonstrate that the derived phase equation predicts intriguing multimodal locking behavior.

Systems biology of cellular rhythms

Rhythms abound in biological systems, particularly at the cellular level where they originate from the feedback loops present in regulatory networks. Cellular rhythms can be investigated both by experimental and modeling approaches, and thus represent a prototypic field of research for systems biology. They have also become a major topic in synthetic biology. We review advances in the study of cellular rhythms of biochemical rather than electrical origin by considering a variety of oscillatory processes such as Ca++ oscillations, circadian rhythms, the segmentation clock, oscillations in p53 and NF-κB, synthetic oscillators, and the oscillatory dynamics of cyclin-dependent kinases driving the cell cycle. Finally we discuss the coupling between cellular rhythms and their robustness with respect to molecular noise.

A network-oriented perspective on cardiac calcium signaling

The normal contractile, electrical, and energetic function of the heart depends on the synchronization of biological oscillators and signal integrators that make up cellular signaling networks. In this review we interpret experimental data from molecular, cellular, and transgenic models of cardiac signaling behavior in the context of established concepts in cell network architecture and organization. Focusing on the cellular Ca(2+) handling machinery, we describe how the plasticity and adaptability of normal Ca(2+) signaling is dependent on dynamic network configurations that operate across a wide range of functional states. We consider how (mal)adaptive changes in signaling pathways restrict the dynamic range of the network such that it cannot respond appropriately to physiologic stimuli or perturbation. Based on these concepts, a model is proposed in which pathologic abnormalities in cardiac rhythm and contractility (e.g., arrhythmias and heart failure) arise as a consequence of progressive desynchronization and reduction in the dynamic range of the Ca(2+) signaling network. We discuss how a systems-level understanding of the network organization, cellular noise, and chaotic behavior may inform the design of new therapeutic modalities that prevent or reverse the disease-linked unraveling of the Ca(2+) signaling network.

Topology and dynamics of the zebrafish segmentation clock core circuit

By combining biochemical, embryological, and mathematical approaches, this work uncovers an important role for protein-protein interactions in determining the dynamics of the somite-forming segmentation clock in vertebrates.

During vertebrate embryogenesis, the rhythmic and sequential segmentation of the body axis is regulated by an oscillating genetic network termed the segmentation clock. We describe a new dynamic model for the core pace-making circuit of the zebrafish segmentation clock based on a systematic biochemical investigation of the network's topology and precise measurements of somitogenesis dynamics in novel genetic mutants. We show that the core pace-making circuit consists of two distinct negative feedback loops, one with Her1 homodimers and the other with Her7:Hes6 heterodimers, operating in parallel. To explain the observed single and double mutant phenotypes of her1, her7, and hes6 mutant embryos in our dynamic model, we postulate that the availability and effective stability of the dimers with DNA binding activity is controlled in a “dimer cloud” that contains all possible dimeric combinations between the three factors. This feature of our model predicts that Hes6 protein levels should oscillate despite constant hes6 mRNA production, which we confirm experimentally using novel Hes6 antibodies. The control of the circuit's dynamics by a population of dimers with and without DNA binding activity is a new principle for the segmentation clock and may be relevant to other biological clocks and transcriptional regulatory networks.

The segmented pattern of the vertebral column, one of the defining features of the vertebrate body, is established during embryogenesis. The embryo's segments, called somites, form sequentially and rhythmically from head to tail. The periodicity of somite formation is regulated by the segmentation clock, a genetic oscillator that ticks in the posterior-most embryonic tissue: for each tick of the clock, one new bilateral pair of segments is made. The period of the clock appears to determine the number and the length of segments, but what controls this periodicity? In this article, we have investigated the interactions of three transcription factors that form the core of the clock's regulatory circuit, and have measured how the period of segmentation changes when these factors are mutated alone or in combination. We find that these three factors contribute to a “dimer cloud” that contains all possible dimeric combinations; however, only two dimers in this cloud can bind DNA, which allows them to directly regulate the oscillatory gene expression that underpins the periodicity of segment formation. Nevertheless, a mathematical model of the clock's dynamics based on our experimental findings indicates that the non-DNA-binding dimers also influence the stability, and hence the function, of the two DNA-binding dimers controlling the segmentation clock's period. Such involvement of non-DNA-binding dimers is a novel regulatory principle for the segmentation clock, which might also be a general mechanism that operates in other biological clocks.

Operator sequence alters gene expression independently of transcription factor occupancy in bacteria

A canonical quantitative view of transcriptional regulation holds that the only role of operator sequence is to set the probability of transcription factor binding, with operator occupancy determining the level of gene expression. In this work, we test this idea by characterizing repression in vivo and the binding of RNA polymerase in vitro in experiments where operators of various sequences were placed either upstream or downstream from the promoter in Escherichia coli. Surprisingly, we find that operators with a weaker binding affinity can yield higher repression levels than stronger operators. Repressor bound to upstream operators modulates promoter escape, and the magnitude of this modulation is not correlated with the repressor-operator binding affinity. This suggests that operator sequences may modulate transcription by altering the nature of the interaction of the bound transcription factor with the transcriptional machinery, implying a new layer of sequence dependence that must be confronted in the quantitative understanding of gene expression.

Chaotic motifs in gene regulatory networks

Chaos should occur often in gene regulatory networks (GRNs) which have been widely described by nonlinear coupled ordinary differential equations, if their dimensions are no less than 3. It is therefore puzzling that chaos has never been reported in GRNs in nature and is also extremely rare in models of GRNs. On the other hand, the topic of motifs has attracted great attention in studying biological networks, and network motifs are suggested to be elementary building blocks that carry out some key functions in the network. In this paper, chaotic motifs (subnetworks with chaos) in GRNs are systematically investigated. The conclusion is that: (i) chaos can only appear through competitions between different oscillatory modes with rivaling intensities. Conditions required for chaotic GRNs are found to be very strict, which make chaotic GRNs extremely rare. (ii) Chaotic motifs are explored as the simplest few-node structures capable of producing chaos, and serve as the intrinsic source of chaos of random few-node GRNs. Several optimal motifs causing chaos with atypically high probability are figured out. (iii) Moreover, we discovered that a number of special oscillators can never produce chaos. These structures bring some advantages on rhythmic functions and may help us understand the robustness of diverse biological rhythms. (iv) The methods of dominant phase-advanced driving (DPAD) and DPAD time fraction are proposed to quantitatively identify chaotic motifs and to explain the origin of chaotic behaviors in GRNs.

Dynamics of TGF-β/Smad signaling

The physiological responses to TGF-β stimulation are diverse and vary amongst different cell types and environmental conditions. Even though the principal molecular components of the canonical and the non-canonical TGF-β signaling pathways have been largely identified, the mechanism that underlies the well-established context dependent physiological responses remains a mystery. Understanding how the components of TGF-β signaling function as a system and how this system functions in the context of the global cellular regulatory network requires a more quantitative and systematic approach. Here, we review the recent progress in understanding TGF-β biology using integration of mathematical modeling and quantitative experimental analysis. These studies reveal many interesting dynamics of TGF-β signaling and how cells quantitatively decode variable doses of TGF-β stimulation.

Inositol 1,4,5-trisphosphate receptors and pacemaker rhythms

Intracellular Ca(2+) plays an important role in the control of the heart rate through the interaction between Ca(2+) release by ryanodine receptors in the sarcoplasmic reticulum (SR) and the extrusion of Ca(2+) by the sodium-calcium exchanger which generates an inward current. A second type of SR Ca(2+) release channel, the inositol 1,4,5-trisphosphate receptor (IP(3)R), can release Ca(2+) from SR stores in many cell types, including cardiac myocytes. However, it is still uncertain whether IP(3)Rs play any functional role in regulating the heart rate. Accumulated evidence shows that IP(3) and IP(3)R are involved in rhythm control in non-cardiac pacemaker tissues and in the embryonic heart. In this review we focus on intracellular Ca(2+) oscillations generated by Ca(2+) release from IP(3)R that initiates membrane depolarization and provides a common mechanism producing spontaneous activity in a range of cells with pacemaker function. Emerging new evidence also suggests that IP(3)/IP(3)Rs play a functional role in normal and diseased hearts and in cardiac rhythm control. Several membrane currents, including a store-operated Ca(2+) current, might be activated by Ca(2+) release from IP(3)Rs. IP(3)/IP(3)R may thus add another dimension to the complex regulation of heart rate.

Spatiotemporal coordinated hierarchical properties of cellular protrusion revealed by multiscale analysis

We present a methodology for integrative multiscale analysis to highlight hierarchical properties of cellular protrusion and mechanochemical interactions in cellular protrusion based on live cell imaging data with high spatiotemporal resolution. As an appropriate experimental system, we selected non-polarized full-moon-shaped keratocytes that present balanced protrusion around the entire cell periphery at the cellular scale simultaneously with active protrusion and retraction at the subcellular scale. We achieved the observation of a whole cell with sub-micrometer spatial precision and sub-second time resolution for three minutes or more. The multiscale characteristics of cell peripheral activity and those of the cell peripheral shape were extracted from an identical image sequence by estimating the cell protrusion rates and the cell peripheral curvatures at various differential intervals. The spatiotemporal maps of the cell protrusion rates demonstrated a spatiotemporally nested structure of travelling waves of active protruding regions at the cellular and subcellular scales. Moreover, correlation analysis demonstrated the relationship between the cell protrusion rate and peripheral curvature at the subcellular scale. The novel integrative methodology presented here well highlighted the hierarchical properties of organized cellular protrusion, and further provided insight about the underlying mechanochemical interactions between the cell membrane and the actin filaments under the membrane.

Fractional dynamics of globally slow transcription and its impact on deterministic genetic oscillation

In dynamical systems theory, a system which can be described by differential equations is called a continuous dynamical system. In studies on genetic oscillation, most deterministic models at early stage are usually built on ordinary differential equations (ODE). Therefore, gene transcription which is a vital part in genetic oscillation is presupposed to be a continuous dynamical system by default. However, recent studies argued that discontinuous transcription might be more common than continuous transcription. In this paper, by appending the inserted silent interval lying between two neighboring transcriptional events to the end of the preceding event, we established that the running time for an intact transcriptional event increases and gene transcription thus shows slow dynamics. By globally replacing the original time increment for each state increment by a larger one, we introduced fractional differential equations (FDE) to describe such globally slow transcription. The impact of fractionization on genetic oscillation was then studied in two early stage models – the Goodwin oscillator and the Rössler oscillator. By constructing a “dual memory” oscillator – the fractional delay Goodwin oscillator, we suggested that four general requirements for generating genetic oscillation should be revised to be negative feedback, sufficient nonlinearity, sufficient memory and proper balancing of timescale. The numerical study of the fractional Rössler oscillator implied that the globally slow transcription tends to lower the chance of a coupled or more complex nonlinear genetic oscillatory system behaving chaotically.

Negative feedback enhances robustness in the yeast polarity establishment circuit

Many cells undergo symmetry-breaking polarization toward a randomly oriented "front" in the absence of spatial cues. In budding yeast, such polarization involves a positive feedback loop that enables amplification of stochastically arising clusters of polarity factors. Previous mathematical modeling suggested that, if more than one cluster were amplified, the clusters would compete for limiting resources and the largest would "win," explaining why yeast cells always make one and only one bud. Here, using imaging with improved spatiotemporal resolution, we show the transient coexistence of multiple clusters during polarity establishment, as predicted by the model. Unexpectedly, we also find that initial polarity factor clustering is oscillatory, revealing the presence of a negative feedback loop that disperses the factors. Mathematical modeling predicts that negative feedback would confer robustness to the polarity circuit and make the kinetics of competition between polarity factor clusters relatively insensitive to polarity factor concentration. These predictions are confirmed experimentally.

A multiphysics model of in vitro transcription coupling enzymatic reaction and precipitation formation

Multiphysics modeling, which integrates the models studied in different disciplines so far, is an indispensable approach toward a comprehensive understanding of biological systems composed of diverse phenomena. However, the variety of the models is narrower than the actual diverse phenomena because of the difficulty in coupling independent models separately studied in different disciplines for the actual coupled phenomena. In this study, we develop a mathematical model coupling an enzymatic reaction and mineralization formation. As a test case, we selected an in vitro transcription system where a transcription reaction occurs along with the precipitation formation of magnesium pyrophosphate (Mg(2)PPi). To begin, we experimentally elucidated how the transcription reaction and the precipitation formation are coupled. In the analysis, we applied a Michaelis-Menten-type equation to the transcription reaction and a semiempirical equation describing the correlation between the induction period and the supersaturation ratio to the precipitation formation, respectively. Based on the experimental results, we then integrated these two models. These models were connected by supersaturation that increases as the transcription reaction proceeds and becomes the driving force of the precipitation. We believe that our modeling approach could significantly contribute to the development of newer multiphysics models in systems biology such as bone metabolic networks.

The yin and yang of yeast transcription: elements of a global feedback system between metabolism and chromatin

When grown in continuous culture, budding yeast cells tend to synchronize their respiratory activity to form a stable oscillation that percolates throughout cellular physiology and involves the majority of the protein-coding transcriptome. Oscillations in batch culture and at single cell level support the idea that these dynamics constitute a general growth principle. The precise molecular mechanisms and biological functions of the oscillation remain elusive. Fourier analysis of transcriptome time series datasets from two different oscillation periods (0.7 h and 5 h) reveals seven distinct co-expression clusters common to both systems (34% of all yeast ORF), which consolidate into two superclusters when correlated with a compilation of 1,327 unrelated transcriptome datasets. These superclusters encode for cell growth and anabolism during the phase of high, and mitochondrial growth, catabolism and stress response during the phase of low oxygen uptake. The promoters of each cluster are characterized by different nucleotide contents, promoter nucleosome configurations, and dependence on ATP-dependent nucleosome remodeling complexes. We show that the ATP:ADP ratio oscillates, compatible with alternating metabolic activity of the two superclusters and differential feedback on their transcription via activating (RSC) and repressive (Isw2) types of promoter structure remodeling. We propose a novel feedback mechanism, where the energetic state of the cell, reflected in the ATP:ADP ratio, gates the transcription of large, but functionally coherent groups of genes via differential effects of ATP-dependent nucleosome remodeling machineries. Besides providing a mechanistic hypothesis for the delayed negative feedback that results in the oscillatory phenotype, this mechanism may underpin the continuous adaptation of growth to environmental conditions.

How cells process information: quantification of spatiotemporal signaling dynamics

Arguably, one of the foremost distinctions between life and non-living matter is the ability to sense environmental changes and respond appropriately--an ability that is invested in every living cell. Within a single cell, this function is largely carried out by networks of signaling molecules. However, the details of how signaling networks help cells make complicated decisions are still not clear. For instance, how do cells read graded, analog stress signals but convert them into digital live-or-die responses? The answer to such questions may originate from the fact that signaling molecules are not static but dynamic entities, changing in numbers and activity over time and space. In the past two decades, researchers have been able to experimentally monitor signaling dynamics and use mathematical techniques to quantify and abstract general principles of how cells process information. In this review, the authors first introduce and discuss various experimental and computational methodologies that have been used to study signaling dynamics. The authors then discuss the different types of temporal dynamics such as oscillations and bistability that can be exhibited by signaling systems and highlight studies that have investigated such dynamics in physiological settings. Finally, the authors illustrate the role of spatial compartmentalization in regulating cellular responses with examples of second-messenger signaling in cardiac myocytes.

What is the core oscillator in the speract-activated pathway of the Strongylocentrotus purpuratus sperm flagellum?

Sperm chemotaxis has an important role in fertilization. Most of our knowledge regarding this phenomenon comes from studies in organisms whose fertilization occurs externally, like sea urchins. Sea urchin spermatozoa respond to sperm-activating peptides, which diffuse from the egg jelly coat and interact with their receptor in the flagellum, triggering several physiological responses: changes in membrane potential, intracellular pH, cyclic nucleotide levels, and intracellular Ca2+ concentration ([Ca2+]). In particular, flagellar [Ca2+] has been shown to oscillate. These [Ca2+] oscillations are correlated with changes in the flagellar shape and so with the regulation of the sperm swimming paths. In this study, we demonstrate, from a mathematical modeling perspective, that the reported speract-activated signaling pathway in Strongylocentrotus purpuratus (speract being a sperm-activating peptide specific to this species) has the necessary elements to replicate the reported [Ca2+] oscillations. We further investigate which elements of this signaling pathway constitute the core oscillator.

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.

Multisite phosphoregulation of Cdc25 activity refines the mitotic entrance and exit switches

Cyclin-dependent kinase 1 (Cdk1) kinase dephosphorylation and activation by Cdc25 phosphatase are essential for mitotic entry. Activated Cdk1 phosphorylates Cdc25 and other substrates, further activating Cdc25 to form a positive feedback loop that drives the abrupt G2/mitosis switch. Conversely, mitotic exit requires Cdk1 inactivation and reversal of Cdk1 substrate phosphorylation. This dephosphorylation is mediated, in part, by Clp1/Cdc14, a Cdk1-antagonizing phosphatase, which reverses Cdk1 phosphorylation of itself, Cdc25, and other Cdk1 substrates. Thus, Cdc25 phosphoregulation is essential for proper G2-M transition, and its contributions to cell cycle control have been modeled based on studies using Xenopus and human cell extracts. Because cell extract systems only approximate in vivo conditions where proteins interact within dynamic cellular environments, here, we use Schizosaccharomyces pombe to characterize, both experimentally and mathematically, the in vivo contributions of Cdk1-mediated phosphorylation of Cdc25 to the mitotic transition. Through comprehensive mapping of Cdk1 phosphosites on Cdc25 and characterization of phosphomutants, we show that Cdc25 hyperphosphorylation by Cdk1 governs Cdc25 catalytic activation, the precision of mitotic entry, and unvarying cell length but not Cdc25 localization or abundance. We propose a mathematical model that explains Cdc25 regulation by Cdk1 through a distributive and disordered phosphorylation mechanism that ultrasensitively activates Cdc25. We also show that Clp1/Cdc14 dephosphorylation of Cdk1 sites on Cdc25 controls the proper timing of cell division, a mechanism that is likely due to the double negative feedback loop between Clp1/Cdc14 and Cdc25 that controls the abruptness of the mitotic exit switch.

Oscillatory dynamics of Cdc42 GTPase in the control of polarized growth

Cells promote polarized growth by activation of Rho-family protein Cdc42 at the cell membrane. We combined experiments and modeling to study bipolar growth initiation in fission yeast. Concentrations of a fluorescent marker for active Cdc42, Cdc42 protein, Cdc42-activator Scd1, and scaffold protein Scd2 exhibited anticorrelated fluctuations and oscillations with a 5-minute average period at polarized cell tips. These dynamics indicate competition for active Cdc42 or its regulators and the presence of positive and delayed negative feedbacks. Cdc42 oscillations and spatial distribution were sensitive to the amounts of Cdc42-activator Gef1 and to the activity of Cdc42-dependent kinase Pak1, a negative regulator. Feedbacks regulating Cdc42 oscillations and spatial self-organization appear to provide a flexible mechanism for fission yeast cells to explore polarization states and to control their morphology.

The slow-scale linear noise approximation: an accurate, reduced stochastic description of biochemical networks under timescale separation conditions

BACKGROUND

It is well known that the deterministic dynamics of biochemical reaction networks can be more easily studied if timescale separation conditions are invoked (the quasi-steady-state assumption). In this case the deterministic dynamics of a large network of elementary reactions are well described by the dynamics of a smaller network of effective reactions. Each of the latter represents a group of elementary reactions in the large network and has associated with it an effective macroscopic rate law. A popular method to achieve model reduction in the presence of intrinsic noise consists of using the effective macroscopic rate laws to heuristically deduce effective probabilities for the effective reactions which then enables simulation via the stochastic simulation algorithm (SSA). The validity of this heuristic SSA method is a priori doubtful because the reaction probabilities for the SSA have only been rigorously derived from microscopic physics arguments for elementary reactions.

RESULTS

We here obtain, by rigorous means and in closed-form, a reduced linear Langevin equation description of the stochastic dynamics of monostable biochemical networks in conditions characterized by small intrinsic noise and timescale separation. The slow-scale linear noise approximation (ssLNA), as the new method is called, is used to calculate the intrinsic noise statistics of enzyme and gene networks. The results agree very well with SSA simulations of the non-reduced network of elementary reactions. In contrast the conventional heuristic SSA is shown to overestimate the size of noise for Michaelis-Menten kinetics, considerably under-estimate the size of noise for Hill-type kinetics and in some cases even miss the prediction of noise-induced oscillations.

CONCLUSIONS

A new general method, the ssLNA, is derived and shown to correctly describe the statistics of intrinsic noise about the macroscopic concentrations under timescale separation conditions. The ssLNA provides a simple and accurate means of performing stochastic model reduction and hence it is expected to be of widespread utility in studying the dynamics of large noisy reaction networks, as is common in computational and systems biology.

Substrate-dependent control of ERK phosphorylation can lead to oscillations

The extracellular signal-regulated kinase (ERK) controls cellular processes by phosphorylating multiple substrates. The ERK protein can use the same domains to interact with phosphatases, which dephosphorylate and deactivate ERK, and with substrates, which connect ERK to its downstream effects. As a consequence, substrates can compete with phosphatases and control the level of ERK phosphorylation. We propose that this effect can qualitatively change the dynamics of a network that controls ERK activation. On its own, this network can be bistable, but in a larger system, where ERK accelerates the degradation of a substrate competing with a phosphatase, this network can oscillate. Previous studies proposed that oscillatory ERK signaling requires a negative feedback in which active ERK reduces the rate at which it is phosphorylated by upstream kinase. We argue that oscillations can also emerge even when this rate is constant, due to substrate-dependent control of ERK phosphorylation.

Bifurcation threshold of the delayed van der Pol oscillator under stochastic modulation

We obtain the location of the Hopf bifurcation threshold for a modified van der Pol oscillator, parametrically driven by a stochastic source and including delayed feedback in both position and velocity. We introduce a multiple scale expansion near threshold, and we solve the resulting Fokker-Planck equation associated with the evolution at the slowest time scale. The analytical results are compared with a direct numerical integration of the model equation. Delay modifies the Hopf frequency at threshold and leads to a stochastic bifurcation that is shifted relative to the deterministic limit by an amount that depends on the delay time, the amplitude of the feedback terms, and the intensity of the noise. Interestingly, stochasticity generally increases the region of stability of the limit cycle.

Classifying the topology of AHL-driven quorum sensing circuits in proteobacterial genomes

Virulence and adaptability of many Gram-negative bacterial species are associated with an N-acylhomoserine lactone (AHL) gene regulation mechanism called quorum sensing (QS). The arrangement of quorum sensing genes is variable throughout bacterial genomes, although there are unifying themes that are common among the various topological arrangements. A bioinformatics survey of 1,403 complete bacterial genomes revealed characteristic gene topologies in 152 genomes that could be classified into 16 topological groups. We developed a concise notation for the patterns and show that the sequences of LuxR regulators and LuxI autoinducer synthase proteins cluster according to the topological patterns. The annotated topologies are deposited online with links to sequences and genome annotations at http://bacteria.itk.ppke.hu/QStopologies/.