Design principles of biochemical oscillators

Transient pulse formation in jasmonate signaling pathway

The jasmonate (JA) signaling pathway in plants is activated as defense response to a number of stresses like attacks by pests or pathogens and wounding by animals. Some recent experiments provide significant new knowledge on the molecular detail and connectivity of the pathway. The pathway has two major components in the form of feedback loops, one negative and the other positive. We construct a minimal mathematical model, incorporating the feedback loops, to study the dynamics of the JA signaling pathway. The model exhibits transient gene expression activity in the form of JA pulses in agreement with experimental observations. The dependence of the pulse amplitude, duration and peak time on the key parameters of the model is determined computationally. The deterministic and stochastic aspects of the pathway dynamics are investigated using both the full mathematical model and a reduced version of it. We also compare the mechanism of pulse formation with the known mechanisms of pulse generation in some bacterial and viral systems.

Efficient modeling, simulation and coarse-graining of biological complexity with NFsim

Managing the overwhelming numbers of molecular states and interactions is a fundamental obstacle to building predictive models of biological systems. Here we introduce the Network-Free Stochastic Simulator (NFsim), a general-purpose modeling platform that overcomes the combinatorial nature of molecular interactions. Unlike standard simulators that represent molecular species as variables in equations, NFsim uses a biologically intuitive representation: objects with binding and modification sites acted on by reaction rules. During simulations, rules operate directly on molecular objects to produce exact stochastic results with performance that scales independently of the reaction network size. Reaction rates can be defined as arbitrary functions of molecular states to provide powerful coarse-graining capabilities, for example to merge Boolean and kinetic representations of biological networks. NFsim enables researchers to simulate many biological systems that were previously inaccessible to general-purpose software, as we illustrate with models of immune system signaling, microbial signaling, cytoskeletal assembly and oscillating gene expression.

Oscillatory control of signalling molecules

The emergence of biological function from the dynamic control of cellular signalling molecules is a fundamental process in biology. Key questions include: How do cells decipher noisy environmental cues, encode these signals to control fate decisions and propagate information through tissues? Recent advances in systems biology, and molecular and cellular biology, exemplified by analyses of signalling via the transcription factor Nuclear Factor kappaB (NF-κB), reveal a critical role of oscillatory control in the regulation of these biological functions. The emerging view is that the oscillatory dynamics of signalling molecules and the epigenetically regulated specificity for target genes contribute to robust regulation of biological function at different levels of cellular organisation through frequency-dependent information encoding.

Architecture-dependent robustness and bistability in a class of genetic circuits

Understanding the relationship between genotype and phenotype is a challenge in systems biology. An interesting yet related issue is why a particular circuit topology is present in a cell when the same function can supposedly be obtained from an alternative architecture. Here we analyzed two topologically equivalent genetic circuits of coupled positive and negative feedback loops, named NAT and ALT circuits, respectively. The computational search for the oscillation volume of the entire biologically reasonable parameter region through large-scale random samplings shows that the NAT circuit exhibits a distinctly larger fraction of the oscillatory region than the ALT circuit. Such a global robustness difference between two circuits is supplemented by analyzing local robustness, including robustness to parameter perturbations and to molecular noise. In addition, detailed dynamical analysis shows that the molecular noise of both circuits can induce transient switching of the different mechanism between a stable steady state and a stable limit cycle. Our investigation on robustness and dynamics through examples provides insights into the relationship between network architecture and its function.

Approaching the molecular origins of collective dynamics in oscillating cell populations

From flocking birds, to organ generation, to swarming bacterial colonies, biological systems often exhibit collective behaviors. Here, we review recent advances in our understanding of collective dynamics in cell populations. We argue that understanding population-level oscillations requires examining the system under consideration at three different levels of complexity: at the level of isolated cells, homogenous populations, and spatially structured populations. We discuss the experimental and theoretical challenges this poses and highlight how new experimental techniques, when combined with conceptual tools adapted from physics, may help us overcome these challenges.

Diffusive coupling can discriminate between similar reaction mechanisms in an allosteric enzyme system

BACKGROUND

A central question for the understanding of biological reaction networks is how a particular dynamic behavior, such as bistability or oscillations, is realized at the molecular level. So far this question has been mainly addressed in well-mixed reaction systems which are conveniently described by ordinary differential equations. However, much less is known about how molecular details of a reaction mechanism can affect the dynamics in diffusively coupled systems because the resulting partial differential equations are much more difficult to analyze.

RESULTS

Motivated by recent experiments we compare two closely related mechanisms for the product activation of allosteric enzymes with respect to their ability to induce different types of reaction-diffusion waves and stationary Turing patterns. The analysis is facilitated by mapping each model to an associated complex Ginzburg-Landau equation. We show that a sequential activation mechanism, as implemented in the model of Monod, Wyman and Changeux (MWC), can generate inward rotating spiral waves which were recently observed as glycolytic activity waves in yeast extracts. In contrast, in the limiting case of a simple Hill activation, the formation of inward propagating waves is suppressed by a Turing instability. The occurrence of this unusual wave dynamics is not related to the magnitude of the enzyme cooperativity (as it is true for the occurrence of oscillations), but to the sensitivity with respect to changes of the activator concentration. Also, the MWC mechanism generates wave patterns that are more stable against long wave length perturbations.

CONCLUSIONS

This analysis demonstrates that amplitude equations, which describe the spatio-temporal dynamics near an instability, represent a valuable tool to investigate the molecular effects of reaction mechanisms on pattern formation in spatially extended systems. Using this approach we have shown that the occurrence of inward rotating spiral waves in glycolysis can be explained in terms of an MWC, but not with a Hill mechanism for the activation of the allosteric enzyme phosphofructokinase. Our results also highlight the importance of enzyme oligomerization for a possible experimental generation of Turing patterns in biological systems.

Systematic quantification of negative feedback mechanisms in the extracellular signal-regulated kinase (ERK) signaling network

Cell responses are actuated by tightly controlled signal transduction pathways. Although the concept of an integrated signaling network replete with interpathway cross-talk and feedback regulation is broadly appreciated, kinetic data of the type needed to characterize such interactions in conjunction with mathematical models are lacking. In mammalian cells, the Ras/ERK pathway controls cell proliferation and other responses stimulated by growth factors, and several cross-talk and feedback mechanisms affecting its activation have been identified. In this work, we take a systematic approach to parse the magnitudes of multiple regulatory mechanisms that attenuate ERK activation through canonical (Ras-dependent) and non-canonical (PI3K-dependent) pathways. In addition to regulation of receptor and ligand levels, we consider three layers of ERK-dependent feedback: desensitization of Ras activation, negative regulation of MEK kinase (e.g. Raf) activities, and up-regulation of dual-specificity ERK phosphatases. Our results establish the second of these as the dominant mode of ERK self-regulation in mouse fibroblasts. We further demonstrate that kinetic models of signaling networks, trained on a sufficient diversity of quantitative data, can be reasonably comprehensive, accurate, and predictive in the dynamical sense.

Fixed point characterization of biological networks with complex graph topology

MOTIVATION

Feedback circuits are important motifs in biological networks and part of virtually all regulation processes that are needed for a reliable functioning of the cell. Mathematically, feedback is connected to complex behavior of the systems, which is often related to bifurcations of fixed points. Therefore, several approaches for the investigation of fixed points in biological networks have been developed in recent years. Many of them assume the fixed point coordinates to be known, and an efficient way to calculate the entire set of fixed points for interrelated feedback structures is highly desirable.

RESULTS

In this article, we consider regulatory network models, which are differential equations with an underlying directed graph that illustrates independencies among variables. We introduce the circuit-breaking algorithm (CBA), a method that constructs one-dimensional characteristics for these network models, which inherit important information about the system. In particular, fixed points are related to the zeros of these characteristics. The CBA operates on the graph topology, and results from graph theory are used in order to make calculations efficient. Our framework provides a general scheme for analyzing network models in terms of interrelated feedback circuits. The efficiency of the approach is demonstrated on a model for calcium oscillations based on experiments in hepatocytes, which consists of several interrelated feedback circuits.

A sequestration feedback determines dynamics and temperature entrainment of the KaiABC circadian clock

The circadian KaiABC oscillator is driven by a sequestration feedback, which is biochemically realized by a strong affinity of KaiA to exclusively serine phosphorylated KaiBC complexes.

A highly non-linear feedback model explains the time courses of the phosphorylation states and the robustness under concerted changes of all Kai proteins.

Native mass spectrometry reveals the existence of two KaiA binding sites on KaiC, confirming the theoretical predictions.

Temperature entrainment arises from a temperature-dependent change in the abundance of KaiAC and KaiBC complexes.

The circadian rhythm of the cyanobacterium Synechococcus elongatus is controlled by three proteins, KaiA, KaiB, and KaiC. In a test tube, these proteins form complexes of various stoichiometry and the average phosphorylation level of KaiC exhibits robust circadian oscillations in the presence of ATP (Nakajima et al, 2005). Although the three component oscillator is apparently simple, it is highly precise and shows in-phase oscillations over several days (Mihalcescu et al, 2004). If we assume that cyanobacteria gain an evolutionary advantage from predicting the time of maximal sunlight intensity in a very reliable way, a highly robust but entrainable oscillator is likely the optimal solution. To elucidate the mechanism of the opposing properties of the clockwork—robustness and tenability—a mathematical modeling approach is necessary. The model must account for both the measured invariance of the phosphorylation level under concerted concentration changes of all Kai proteins (Kageyama et al, 2006) and for the experimentally observed temperature entrainment (Yoshida et al, 2009).

Here, we show by mathematical modeling, in combination with the measurements of the KaiABC complex formation dynamics from native mass spectrometry, that oscillations in the Kai system are a consequence of KaiA sequestration by KaiC hexamers and KaiBC complexes. Our in vitro model includes the characteristic KaiC phospho-form cycle, originating from two KaiC residues, serine (431) and threonine (432) (Nishiwaki et al, 2000). We allow for three pools that represent different forms of the KaiC complex. Together with the four phosphorylation states of KaiC, this gives a 3 × 4-dimensional model. We used data from the time course experiments of the O'Shea laboratory (Rust et al, 2007) to determine the unknown parameters of the KaiABC system. A global optimization algorithm is used to scan a large range of parameter values, resulting in a mathematical quantitative model of the KaiABC clockwork, see Figure 5. The KaiB response experiment (Rust et al, 2007) turned out to be crucial for identifying the mechanism by which individual KaiC hexamers can be synchronized in their phosphorylation dynamics. Importantly, the dephosphorylation phase can only be explained by a highly non-linear dependency of the KaiBC complex formation on the actual phosphorylation state. This can be realized by allowing only KaiBC complexes with exclusively serine phosphorylated KaiC, [S–KaiBC]₆, to inactivate KaiA with a high efficiency.

This theoretical prediction is confirmed by native mass spectrometry, generating semi-quantitative time courses of the KaiABC complex formation dynamics. Our experiments show the existence of two different KaiC binding sites to KaiA. The constant sequestration of free KaiA by the KaiA₂C₆ complexes is the molecular realization of the dynamic invariance condition because it requires most of the KaiA to be inactive at every instant of time, regardless of the phosphorylation state. The second binding site KaiA₄C₆ reflects the KaiA-binding domain at the catalytic active center of the KaiC hexamer. This hypothesis is confirmed by comparison of the mass spectrometry signal for KaiA₄C₆ with predictions from the mathematical model (Figure 7A and B). In the late phosphorylation phase, KaiBC complexes rapidly build up and sequestrate KaiA (Figure 7C), which represent an additional binding site. The relative amount of KaiA₆B₆C₆ confirms the sequestration hypotheses and corresponds to the theoretically estimated amount of sequestrated KaiA. The time of maximum sequestration—as defined by the appearance of the largest observed sequestration complex KaiA₁₀B₆C₆ (Figure 7E)—agrees with the theoretical expected sequestration maximum (Figure 7F) where [S-KaiBC]₆ is maximal.

To test further the predictive power of the mathematical model, we reproduced the observed phase synchronization dynamics on entrainment by temperature cycles. The response to a sudden temperature change results in a phase shift of phospho-KaiC, whereas the circadian period does not show any temperature dependency within a physiologically relevant range (Yoshida et al, 2009). As phosphorylation and dephosphorylation dynamics of KaiC alone and incubated with KaiA do not show significant temperature dependence (Tomita et al, 2005), phase entrainment is likely a consequence of temperature-induced changes in binding constants associated with the various KaiABC complexes. From thermodynamic arguments, we expect that an increasing temperature will enhance dissociation of KaiA and KaiB from KaiC. Indeed, a reduction in the net complex formation rate for S-KaiBC, D-KaiBC, and for KaiAC on temperature increase results in the experimentally observed differences in phase response, which compensate during the circadian cycle to assure temperature invariance of the ciracadian period.

The circadian rhythm of the cyanobacterium Synechococcus elongatus is controlled by three proteins, KaiA, KaiB, and KaiC. In a test tube, these proteins form complexes of various stoichiometry and the average phosphorylation level of KaiC exhibits robust circadian oscillations in the presence of ATP. Using mathematical modeling, we were able to reproduce quantitatively the experimentally observed phosphorylation dynamics of the KaiABC clockwork in vitro. We thereby identified a highly non-linear feedback loop through KaiA inactivation as the key synchronization mechanism of KaiC phosphorylation. By using the novel method of native mass spectrometry, we confirm the theoretically predicted complex formation dynamics and show that inactivation of KaiA is a consequence of sequestration by KaiC hexamers and KaiBC complexes. To test further the predictive power of the mathematical model, we reproduced the observed phase synchronization dynamics on entrainment by temperature cycles. Our model gives strong evidence that the underlying entrainment mechanism arises from a temperature-dependent change in the abundance of KaiAC and KaiBC complexes.

Understanding the temporal codes of intra-cellular signals

The health of organisms and cells depends on appropriate responses to diverse internal and external cues, stimuli, or challenges, such as changes in hormone or cytokine levels, or exposure to a pathogen. Cellular responses must be tailored to the identity and intensity of the stimulus and therefore intra-cellular signals must carry information about both. However, signaling mediators often form intricate networks that react to multiple stimuli yet manage to produce stimulus-specific responses. The multi-functionality ('functional pleiotropism') of signaling nodes suggests that biological networks have evolved ways of passing physiologically relevant stimulus information through shared channels. Increasing evidence supports the notion that this is achieved in part through temporal regulation of signaling mediators' activities. The present challenge is to identify the features of temporal activity profile that represent information about a given stimulus and understand how cells read the temporal codes to control their responses.

Frequency control of cell cycle oscillators

The cell cycle oscillator, based on a core negative feedback loop and modified extensively by positive feedback, cycles with a frequency that is regulated by environmental and developmental programs to encompass a wide range of cell cycle times. We discuss how positive feedback allows frequency tuning, how size and morphogenetic checkpoints regulate oscillator frequency, and how extrinsic oscillators such as the circadian clock gate cell cycle frequency. The master cell cycle regulatory oscillator in turn controls the frequency of peripheral oscillators controlling essential events. A recently proposed phase-locking model accounts for this coupling.

Quantitative analysis of cellular metabolic dissipative, self-organized structures

One of the most important goals of the postgenomic era is understanding the metabolic dynamic processes and the functional structures generated by them. Extensive studies during the last three decades have shown that the dissipative self-organization of the functional enzymatic associations, the catalytic reactions produced during the metabolite channeling, the microcompartmentalization of these metabolic processes and the emergence of dissipative networks are the fundamental elements of the dynamical organization of cell metabolism. Here we present an overview of how mathematical models can be used to address the properties of dissipative metabolic structures at different organizational levels, both for individual enzymatic associations and for enzymatic networks. Recent analyses performed with dissipative metabolic networks have shown that unicellular organisms display a singular global enzymatic structure common to all living cellular organisms, which seems to be an intrinsic property of the functional metabolism as a whole. Mathematical models firmly based on experiments and their corresponding computational approaches are needed to fully grasp the molecular mechanisms of metabolic dynamical processes. They are necessary to enable the quantitative and qualitative analysis of the cellular catalytic reactions and also to help comprehend the conditions under which the structural dynamical phenomena and biological rhythms arise. Understanding the molecular mechanisms responsible for the metabolic dissipative structures is crucial for unraveling the dynamics of cellular life.

Timing control in regulatory networks by multisite protein modifications

Computational and experimental studies have yielded quantitative insights into the role for multisite phosphorylation, and other protein modifications, in cell function. This work has emphasized the creation of thresholds and switches for cellular decisions. To date, the dynamics of phosphorylation events have been disregarded yet could be equally relevant for cell function. Here, we discuss theoretical predictions about the kinetic functions of multisite phosphorylation in regulatory networks and how these predictions relate to experimental findings. Using DNA replication as an example, we demonstrate that multisite phosphorylations can support coherent origin firing and robustness against rereplication. We suggest that multisite protein modifications provide a molecular mechanism to robustly time cellular events in the cell cycle, the circadian clock and signal transduction.

A synthetic-natural hybrid oscillator in human cells

Recent studies have shown that many cell-signaling networks contain interactions and feedback loops that give rise to complex dynamics. Synthetic biology has allowed researchers to construct and analyze well-defined signaling circuits exhibiting behavior that can be predicted and quantitatively understood. Combining these approaches--wiring natural network components together with engineered interactions--has the potential to precisely modulate the dynamics of endogenous signaling processes and control the cell decisions they influence. Here, we focus on the p53 signaling pathway as a template for constructing a tunable oscillator comprised of both natural and synthetic components in mammalian cells. We find that a reduced p53 circuit implementing a single feedback loop preserves some features of the full network's dynamics, exhibiting pulses of p53 with tightly controlled timing. However, in contrast to the full natural p53 network, these pulses are damped in individual cells, with amplitude that depends on the input strength. Guided by a computational model of the reduced circuit, we constructed and analyzed circuit variants supplemented with synthetic positive and negative feedback loops and subjected to chemical perturbation. Our work demonstrates that three important features of oscillator dynamics--amplitude, period, and the rate of damping--can be controlled by manipulating stimulus level, interaction strength, and feedback topology. The approaches taken here may be useful for the rational design of synthetic networks with defined dynamics, and for identifying perturbations that control dynamics in natural biological circuits for research or therapeutic purposes.

Parameter optimization by using differential elimination: a general approach for introducing constraints into objective functions

Background

The investigation of network dynamics is a major issue in systems and synthetic biology. One of the essential steps in a dynamics investigation is the parameter estimation in the model that expresses biological phenomena. Indeed, various techniques for parameter optimization have been devised and implemented in both free and commercial software. While the computational time for parameter estimation has been greatly reduced, due to improvements in calculation algorithms and the advent of high performance computers, the accuracy of parameter estimation has not been addressed.

Results

We propose a new approach for parameter optimization by using differential elimination, to estimate kinetic parameter values with a high degree of accuracy. First, we utilize differential elimination, which is an algebraic approach for rewriting a system of differential equations into another equivalent system, to derive the constraints between kinetic parameters from differential equations. Second, we estimate the kinetic parameters introducing these constraints into an objective function, in addition to the error function of the square difference between the measured and estimated data, in the standard parameter optimization method. To evaluate the ability of our method, we performed a simulation study by using the objective function with and without the newly developed constraints: the parameters in two models of linear and non-linear equations, under the assumption that only one molecule in each model can be measured, were estimated by using a genetic algorithm (GA) and particle swarm optimization (PSO). As a result, the introduction of new constraints was dramatically effective: the GA and PSO with new constraints could successfully estimate the kinetic parameters in the simulated models, with a high degree of accuracy, while the conventional GA and PSO methods without them frequently failed.

Conclusions

The introduction of new constraints in an objective function by using differential elimination resulted in the drastic improvement of the estimation accuracy in parameter optimization methods. The performance of our approach was illustrated by simulations of the parameter optimization for two models of linear and non-linear equations, which included unmeasured molecules, by two types of optimization techniques. As a result, our method is a promising development in parameter optimization.

Vesicle-to-micelle oscillations and spatial patterns

A pH oscillator is coupled to and controls rhythmic interconversion of nanoscopic vesicles and micelles made of fatty acids. When changes in pH are combined with diffusion, self-assembly produces spatially extended patterns of vesicle/micelle "stripes" or concentric "shells".

Gene expression model (in)validation by Fourier analysis

Background

The determination of the right model structure describing a gene regulation network and the identification of its parameters are major goals in systems biology. The task is often hampered by the lack of relevant experimental data with sufficiently low noise level, but the subset of genes whose concentration levels exhibit an oscillatory behavior in time can readily be analyzed on the basis of their Fourier spectrum, known to turn complex signals into few relatively noise-free parameters. Such genes therefore offer opportunities of understanding gene regulation quantitatively.

Results

Fourier analysis is applied to data on gene expression levels in mouse liver cells that oscillate according to the circadian rhythm. Several model structures in the form of linear and nonlinear differential equations are matched to the data and it is shown that although the considered models can reproduce many features of the oscillatory patterns, some can be excluded on the basis of Fourier analysis without appeal to prior knowledge of regulatory pathways. A systematic method for testing models is also proposed based on measuring the effects of variations in gene copy-number on the expression levels of coupled genes.

Conclusions

Fourier analysis is a technique that is well-adapted to the study of biological oscillators and can be used instead or in addition to conventional modeling techniques. Its usefulness will increase as more high-resolution data become available.

Oscillations in Cdc14 release and sequestration reveal a circuit underlying mitotic exit

The phosphatase Cdc14 exerts negative feedback on its upstream regulators to limit its release from the nucleolus to once per cell cycle.

In budding yeast, the phosphatase Cdc14 orchestrates progress through anaphase and mitotic exit, thereby resetting the cell cycle for a new round of cell division. Two consecutive pathways, Cdc fourteen early anaphase release (FEAR) and mitotic exit network (MEN), contribute to the progressive activation of Cdc14 by regulating its release from the nucleolus, where it is kept inactive by Cfi1. In this study, we show that Cdc14 activation requires the polo-like kinase Cdc5 together with either Clb–cyclin-dependent kinase (Cdk) or the MEN kinase Dbf2. Once active, Cdc14 triggers a negative feedback loop that, in the presence of stable levels of mitotic cyclins, generates periodic cycles of Cdc14 release and sequestration. Similar phenotypes have been described for yeast bud formation and centrosome duplication. A common theme emerges where events that must happen only once per cycle, although intrinsically capable of oscillations, are limited to one occurrence by the cyclin–Cdk cell cycle engine.

Cytokinesis through biochemical-mechanical feedback loops

Cytokinesis is emerging as a control system defined by interacting biochemical and mechanical modules, which form a system of feedback loops. This integrated system accounts for the regulation and kinetics of cytokinesis furrowing and demonstrates that cytokinesis is a whole-cell process in which the global and equatorial cortices and cytoplasm are active players in the system. Though originally defined in Dictyostelium, features of the control system are recognizable in other organisms, suggesting a universal mechanism for cytokinesis regulation and contractility.

Reliability of transcriptional cycles and the yeast cell-cycle oscillator

A recently published transcriptional oscillator associated with the yeast cell cycle provides clues and raises questions about the mechanisms underlying autonomous cyclic processes in cells. Unlike other biological and synthetic oscillatory networks in the literature, this one does not seem to rely on a constitutive signal or positive auto-regulation, but rather to operate through stable transmission of a pulse on a slow positive feedback loop that determines its period. We construct a continuous-time Boolean model of this network, which permits the modeling of noise through small fluctuations in the timing of events, and show that it can sustain stable oscillations. Analysis of simpler network models shows how a few building blocks can be arranged to provide stability against fluctuations. Our findings suggest that the transcriptional oscillator in yeast belongs to a new class of biological oscillators.

Technologies such as gene arrays enable acquisition of large amounts of data on gene expression variations, which reveal the structures of gene regulatory networks that govern the metabolic and developmental machinery in the cell. We study a model of an oscillatory gene regulatory network that has been recently suggested to play an integral role in maintaining the cell cycle in yeast. The oscillator differs from other known biological and synthetic oscillatory networks in that it seems to rely on a long positive feedback loop. We show that the presence of certain stabilizing sub-networks can account for the robustness and the unusual architecture of this oscillator. Our modeling approach elucidates both the logical structure of the system and the importance of the timing of update events.

A comparative analysis of synthetic genetic oscillators

Synthetic biology is a rapidly expanding discipline at the interface between engineering and biology. Much research in this area has focused on gene regulatory networks that function as biological switches and oscillators. Here we review the state of the art in the design and construction of oscillators, comparing the features of each of the main networks published to date, the models used for in silico design and validation and, where available, relevant experimental data. Trends are apparent in the ways that network topology constrains oscillator characteristics and dynamics. Also, noise and time delay within the network can both have constructive and destructive roles in generating oscillations, and stochastic coherence is commonplace. This review can be used to inform future work to design and implement new types of synthetic oscillators or to incorporate existing oscillators into new designs.

Quantitative genetic interactions reveal biological modularity

Traditionally, research has been reductionist, characterizing the individual components of biological systems. But new technologies have increased the size and scope of biological data, and systems approaches have broadened the view of how these components are interconnected. Here, we discuss how quantitative mapping of genetic interactions enhances our view of biological systems, allowing their deeper interrogation across different biological scales.

Functional motifs in biochemical reaction networks

The signal-response characteristics of a living cell are determined by complex networks of interacting genes, proteins, and metabolites. Understanding how cells respond to specific challenges, how these responses are contravened in diseased cells, and how to intervene pharmacologically in the decision-making processes of cells requires an accurate theory of the information-processing capabilities of macromolecular regulatory networks. Adopting an engineer's approach to control systems, we ask whether realistic cellular control networks can be decomposed into simple regulatory motifs that carry out specific functions in a cell. We show that such functional motifs exist and review the experimental evidence that they control cellular responses as expected.

Harmonic oscillations in homeostatic controllers: Dynamics of the p53 regulatory system

Homeostatic mechanisms are essential for the protection and adaptation of organisms in a changing and challenging environment. Previously, we have described molecular mechanisms that lead to robust homeostasis/adaptation under inflow or outflow perturbations. Here we report that harmonic oscillations occur in models of such homeostatic controllers and that a close relationship exists between the control of the p53/Mdm2 system and that of a homeostatic inflow controller. This homeostatic control model of the p53 system provides an explanation why large fluctuations in the amplitude of p53/Mdm2 oscillations may arise as part of the homeostatic regulation of p53 by Mdm2 under DNA-damaging conditions. In the presence of DNA damage p53 is upregulated, but is subject to a tight control by Mdm2 and other factors to avoid a premature apoptotic response of the cell at low DNA damage levels. One of the regulatory steps is the Mdm2-mediated degradation of p53 by the proteasome. Oscillations in the p53/Mdm2 system are considered to be part of a mechanism by which a cell decides between cell cycle arrest/DNA repair and apoptosis. In the homeostatic inflow control model, harmonic oscillations in p53/Mdm2 levels arise when the binding strength of p53 to degradation complexes increases. Due to the harmonic character of the oscillations rapid fluctuating noise can lead, as experimentally observed, to large variations in the amplitude of the oscillation but not in their period, a behavior which has been difficult to simulate by deterministic limit-cycle models. In conclusion, the oscillatory response of homeostatic controllers may provide new insights into the origin and role of oscillations observed in homeostatically controlled molecular networks.

Avoiding transcription factor competition at promoter level increases the chances of obtaining oscillation

Background

The ultimate goal of synthetic biology is the conception and construction of genetic circuits that are reliable with respect to their designed function (e.g. oscillators, switches). This task remains still to be attained due to the inherent synergy of the biological building blocks and to an insufficient feedback between experiments and mathematical models. Nevertheless, the progress in these directions has been substantial.

Results

It has been emphasized in the literature that the architecture of a genetic oscillator must include positive (activating) and negative (inhibiting) genetic interactions in order to yield robust oscillations. Our results point out that the oscillatory capacity is not only affected by the interaction polarity but by how it is implemented at promoter level. For a chosen oscillator architecture, we show by means of numerical simulations that the existence or lack of competition between activator and inhibitor at promoter level affects the probability of producing oscillations and also leaves characteristic fingerprints on the associated period/amplitude features.

Conclusions

In comparison with non-competitive binding at promoters, competition drastically reduces the region of the parameters space characterized by oscillatory solutions. Moreover, while competition leads to pulse-like oscillations with long-tail distribution in period and amplitude for various parameters or noisy conditions, the non-competitive scenario shows a characteristic frequency and confined amplitude values. Our study also situates the competition mechanism in the context of existing genetic oscillators, with emphasis on the Atkinson oscillator.

Immune response modeling of interferon beta-pretreated influenza virus-infected human dendritic cells

The pretreatment of human dendritic cells with interferon-beta enhances their immune response to influenza virus infection. We measured the expression levels of several key players in that response over a period of 13 h both during pretreatment and after viral infection. Their activation profiles reflect the presence of both negative and positive feedback loops in interferon induction and interferon signaling pathway. Based on these measurements, we have developed a comprehensive computational model of cellular immune response that elucidates its mechanism and its dynamics in interferon-pretreated dendritic cells, and provides insights into the effects of duration and strength of pretreatment.

Costs and constraints from time-delayed feedback in small gene regulatory motifs

The multistep character of transcription, translation, and protein modification inevitably leads to time delays between sensing gene regulatory signals and responding with changed concentrations of functional proteins. However, the interplay between the time-delayed and the stochastic nature of gene regulation has been poorly investigated. Here we present an extension of the linear noise approximation which makes it possible to estimate second moments--variances and covariances--of fluctuations around stationary states in time-delayed systems. The usefulness of the method is exemplified by analyzing two ubiquitous regulatory motifs. In the first system, we show that there is an optimal combination of transcriptional repression and direct product inhibition in determining the activity of an enzyme system. In particular, we demonstrate that direct product inhibition is necessary to avoid deleterious fluctuations in a system when the gene regulatory response is delayed. The second system is an anabolic motif where the substrate fluxes are balanced by time-delayed regulation responding to the substrate concentrations. The extended linear noise approximation makes it possible to show analytically that increased association rate between the substrates leads to a lower product flux because of increasing unbalance in substrate pools.

Modelling the molecular mechanisms of synaptic plasticity using systems biology approaches

Synaptic plasticity is thought to underlie learning and memory, but the complexity of the interactions between the ion channels, enzymes and genes that are involved in synaptic plasticity impedes a deep understanding of this phenomenon. Computer modelling has been used to investigate the information processing that is performed by the signalling pathways involved in synaptic plasticity in principal neurons of the hippocampus, striatum and cerebellum. In the past few years, new software developments that combine computational neuroscience techniques with systems biology techniques have allowed large-scale, kinetic models of the molecular mechanisms underlying long-term potentiation and long-term depression. We highlight important advancements produced by these quantitative modelling efforts and introduce promising approaches that use advancements in live-cell imaging.

A modular positive feedback-based gene amplifier

Background

Positive feedback is a common mechanism used in the regulation of many gene circuits as it can amplify the response to inducers and also generate binary outputs and hysteresis. In the context of electrical circuit design, positive feedback is often considered in the design of amplifiers. Similar approaches, therefore, may be used for the design of amplifiers in synthetic gene circuits with applications, for example, in cell-based sensors.

Results

We developed a modular positive feedback circuit that can function as a genetic signal amplifier, heightening the sensitivity to inducer signals as well as increasing maximum expression levels without the need for an external cofactor. The design utilizes a constitutively active, autoinducer-independent variant of the quorum-sensing regulator LuxR. We experimentally tested the ability of the positive feedback module to separately amplify the output of a one-component tetracycline sensor and a two-component aspartate sensor. In each case, the positive feedback module amplified the response to the respective inducers, both with regards to the dynamic range and sensitivity.

Conclusions

The advantage of our design is that the actual feedback mechanism depends only on a single gene and does not require any other modulation. Furthermore, this circuit can amplify any transcriptional signal, not just one encoded within the circuit or tuned by an external inducer. As our design is modular, it can potentially be used as a component in the design of more complex synthetic gene circuits.

Computational systems biology and dose-response modeling in relation to new directions in toxicity testing

The new paradigm envisioned for toxicity testing in the 21st century advocates shifting from the current animal-based testing process to a combination of in vitro cell-based studies, high-throughput techniques, and in silico modeling. A strategic component of the vision is the adoption of the systems biology approach to acquire, analyze, and interpret toxicity pathway data. As key toxicity pathways are identified and their wiring details elucidated using traditional and high-throughput techniques, there is a pressing need to understand their qualitative and quantitative behaviors in response to perturbation by both physiological signals and exogenous stressors. The complexity of these molecular networks makes the task of understanding cellular responses merely by human intuition challenging, if not impossible. This process can be aided by mathematical modeling and computer simulation of the networks and their dynamic behaviors. A number of theoretical frameworks were developed in the last century for understanding dynamical systems in science and engineering disciplines. These frameworks, which include metabolic control analysis, biochemical systems theory, nonlinear dynamics, and control theory, can greatly facilitate the process of organizing, analyzing, and understanding toxicity pathways. Such analysis will require a comprehensive examination of the dynamic properties of "network motifs"--the basic building blocks of molecular circuits. Network motifs like feedback and feedforward loops appear repeatedly in various molecular circuits across cell types and enable vital cellular functions like homeostasis, all-or-none response, memory, and biological rhythm. These functional motifs and associated qualitative and quantitative properties are the predominant source of nonlinearities observed in cellular dose response data. Complex response behaviors can arise from toxicity pathways built upon combinations of network motifs. While the field of computational cell biology has advanced rapidly with increasing availability of new data and powerful simulation techniques, a quantitative orientation is still lacking in life sciences education to make efficient use of these new tools to implement the new toxicity testing paradigm. A revamped undergraduate curriculum in the biological sciences including compulsory courses in mathematics and analysis of dynamical systems is required to address this gap. In parallel, dissemination of computational systems biology techniques and other analytical tools among practicing toxicologists and risk assessment professionals will help accelerate implementation of the new toxicity testing vision.

Controlling biological networks by time-delayed signals

This paper describes the use of time-delayed feedback to regulate the behaviour of biological networks. The general ideas on specific transcriptional regulatory and neural networks are demonstrated. It is shown that robust yet tunable controllers can be constructed that provide the biological systems with model-engineered inputs. The results indicate that time delay modulation may serve as an efficient biocompatible control tool.

The molecular clockwork of a protein-based circadian oscillator

The circadian clock of the cyanobacterium Synechococcus elongatus PCC 7942 is governed by a core oscillator consisting of the proteins KaiA, KaiB, and KaiC. Remarkably, circadian oscillations in the phosphorylation state of KaiC can be reconstituted in a test tube by mixing the three Kai proteins and adenosine triphosphate. The in vitro oscillator provides a well-defined system in which experiments can be combined with mathematical analysis to understand the mechanism of a highly robust biological oscillator. In this Review, we summarize the biochemistry of the Kai proteins and examine models that have been proposed to explain how oscillations emerge from the properties of the oscillator's constituents.

Cycling expression and cooperative operator interaction in the trp operon of Escherichia coli

Oscillatory behaviour in the tryptophan operon of an Escherichia coli mutant strain lacking the enzyme-inhibition regulatory mechanism has been observed by Bliss et al. but not confirmed by others. This behaviour could be important from the standpoint of synthetic biology, whose goals include the engineering of intracellular genetic oscillators. This work is devoted to investigating, from a mathematical modelling point of view, the possibility that the trp operon of the E. coli inhibition-free strain expresses cyclically. For that we extend a previously introduced model for the regulatory pathway of the tryptophan operon in Escherichia coli to account for the observed multiplicity and cooperativity of repressor binding sites. Thereafter we investigate the model dynamics using deterministic numeric solutions, stochastic simulations, and analytic studies. Our results suggest that a quasi-periodic behaviour could be observed in the trp operon expression level of single bacteria.

How to turn a genetic circuit into a synthetic tunable oscillator, or a bistable switch

Systems and Synthetic Biology use computational models of biological pathways in order to study in silico the behaviour of biological pathways. Mathematical models allow to verify biological hypotheses and to predict new possible dynamical behaviours. Here we use the tools of non-linear analysis to understand how to change the dynamics of the genes composing a novel synthetic network recently constructed in the yeast Saccharomyces cerevisiae for In-vivo Reverse-engineering and Modelling Assessment (IRMA). Guided by previous theoretical results that make the dynamics of a biological network depend on its topological properties, through the use of simulation and continuation techniques, we found that the network can be easily turned into a robust and tunable synthetic oscillator or a bistable switch. Our results provide guidelines to properly re-engineering in vivo the network in order to tune its dynamics.

Model of gene transcription including the return of a RNA polymerase to the beginning of a transcriptional cycle

The gene transcription occurs via the RNA polymerase (RNAP) recruitment on the DNA promoter sequence, formation of a locally open DNA chain, promoter escape, steps of the RNA synthesis, and RNA and RNAP release after reading the final DNA base. Just after the end of the RNA synthesis, RNAP surrounds the closed DNA chain and may diffuse along DNA, desorb, or reach the promoter and start the RNA-synthesis cycle again. We present a generic kinetic model taking the latter steps into account and show analytically and by Monte Carlo simulations that it predicts transcriptional bursts even in the absence of explicit regulation of the transcription by master proteins.

Positive-feedback loops in cell cycle progression

A positive-feedback loop is a simple motif that is ubiquitous to the modules and networks that comprise cellular signaling systems. Signaling behaviors that are synonymous with positive feedback include amplification and rapid switching, maintenance, and the coherence of outputs. Recent advances have been made towards understanding how positive-feedback loops function, as well as their mechanistic basis in controlling eukaryotic cell cycle progression. Some of these advances will be reviewed here, including: how cyclin controls passage through Start and maintains coherence of G1/S regulon expression in yeast; how Polo-like kinase 1 activation is driven by Bora and Aurora A, and its expression is stimulated by Forkhead Box M1 in mammalian cells; and how some of the various dynamic behaviors of spindle assembly and anaphase onset can be produced.

Features of anti-inflammatory effects of modulated extremely high-frequency electromagnetic radiation

Using a model of acute zymosan-induced paw edema in NMRI mice, we test the hypothesis that anti-inflammatory effects of extremely high-frequency electromagnetic radiation (EHF EMR) can be essentially modified by application of pulse modulation with certain frequencies. It has been revealed that a single exposure of animals to continuous EHF EMR for 20 min reduced the exudative edema of inflamed paw on average by 19% at intensities of 0.1-0.7 mW/cm(2) and frequencies from the range of 42.2-42.6 GHz. At fixed effective carrier frequency of 42.2 GHz, the anti-inflammatory effect of EHF EMR did not depend on modulation frequencies, that is, application of different modulation frequencies from the range of 0.03-100 Hz did not lead to considerable changes in the effect level. On the contrary, at "ineffective" carrier frequencies of 43.0 and 61.22 GHz, the use of modulation frequencies of 0.07-0.1 and 20-30 Hz has allowed us to restore the effect up to a maximal level. The results obtained show the critical dependence of anti-inflammatory action of low-intensity EHF EMR on carrier and modulation frequencies. Within the framework of this study, the possibility of changing the level of expected biological effect of modulated EMR by a special selection of combination of carrier and modulation frequencies is confirmed.

Regulation of copper transport in Arabidopsis thaliana: a biochemical oscillator?

Plants are among the most versatile higher eukaryotes in accommodating environmental copper availability to largely variable demands. In particular, copper deficiency in soils is a threat for plant survival since it mostly affects reproductive structures. One of the strategies that plant cells use to overcome this situation is to increase copper levels by expressing high-affinity copper transporters delivering the metal to the cytosol. In this minireview, we discuss recent advances in the structure, function, and regulation of the CTR/COPT family of copper transporters, and pay special attention to the Arabidopsis thaliana counterparts. These are constituted by transmembrane polypeptides, containing several copper-binding sequences of functional and/or regulatory value, and assembling as trimers. Copper deficiency activates the expression of some members of the COPT family via the interaction of the SPL7 transcription factor with reiterative GTAC motifs present in their promoters. Interestingly, the regulation of the synthesis of these transporters by copper itself constitutes a negative-feedback loop that could cause a sustained oscillation in the cytosolic copper levels. We analyze the theoretical conditions required for this hypothetical copper oscillation and the potential advantages of synchronization with other cycles. Diverse data in other organisms point to the relationship between copper homeostasis and circadian cycles.

Exciting traffic jams: nonlinear phenomena behind traffic jam formation on highways

A nonlinear car-following model is studied with driver reaction time delay by using state-of-the-art numerical continuations techniques. These allow us to unveil the detailed microscopic dynamics as well as to extract macroscopic properties of traffic flow. Parameter domains are determined where the uniform flow equilibrium is stable but sufficiently large excitations may trigger traffic jams. This behavior becomes more robust as the reaction time delay is increased.

Frequency spectrum of transepithelial potential difference reveals transport-related oscillations

How epithelia transport fluid is a fundamental issue that is unresolved. Explanations offered include molecular engines, local transcellular osmosis, local paracellular osmosis, and paracellular fluid transport. On the basis of experimental and theoretical work done on corneal endothelium, a fluid transporting epithelium, we suggest electroosmotic coupling at the level of the intercellular junctions driven by the transendothelial electrical potential difference as an explanation of paracellular fluid transport. We collect frequency spectra of that potential difference in real-time. For what we believe is the first time for any epithelium, we report that, unexpectedly, the potential difference displays oscillations at many characteristic frequencies. We also show that on both stimulating cell activity and inhibiting ion transport mechanisms, there are corresponding changes in the oscillations amplitudes that mirror changes known previously in rates of fluid transport. We believe these findings provide a novel tool to study the kinetics of electrogenic elements such as channels and transporters, which from this evidence would give rise to current oscillations with characteristic periods going from 150 ms to 8 s.

Interlinking positive and negative feedback loops creates a tunable motif in gene regulatory networks

Positive and negative feedback loops are often coupled to perform various functions in gene regulatory networks, acting as bistable switches, oscillators, and excitable devices. It is implied that such a system with interlinked positive and negative feedback loops is a flexible motif that can modulate itself among various functions. Here, we developed a minimal model for the system and systematically explored its dynamics and performance advantage in response to stimuli in a unifying framework. The system indeed displays diverse behaviors when the strength of feedback loops is changed. First, the system can be tunable from monostability to bistability by increasing the strength of positive feedback, and the bistability regime is modulated by the strength of negative feedback. Second, the system undergoes transitions from bistability to excitability and to oscillation with increasing the strength of negative feedback, and the reverse conversion occurs by enhancing the strength of positive feedback. Third, the system is more flexible than a single feedback loop; it can produce robust larger-amplitude oscillations over a wider stimulus regime compared with a single time-delayed negative feedback loop. Furthermore, the tunability of the system depends mainly on the topology of coupled feedback loops but less on the exact parameter values or the mode of interactions between model components. Thus, our results interpret why such a system represents a tunable motif and can accomplish various functions. These also suggest that coupled feedback loops can act as toolboxes for engineering diverse functional circuits in synthetic biology.

The ups and downs of p53: understanding protein dynamics in single cells

Cells living in a complex environment must constantly detect, process and appropriately respond to changing signals. Therefore, all cellular information processing is dynamic in nature. As a consequence, understanding the process of signal transduction often requires detailed quantitative analysis of dynamic behaviours. Here, we focus on the oscillatory dynamics of the tumour suppressor protein p53 as a model for studying protein dynamics in single cells to better understand its regulation and function.