Frustrated bistability as a means to engineer oscillations in biological systems

Autoregulation of Transcription and Translation: A Qualitative Analysis

The regulation of both mRNA transcription and translation by down-stream gene products allows for a range of rich dynamical behaviours (e.g. homeostatic, oscillatory, excitability and intermittent solutions). Here, qualitative analysis is applied to an existing model of a gene regulatory network in which a protein dimer inhibits its own transcription and upregulates its own translation rate. It is demonstrated that the model possesses a unique steady state, conditions are derived under which limit cycle solutions arise and estimates are provided for the oscillator period in the limiting case of a relaxation oscillator. The analysis demonstrates that oscillations can arise only if mRNA is more stable than protein and the effect of nonlinear translation inhibition is sufficiently strong. Moreover, it is shown that the oscillation period can vary non-monotonically with transcription rate. Thus the proposed framework can provide an explanation for observed species-specific dependency of segmentation clock period on Notch signalling activity. Finally, this study facilitates the application of the proposed model to more general biological settings where post transcriptional regulation effects are likely important.

A nested bistable module within a negative feedback loop ensures different types of oscillations in signaling systems

In this article, we consider a double phosphorylation cycle, a ubiquitous signaling component, having the ability to display bistability, a behavior strongly related to the existence of positive feedback loops. If this component is connected to other signaling elements, it very likely undergoes some sort of protein-protein interaction. In several cases, these interactions result in a non-explicit negative feedback effect, leading to interlinked positive and negative feedbacks. This combination was studied in the literature as a way to generate relaxation-type oscillations. Here, we show that the two feedbacks together ensure two types of oscillations, the relaxation-type ones and a smoother type of oscillations functioning in a very narrow range of frequencies, in such a way that outside that range, the amplitude of the oscillations is severely compromised. Even more, we show that the two feedbacks are essential for both oscillatory types to emerge, and it is their hierarchy what determines the type of oscillation at work. We used bifurcation analyses and amplitude vs. frequency curves to characterize and classify the oscillations. We also applied the same ideas to another simple model, with the goal of generalizing what we learned from signaling models. The results obtained display the wealth of oscillatory dynamics that exists in a system with a bistable module nested within a negative feedback loop, showing how to transition between different types of oscillations and other dynamical behaviors such as excitability. Our work provides a framework for the study of other oscillatory systems based on bistable modules, from simple two-component models to more complex examples like the MAPK cascade and experimental cases like cell cycle oscillators.

Multi-stability in cellular differentiation enabled by a network of three mutually repressing master regulators

Identifying the design principles of complex regulatory networks driving cellular decision-making remains essential to decode embryonic development as well as enhance cellular reprogramming. A well-studied network motif involved in cellular decision-making is a toggle switch-a set of two opposing transcription factors A and B, each of which is a master regulator of a specific cell fate and can inhibit the activity of the other. A toggle switch can lead to two possible states-(high A, low B) and (low A, high B)-and drives the 'either-or' choice between these two cell fates for a common progenitor cell. However, the principles of coupled toggle switches remain unclear. Here, we investigate the dynamics of three master regulators A, B and C inhibiting each other, thus forming three-coupled toggle switches to form a toggle triad. Our simulations show that this toggle triad can lead to co-existence of cells into three differentiated 'single positive' phenotypes-(high A, low B, low C), (low A, high B, low C) and (low A, low B, high C). Moreover, the hybrid or 'double positive' phenotypes-(high A, high B, low C), (low A, high B, high C) and (high A, low B, high C)-can coexist together with 'single positive' phenotypes. Including self-activation loops on A, B and C can increase the frequency of 'double positive' states. Finally, we apply our results to understand cellular decision-making in terms of differentiation of naive CD4⁺ T cells into Th1, Th2 and Th17 states, where hybrid Th1/Th2 and hybrid Th1/Th17 cells have been reported in addition to the Th1, Th2 and Th17 ones. Our results offer novel insights into the design principles of a multi-stable network topology and provide a framework for synthetic biology to design tristable systems.

The role of cooperativity in a p53-miR34 dynamical mathematical model

The objective of this study is to evaluate the role of cooperativity, captured by the Hill coefficient, in a minimal mathematical model describing the interactions between p53 and miR-34a. The model equations are analyzed for negative, none and normal cooperativity using a specific version of bifurcation theory and they are solved numerically. Special attention is paid to the sign of so-called first Lyapunov value. Interpretations of the results are given, both according to dynamic theory and in biological terms. In terms of cell signaling, we propose the hypothesis that when the outgoing signal of a system spends a physiologically significant amount of time outside of its equilibrium state, then the value of that signal can be sampled at any point along the trajectory towards that equilibrium and indeed, at multiple points. Coupled with non-linear behavior, such as that caused by cooperativity, this feature can account for a complex and varied response, which p53 is known for. From dynamical point of view, we found that when cooperativity is negative, the system has only one stable equilibrium point. In the absence of cooperativity, there is a single unstable equilibrium point with a critical boundary of stability. In the case with normal cooperativity, the system can have one, two, or three steady states with both, bi-stability and bi-instability occurring.

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

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

Combining a Toggle Switch and a Repressilator within the AC-DC Circuit Generates Distinct Dynamical Behaviors

Although the structure of a genetically encoded regulatory circuit is an important determinant of its function, the relationship between circuit topology and the dynamical behaviors it can exhibit is not well understood. Here, we explore the range of behaviors available to the AC-DC circuit. This circuit consists of three genes connected as a combination of a toggle switch and a repressilator. Using dynamical systems theory, we show that the AC-DC circuit exhibits both oscillations and bistability within the same region of parameter space; this generates emergent behaviors not available to either the toggle switch or the repressilator alone. The AC-DC circuit can switch on oscillations via two distinct mechanisms, one of which induces coherence into ensembles of oscillators. In addition, we show that in the presence of noise, the AC-DC circuit can behave as an excitable system capable of spatial signal propagation or coherence resonance. Together, these results demonstrate how combinations of simple motifs can exhibit multiple complex behaviors.

Revealing determinants of two-phase dynamics of P53 network under gamma irradiation based on a reduced 2D relaxation oscillator model

This study proposes a two-dimensional (2D) oscillator model of p53 network, which is derived via reducing the multidimensional two-phase dynamics model into a model of ataxia telangiectasia mutated (ATM) and Wip1 variables, and studies the impact of p53-regulators on cell fate decision. First, the authors identify a 6D core oscillator module, then reduce this module into a 2D oscillator model while preserving the qualitative behaviours. The introduced 2D model is shown to be an excitable relaxation oscillator. This oscillator provides a mechanism that leads diverse modes underpinning cell fate, each corresponding to a cell state. To investigate the effects of p53 inhibitors and the intrinsic time delay of Wip1 on the characteristics of oscillations, they introduce also a delay differential equation version of the 2D oscillator. They observe that the suppression of p53 inhibitors decreases the amplitudes of p53 oscillation, though the suppression increases the sustained level of p53. They identify Wip1 and P53DINP1 as possible targets for cancer therapies considering their impact on the oscillator, supported by biological findings. They model some mutations as critical changes of the phase space characteristics. Possible cancer therapeutic strategies are then proposed for preventing these mutations' effects using the phase space approach.

Genetically tunable frustration controls allostery in an intrinsically disordered transcription factor

Intrinsically disordered proteins (IDPs) present a functional paradox because they lack stable tertiary structure, but nonetheless play a central role in signaling, utilizing a process known as allostery. Historically, allostery in structured proteins has been interpreted in terms of propagated structural changes that are induced by effector binding. Thus, it is not clear how IDPs, lacking such well-defined structures, can allosterically affect function. Here, we show a mechanism by which an IDP can allosterically control function by simultaneously tuning transcriptional activation and repression, using a novel strategy that relies on the principle of ‘energetic frustration’. We demonstrate that human glucocorticoid receptor tunes this signaling in vivo by producing translational isoforms differing only in the length of the disordered region, which modulates the degree of frustration. We expect this frustration-based model of allostery will prove to be generally important in explaining signaling in other IDPs.

Proteins carry out most of the key tasks inside cells. To perform these roles, proteins must fold up to form complex three-dimensional structures. Researchers used to think that the useful parts of proteins all had set structures. However, we now know that ‘disordered’ proteins with variable structures are common and disordered parts of proteins can have vital roles.

In a process called allosteric regulation, regulator molecules can increase or decrease the activity of a protein by binding to it. This binding was thought to work by changing the structure of the protein, but it was not clear how this works in disordered proteins. To investigate, Li et al. studied a disordered protein called glucocorticoid receptor, and found that disordered regions can have opposing effects on other regions of the protein. This creates a ‘tug-of-war’ that Li et al. term “energetic frustration”, whereby the activity of the protein results from the combination of the opposing interactions.

Further investigation revealed that the glucorticoid receptor produces different versions of itself that have different degrees of energetic frustration, which alters how effectively the proteins perform their tasks. This means that the protein can regulate its own activity even in the absence of binding to regulator molecules.

The concept of energetic frustration could enhance our understanding of the many different proteins that contain disordered regions. Eventually, this knowledge could be used to develop drugs that alter the activity of these proteins and so could form part of treatments for a wide range of conditions including autoimmune diseases (such as rheumatoid arthritis and lupus), cancers, and organ rejection for transplant patients. The results presented by Li et al. suggest where more research is needed to achieve this goal. For example, we need to understand more about the stability of disordered protein regions, and to identify which surfaces of the proteins interact with each other.

Cell signaling as a cognitive process

Cellular identity as defined through morphology and function emerges from intracellular signaling networks that communicate between cells. Based on recursive interactions within and among these intracellular networks, dynamical solutions in terms of biochemical behavior are generated that can differ from those in isolated cells. In this way, cellular heterogeneity in tissues can be established, implying that cell identity is not intrinsically predetermined by the genetic code but is rather dynamically maintained in a cognitive manner. We address how to experimentally measure the flow of information in intracellular biochemical networks and demonstrate that even simple causality motifs can give rise to rich, context-dependent dynamic behavior. The concept how intercellular communication can result in novel dynamical solutions is applied to provide a contextual perspective on cell differentiation and tumorigenesis.

Relaxation oscillations and hierarchy of feedbacks in MAPK signaling

We formulated a computational model for a MAPK signaling cascade downstream of the EGF receptor to investigate how interlinked positive and negative feedback loops process EGF signals into ERK pulses of constant amplitude but dose-dependent duration and frequency. A positive feedback loop involving RAS and SOS, which leads to bistability and allows for switch-like responses to inputs, is nested within a negative feedback loop that encompasses RAS and RAF, MEK, and ERK that inhibits SOS via phosphorylation. This negative feedback, operating on a longer time scale, changes switch-like behavior into oscillations having a period of 1 hour or longer. Two auxiliary negative feedback loops, from ERK to MEK and RAF, placed downstream of the positive feedback, shape the temporal ERK activity profile but are dispensable for oscillations. Thus, the positive feedback introduces a hierarchy among negative feedback loops, such that the effect of a negative feedback depends on its position with respect to the positive feedback loop. Furthermore, a combination of the fast positive feedback involving slow-diffusing membrane components with slower negative feedbacks involving faster diffusing cytoplasmic components leads to local excitation/global inhibition dynamics, which allows the MAPK cascade to transmit paracrine EGF signals into spatially non-uniform ERK activity pulses.

Collective dynamics of time-delay-coupled phase oscillators in a frustrated geometry

We study the effect of time delay on the dynamics of a system of repulsively coupled nonlinear oscillators that are configured as a geometrically frustrated network. In the absence of time delay, frustrated systems are known to possess a high degree of multistability among a large number of coexisting collective states except for the fully synchronized state that is normally obtained for attractively coupled systems. Time delay in the coupling is found to remove this constraint and to lead to such a synchronized ground state over a range of parameter values. A quantitative study of the variation of frustration in a system with the amount of time delay has been made and a universal scaling behavior is found. The variation in frustration as a function of the product of time delay and the collective frequency of the system is seen to lie on a characteristic curve that is common for all natural frequencies of the identical oscillators and coupling strengths. Thus time delay can be used as a tuning parameter to control the amount of frustration in a system and thereby influence its collective behavior. Our results can be of potential use in a host of practical applications in physical and biological systems in which frustrated configurations and time delay are known to coexist.

The role of mRNA and protein stability in the function of coupled positive and negative feedback systems in eukaryotic cells

Oscillators and switches are important elements of regulation in biological systems. These are composed of coupling negative feedback loops, which cause oscillations when delayed, and positive feedback loops, which lead to memory formation. Here, we examine the behavior of a coupled feedback system, the Negative Autoregulated Frustrated bistability motif (NAF). This motif is a combination of two previously explored motifs, the frustrated bistability motif (FBM) and the negative auto regulation motif (NAR), which both can produce oscillations. The NAF motif was previously suggested to govern long term memory formation in animals, and was used as a synthetic oscillator in bacteria. We build a mathematical model to analyze the dynamics of the NAF motif. We show analytically that the NAF motif requires an asymmetry in the strengths of activation and repression links in order to produce oscillations. We show that the effect of time delays in eukaryotic cells, originating from mRNA export and protein import, are negligible in this system. Based on the reported protein and mRNA half-lives in eukaryotic cells, we find that even though the NAF motif possesses the ability for oscillations, it mostly promotes constant protein expression at the biologically relevant parameter regimes.

Effects of Four Different Regulatory Mechanisms on the Dynamics of Gene Regulatory Cascades

Gene regulatory cascades (GRCs) are common motifs in cellular molecular networks. A given logical function in these cascades, such as the repression of the activity of a transcription factor, can be implemented by a number of different regulatory mechanisms. The potential consequences for the dynamic performance of the GRC of choosing one mechanism over another have not been analysed systematically. Here, we report the construction of a synthetic GRC in Escherichia coli, which allows us for the first time to directly compare and contrast the dynamics of four different regulatory mechanisms, affecting the transcription, translation, stability, or activity of a transcriptional repressor. We developed a biologically motivated mathematical model which is sufficient to reproduce the response dynamics determined by experimental measurements. Using the model, we explored the potential response dynamics that the constructed GRC can perform. We conclude that dynamic differences between regulatory mechanisms at an individual step in a GRC are often concealed in the overall performance of the GRC, and suggest that the presence of a given regulatory mechanism in a certain network environment does not necessarily mean that it represents a single optimal evolutionary solution.

Coupled positive and negative feedbacks produce diverse gene expression patterns in colonies

Formation of patterns is a common feature in the development of multicellular organism as well as of microbial communities. To investigate the formation of gene expression patterns in colonies, we build a mathematical model of two-dimensional colony growth, where cells carry a coupled positive-and-negative-feedback circuit. We demonstrate that the model can produce sectored, target (concentric), uniform, and scattered expression patterns of regulators, depending on gene expression dynamics and nutrient diffusion. We reconstructed the same regulatory structure in Escherichia coli cells and found gene expression patterns on the surface of colonies similar to the ones produced by the computer simulations. By comparing computer simulations and experimental results, we observed that very simple rules of gene expression can yield a spectrum of well-defined patterns in a growing colony. Our results suggest that variations of the protein content among cells lead to a high level of heterogeneity in colonies.

Importance Formation of patterns is a common feature in the development of microbial communities. In this work, we show that a simple genetic circuit composed of a positive-feedback loop and a negative-feedback loop can produce diverse expression patterns in colonies. We obtained similar sets of gene expression patterns in the simulations and in the experiments. Because the combination of positive feedback and negative feedback is common in intracellular molecular networks, our results suggest that the protein content of cells is highly diversified in colonies.

Formation of patterns is a common feature in the development of microbial communities. In this work, we show that a simple genetic circuit composed of a positive-feedback loop and a negative-feedback loop can produce diverse expression patterns in colonies. We obtained similar sets of gene expression patterns in the simulations and in the experiments. Because the combination of positive feedback and negative feedback is common in intracellular molecular networks, our results suggest that the protein content of cells is highly diversified in colonies.

Networks of coupled circuits: from a versatile toggle switch to collective coherent behavior

We study the versatile performance of networks of coupled circuits. Each of these circuits is composed of a positive and a negative feedback loop in a motif that is frequently found in genetic and neural networks. When two of these circuits are coupled with mutual repression, the system can function as a toggle switch. The variety of its states can be controlled by two parameters as we demonstrate by a detailed bifurcation analysis. In the bistable regimes, switches between the coexisting attractors can be induced by noise. When we couple larger sets of these units, we numerically observe collective coherent modes of individual fixed-point and limit-cycle behavior. It is there that the monotonic change of a single bifurcation parameter allows one to control the onset and arrest of the synchronized oscillations. This mechanism may play a role in biological applications, in particular, in connection with the segmentation clock. While tuning the bifurcation parameter, also a variety of transient patterns emerges upon approaching the stationary states, in particular, a self-organized pacemaker in a completely uniformly equipped ensemble, so that the symmetry breaking happens dynamically.

Control of the G-protein cascade dynamics by GDP dissociation inhibitors

A network of the Rho family GTPases, which cycle between inactive GDP-bound and active GTP-bound states, controls key cellular processes, including proliferation and migration. Activating and deactivating GTPase transitions are controlled by guanine nucleotide exchange factors (GEFs), GTPase activating proteins (GAPs) and GDP dissociation inhibitors (GDIs) that sequester GTPases from the membrane to the cytoplasm. Here we show that a cascade of two Rho family GTPases, RhoA and Rac1, regulated by RhoGDI1, exhibits distinct modes of the dynamic behavior, including abrupt, bistable switches, excitable overshoot transitions and oscillations. The RhoGDI1 abundance and signal-induced changes in the RhoGDI1 affinity for GTPases control these different dynamics, enabling transitions from a single stable steady state to bistability, to excitable pulses and to sustained oscillations of GTPase activities. These RhoGDI1-controlled dynamic modes of RhoA and Rac1 activities form the basis of cell migration behaviors, including protrusion-retraction cycles at the leading edge of migrating cells.

Generation of oscillating gene regulatory network motifs

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

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

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

Modeling the NF-κB mediated inflammatory response predicts cytokine waves in tissue

Background

Waves propagating in "excitable media" is a reliable way to transmit signals in space. A fascinating example where living cells comprise such a medium is Dictyostelium D. which propagates waves of chemoattractant to attract distant cells. While neutrophils chemotax in a similar fashion as Dictyostelium D., it is unclear if chemoattractant waves exist in mammalian tissues and what mechanisms could propagate them.

Results

We propose that chemoattractant cytokine waves may naturally develop as a result of NF-κB response. Using a heuristic mathematical model of NF-κB-like circuits coupled in space we show that the known characteristics of NF-κB response favor cytokine waves.

Conclusions

While the propagating wave of cytokines is generally beneficial for inflammation resolution, our model predicts that there exist special conditions that can cause chronic inflammation and re-occurrence of acute inflammatory response.

Phase and frequency entrainment in locally coupled phase oscillators with repulsive interactions

Recent experiments in one- and two-dimensional microfluidic arrays of droplets containing Belousov-Zhabotinsky reactants show a rich variety of spatial patterns [M. Toiya et al., J. Phys. Chem. Lett. 1, 1241 (2010)]. The dominant coupling between these droplets is inhibitory. Motivated by this experimental system, we study repulsively coupled Kuramoto oscillators with nearest-neighbor interactions, on a linear chain as well as a ring in one dimension, and on a triangular lattice in two dimensions. In one dimension, we show using linear stability analysis as well as numerical study that the stable phase patterns depend on the geometry of the lattice. We show that a transition to the ordered state does not exist in the thermodynamic limit. In two dimensions, we show that the geometry of the lattice constrains the phase difference between two neighboring oscillators to 2π/3. We report the existence of domains with either clockwise or anticlockwise helicity, leading to defects in the lattice. We study the time dependence of these domains and show that at large coupling strengths the domains freeze due to frequency synchronization. Signatures of the above phenomena can be seen in the spatial correlation functions.

On the role of frustration in excitable systems

We study the role of frustration in excitable systems that allow for oscillations either by construction or in an induced way. We first generalize the notion of frustration to systems whose dynamical equations do not derive from a Hamiltonian. Their couplings can be directed or undirected; they should come in pairs of opposing effects like attractive and repulsive, or activating and repressive, ferromagnetic and antiferromagnetic. As examples we then consider bistable frustrated units as elementary building blocks of our motifs of coupled units. Frustration can be implemented in these systems in various ways: on the level of a single unit via the coupling of a self-loop of positive feedback to a negative feedback loop, on the level of coupled units via the topology or via the type of coupling which may be repressive or activating. In comparison to systems without frustration, we analyze the impact of frustration on the type and number of attractors and observe a considerable enrichment of phase space, ranging from stable fixed-point behavior over different patterns of coexisting options for phase-locked motion to chaotic behavior. In particular we find multistable behavior even for the smallest motifs as long as they are frustrated. Therefore we confirm an enrichment of phase space here for excitable systems with their many applications in biological systems, a phenomenon that is familiar from frustrated spin systems and less known from frustrated phase oscillators. So the enrichment of phase space seems to be a generic effect of frustration in dynamical systems. For a certain range of parameters our systems may be realized in cell tissues. Our results point therefore on a possible generic origin for dynamical behavior that is flexible and functionally stable at the same time, since frustrated systems provide alternative paths for the same set of parameters and at the same "energy costs."

Modeling oscillatory control in NF-κB, p53 and Wnt signaling

Oscillations are commonly observed in cellular behavior and span a wide range of timescales, from seconds in calcium signaling to 24 hours in circadian rhythms. In between lie oscillations with time periods of 1-5 hours seen in NF-κB, p53 and Wnt signaling, which play key roles in the immune system, cell growth/death and embryo development, respectively. In the first part of this article, we provide a brief overview of simple deterministic models of oscillations. In particular, we explain the mechanism of saturated degradation that has been used to model oscillations in the NF-κB, p53 and Wnt systems. The second part deals with the potential physiological role of oscillations. We use the simple models described earlier to explore whether oscillatory signals can encode more information than steady-state signals. We then discuss a few simple genetic circuits that could decode information stored in the average, amplitude or frequency of oscillations. The presence of frequency-detector circuit downstream of NF-κB or p53 would be a strong clue that oscillations are important for the physiological response of these signaling systems.

Watch the clock-engineering biological systems to be on time

Inspired by natural time-keeping devices controlling the circadian clock, managing information processing in the brain and coordinating physiological activities on a daily (feeding and sleeping) or seasonal timescale (reproductive activity or hibernation), synthetic biologists have successfully assembled functional synthetic clocks from cataloged genetic components with standardized activities and arranging them in transcription circuits containing positive and negative feedback loops with integrated time-delay dynamics. While the positive feedback loop drives the clock like the (balance) spring in a mechanical watch the negative time-delay circuit represents the pulse generator defining a minimal time unit and precision of the clock like the pendulum fallback or the movement of the balance wheel in a classical mechanic watch. This basic design principle enabled the construction of a variety of synthetic oscillators whose design details are concisely covered in this review.

Simplified models of biological networks

The function of living cells is controlled by complex regulatory networks that are built of a wide diversity of interacting molecular components. The sheer size and intricacy of molecular networks of even the simplest organisms are obstacles toward understanding network functionality. This review discusses the achievements and promise of a bottom-up approach that uses well-characterized subnetworks as model systems for understanding larger networks. It highlights the interplay between the structure, logic, and function of various types of small regulatory circuits. The bottom-up approach advocates understanding regulatory networks as a collection of entangled motifs. We therefore emphasize the potential of negative and positive feedback, as well as their combinations, to generate robust homeostasis, epigenetics, and oscillations.

The combination of positive and negative feedback loops confers exquisite flexibility to biochemical switches

A wide range of cellular processes require molecular regulatory pathways to convert a graded signal into a discrete response. One prevalent switching mechanism relies on the coexistence of two stable states (bistability) caused by positive feedback regulations. Intriguingly, positive feedback is often supplemented with negative feedback, raising the question of whether and how these two types of feedback can cooperate to control discrete cellular responses. To address this issue, we formulate a canonical model of a protein-protein interaction network and analyze the dynamics of a prototypical two-component circuit. The appropriate combination of negative and positive feedback loops can bring a bistable circuit close to the oscillatory regime. Notably, sharply activated negative feedback can give rise to a bistable regime wherein two stable fixed points coexist and may collide pairwise with two saddle points. This specific type of bistability is found to allow for separate and flexible control of switch-on and switch-off events, for example (i) to combine fast and reversible transitions, (ii) to enable transient switching responses and (iii) to display tunable noise-induced transition rates. Finally, we discuss the relevance of such bistable switching behavior, and the circuit topologies considered, to specific biological processes such as adaptive metabolic responses, stochastic fate decisions and cell-cycle transitions. Taken together, our results suggest an efficient mechanism by which positive and negative feedback loops cooperate to drive the flexible and multifaceted switching behaviors arising in biological systems.

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.