Design principles of biochemical oscillators

Intrathyroidal feedforward and feedback network regulating thyroid hormone synthesis and secretion

Thyroid hormones (THs), including T4 and T3, are produced and released by the thyroid gland under the stimulation of thyroid-stimulating hormone (TSH). The homeostasis of THs is regulated via the coordination of the hypothalamic-pituitary-thyroid axis, plasma binding proteins, and local metabolism in tissues. TH synthesis and secretion in the thyrocytes-containing thyroid follicles are exquisitely regulated by an elaborate molecular network comprising enzymes, transporters, signal transduction machineries, and transcription factors. In this article, we synthesized the relevant literature, organized and dissected the complex intrathyroidal regulatory network into structures amenable to functional interpretation and systems-level modeling. Multiple intertwined feedforward and feedback motifs were identified and described, centering around the transcriptional and posttranslational regulations involved in TH synthesis and secretion, including those underpinning the Wolff-Chaikoff and Plummer effects and thyroglobulin-mediated feedback regulation. A more thorough characterization of the intrathyroidal network from a systems biology perspective, including its topology, constituent network motifs, and nonlinear quantitative properties, can help us to better understand and predict the thyroidal dynamics in response to physiological signals, therapeutic interventions, and environmental disruptions.

Degradation-driven protein level oscillation in the yeast Saccharomyces cerevisiae

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

Rhythmic transcription of Bmal1 stabilizes the circadian timekeeping system in mammals

In mammals, the circadian clock consists of transcriptional and translational feedback loops through DNA cis-elements such as E-box and RRE. The E-box-mediated core feedback loop is interlocked with the RRE-mediated feedback loop, but biological significance of the RRE-mediated loop has been elusive. In this study, we established mutant cells and mice deficient for rhythmic transcription of Bmal1 gene by deleting its upstream RRE elements and hence disrupted the RRE-mediated feedback loop. We observed apparently normal circadian rhythms in the mutant cells and mice, but a combination of mathematical modeling and experiments revealed that the circadian period and amplitude of the mutants were more susceptible to disturbance of CRY1 protein rhythm. Our findings demonstrate that the RRE-mediated feedback regulation of Bmal1 underpins the E-box-mediated rhythm in cooperation with CRY1-dependent posttranslational regulation of BMAL1 protein, thereby conferring the perturbation-resistant oscillation and chronologically-organized output of the circadian clock.

Feedback between a retinoid-related nuclear receptor and the let-7 microRNAs controls the pace and number of molting cycles in C. elegans

Animal development requires coordination among cyclic processes, sequential cell fate specifications, and once-a-lifetime morphogenic events, but the underlying timing mechanisms are not well understood. Caenorhabditis elegans undergoes four molts at regular 8 to 10 hour intervals. The pace of the cycle is governed by PERIOD/lin-42 and other as-yet unknown factors. Cessation of the cycle in young adults is controlled by the let-7 family of microRNAs and downstream transcription factors in the heterochronic pathway. Here, we characterize a negative feedback loop between NHR-23, the worm homolog of mammalian retinoid-related orphan receptors (RORs), and the let-7 family of microRNAs that regulates both the frequency and finite number of molts. The molting cycle is decelerated in nhr-23 knockdowns and accelerated in let-7(-) mutants, but timed similarly in let-7(-) nhr-23(-) double mutants and wild-type animals. NHR-23 binds response elements (ROREs) in the let-7 promoter and activates transcription. In turn, let-7 dampens nhr-23 expression across development via a complementary let-7-binding site (LCS) in the nhr-23 3' UTR. The molecular interactions between NHR-23 and let-7 hold true for other let-7 family microRNAs. Either derepression of nhr-23 transcripts by LCS deletion or high gene dosage of nhr-23 leads to protracted behavioral quiescence and extra molts in adults. NHR-23 and let-7 also coregulate scores of genes required for execution of the molts, including lin-42. In addition, ROREs and LCSs isolated from mammalian ROR and let-7 genes function in C. elegans, suggesting conservation of this feedback mechanism. We propose that this feedback loop unites the molting timer and the heterochronic gene regulatory network, possibly by functioning as a cycle counter.

Bacterial degrons in synthetic circuits

Bacterial proteases are a promising post-translational regulation strategy in synthetic circuits because they recognize specific amino acid degradation tags (degrons) that can be fine-tuned to modulate the degradation levels of tagged proteins. For this reason, recent efforts have been made in the search for new degrons. Here we review the up-to-date applications of degradation tags for circuit engineering in bacteria. In particular, we pay special attention to the effects of degradation bottlenecks in synthetic oscillators and introduce mathematical approaches to study queueing that enable the quantitative modelling of proteolytic queues.

Entrainment within hierarchical circadian oscillator networks

Circadian rhythms are endogenous oscillations, widely found across biological species, that have the capability of entraining to the 24-h light-dark cycle. Circadian systems often consist of both central oscillators that receive direct light-dark input and peripheral oscillators that receive input from the central oscillators. In this paper, we address questions related to what governs the time to and pattern of entrainment of these hierarchical circadian systems after an abrupt switch in the light-dark phasing. For a network consisting of a single central oscillator coupled to a chain of N feed-forward peripheral oscillators, we introduce a systematic way to derive an N-dimensional entrainment map whose fixed points correspond to entrained solutions. Using the map, we explain that the direction of reentrainment can involve fairly complicated phase advancing and delaying behavior as well as reentrainment times that depend sensitively on the nature of the perturbation. We also study the dynamics of a hierarchical system in which the peripheral oscillators are mutually coupled. We study how reentrainment times vary as a function of the degree to which the oscillators are desynchronized at the time of the change in light-dark phasing. We show that desynchronizing the peripheral oscillators can, in some circumstances, speed up their ultimate reentrainment following perturbations.

Electrifying rhythms in plant cells

Physiological oscillations (or rhythms) pervade all spatiotemporal scales of biological organization, either because they perform critical functions or simply because they can arise spontaneously and may be difficult to prevent. Regardless of the case, they reflect regulatory relationships between control points of a given system and offer insights as read-outs of the concerted regulation of a myriad of biological processes. Here we review recent advances in understanding ultradian oscillations (period < 24h) in plant cells, with a special focus on single-cell oscillations. Ion channels are at the center stage due to their involvement in electrical/excitabile phenomena associated with oscillations and cell-cell communication. We highlight the importance of quantitative approaches to measure oscillations in appropriate physiological conditions, which are essential strategies to deal with the complexity of biological rhythms. Future development of optogenetics techniques in plants will further boost research on the role of membrane potential in oscillations and waves across multiple cell types.

Universal minimal cost of coherent biochemical oscillations

Biochemical clocks are essential for virtually all living systems. A biochemical clock that is isolated from an external periodic signal and subjected to fluctuations can oscillate coherently only for a finite number of oscillations. Furthermore, such an autonomous clock can oscillate only if it consumes free energy. What is the minimum amount of free-energy consumption required for a certain number of coherent oscillations? We conjecture a universal bound that answers this question. A system that oscillates coherently for N oscillations has a minimal free-energy cost per oscillation of 4π^{2}Nk_{B}T. Our bound is valid for general finite Markov processes, is conjectured based on extensive numerical evidence, is illustrated with numerical simulations of a known model for a biochemical oscillator, and applies to existing experimental data.

Improved estimation for energy dissipation in biochemical oscillations

Biochemical oscillations, regulating the timing of life processes, need consume energy to achieve good performance on crucial functions, such as high accuracy of phase period and high sensitivity to external signals. However, it is a great challenge to precisely estimate the energy dissipation in such systems. Here, based on the stochastic normal form theory (SNFT), we calculate the Pearson correlation coefficient between the oscillatory amplitude and phase, and a trade-off relation between transport efficiency and phase sensitivity can then be derived, which serves as a tighter form than the estimator resulting from the conventional thermodynamic uncertainty relation (TUR). Our findings demonstrate that a more precise energy dissipation estimation can be obtained by enhancing the sensitivity of the biochemical oscillations. Moreover, the internal noise and amplitude power effects have also been discovered.

Time-keeping and decision-making in the cell cycle

Cell growth, DNA replication, mitosis and division are the fundamental processes by which life is passed on from one generation of eukaryotic cells to the next. The eukaryotic cell cycle is intrinsically a periodic process but not so much a ‘clock’ as a ‘copy machine’, making new daughter cells as warranted. Cells growing under ideal conditions divide with clock-like regularity; however, if they are challenged with DNA-damaging agents or mitotic spindle disrupters, they will not progress to the next stage of the cycle until the damage is repaired. These ‘decisions’ (to exit and re-enter the cell cycle) are essential to maintain the integrity of the genome from generation to generation. A crucial challenge for molecular cell biologists in the 1990s was to unravel the genetic and biochemical mechanisms of cell cycle control in eukaryotes. Central to this effort were biochemical studies of the clock-like regulation of ‘mitosis promoting factor’ during synchronous mitotic cycles of fertilized frog eggs and genetic studies of the switch-like regulation of ‘cyclin-dependent kinases' in yeast cells. In this review, we uncover some secrets of cell cycle regulation by mathematical modelling of increasingly more complex molecular regulatory networks of cell cycle ‘clocks’ and ‘switches’.

Dissipative DNA nanotechnology

DNA nanotechnology has emerged as a powerful tool to precisely design and control molecular circuits, machines and nanostructures. A major goal in this field is to build devices with life-like properties, such as directional motion, transport, communication and adaptation. Here we provide an overview of the nascent field of dissipative DNA nanotechnology, which aims at developing life-like systems by combining programmable nucleic-acid reactions with energy-dissipating processes. We first delineate the notions, terminology and characteristic features of dissipative DNA-based systems and then we survey DNA-based circuits, devices and materials whose functions are controlled by chemical fuels. We emphasize how energy consumption enables these systems to perform work and cyclical tasks, in contrast with DNA devices that operate without dissipative processes. The ability to take advantage of chemical fuel molecules brings dissipative DNA systems closer to the active molecular devices that exist in nature.

Misaligned hormonal rhythmicity: Mechanisms of origin and their clinical significance

Rhythmic hormonal secretion is key for sustaining health. While a central pacemaker in the hypothalamus is the main driver of circadian periodicity, many hormones oscillate with different frequencies and amplitudes. These rhythms carry information about healthy physiological functions, while at the same time they must be able to respond to external cues and maintain their robustness against severe perturbations. Since endocrine disruptions can lead to hormonal misalignment and disease, understanding the clinical significance of these rhythms can help support diagnosis and disease management. While the misalignment of dynamic hormone profiles can be quantitatively analysed though statistical and computational techniques, mathematical modelling can provide fundamental understanding about the mechanisms underpinning endocrine rhythms, particularly around the question of what makes them robust to some perturbations but fragile to others. In this study, I will review the key challenges of understanding hormonal rhythm misalignment from a mathematical perspective, including their causes and clinical significance. By reviewing modelling examples of coupled endocrine axes, I will address the question of how perturbations in one endocrine axis propagate to another, leading to the more complex issue of disentangling the contribution of each endocrine system to a robust dynamic environment.

Coherence of oscillations in the weak-noise limit

In a noisy environment, oscillations lose their coherence, which can be characterized by a quality factor. We determine this quality factor for oscillations arising from a driven Fokker-Planck dynamics along a periodic one-dimensional potential analytically in the weak-noise limit. With this expression, we can prove for this continuum model the analog of an upper bound that has been conjectured for the coherence of oscillations in discrete Markov network models. We show that our approach can also be adapted to motion along a noisy two-dimensional limit cycle. Specifically, we apply our scheme to the noisy Stuart-Landau oscillator and the thermodynamically consistent Brusselator as a simple model for a chemical clock. Our approach thus complements the fairly sophisticated extant general framework based on techniques from Hamilton-Jacobi theory with which we compare our results numerically.

Centrioles generate a local pulse of Polo/PLK1 activity to initiate mitotic centrosome assembly

Mitotic centrosomes are formed when centrioles start to recruit large amounts of pericentriolar material (PCM) around themselves in preparation for mitosis. This centrosome "maturation" requires the centrioles and also Polo/PLK1 protein kinase. The PCM comprises several hundred proteins and, in Drosophila, Polo cooperates with the conserved centrosome proteins Spd-2/CEP192 and Cnn/CDK5RAP2 to assemble a PCM scaffold around the mother centriole that then recruits other PCM client proteins. We show here that in Drosophila syncytial blastoderm embryos, centrosomal Polo levels rise and fall during the assembly process-peaking, and then starting to decline, even as levels of the PCM scaffold continue to rise and plateau. Experiments and mathematical modelling indicate that a centriolar pulse of Polo activity, potentially generated by the interaction between Polo and its centriole receptor Ana1 (CEP295 in humans), could explain these unexpected scaffold assembly dynamics. We propose that centrioles generate a local pulse of Polo activity prior to mitotic entry to initiate centrosome maturation, explaining why centrioles and Polo/PLK1 are normally essential for this process.

Spatiotemporal feedforward between PKM2 tetramers and mTORC1 prompts mTORC1 activation

Most mammalian cells couple glucose availability to anabolic processes via the mTORC1 pathway. However, the mechanism by which fluctuations in glucose availability are rapidly translated into mTORC1 signals remains elusive. Here, we show that cells rapidly respond to changes in glucose availability through the spatial coupling of mTORC1 and tetramers of the key glycolytic enzyme pyruvate kinase M2 (PKM2) on lysosomal surfaces in the late G1/S phases. The lysosomal localization of PKM2 tetramers enables rapid increases in local ATP concentrations around lysosomes to activate mTORC1, while bypassing the need to elevate global ATP levels in the entire cell. In essence, this spatial coupling establishes a feedforward loop to enable mTORC1 to rapidly sense and respond to changes in glucose availability. We further demonstrate that this mechanism ensures robust cell proliferation upon fluctuating glucose availability. Thus, we present mechanistic insights into the rapid response of the mTORC1 pathway to changes in glucose availability. The underlying mechanism may be applicable to the control of other cellular processes.

Complex dynamics in a synchronized cell-free genetic clock

Complex dynamics such as period doubling and chaos occur in a wide variety of non-linear dynamical systems. In the context of biological circadian clocks, such phenomena have been previously found in computational models, but their experimental study in biological systems has been challenging. Here, we present experimental evidence of period doubling in a forced cell-free genetic oscillator operated in a microfluidic reactor, where the system is periodically perturbed by modulating the concentration of one of the oscillator components. When the external driving matches the intrinsic period, we experimentally find period doubling and quadrupling in the oscillator dynamics. Our results closely match the predictions of a theoretical model, which also suggests conditions under which our system would display chaotic dynamics. We show that detuning of the external and intrinsic period leads to more stable entrainment, suggesting a simple design principle for synchronized synthetic and natural genetic clocks.

A Bayesian Approach to Extracting Kinetic Information from Artificial Enzymatic Networks

In order to create artificial enzymatic networks capable of increasingly complex behavior, an improved methodology in understanding and controlling the kinetics of these networks is needed. Here, we introduce a Bayesian analysis method allowing for the accurate inference of enzyme kinetic parameters and determination of most likely reaction mechanisms, by combining data from different experiments and network topologies in a single probabilistic analysis framework. This Bayesian approach explicitly allows us to continuously improve our parameter estimates and behavior predictions by iteratively adding new data to our models, while automatically taking into account uncertainties introduced by the experimental setups or the chemical processes in general. We demonstrate the potential of this approach by characterizing systems of enzymes compartmentalized in beads inside flow reactors. The methods we introduce here provide a new approach to the design of increasingly complex artificial enzymatic networks, making the design of such networks more efficient, and robust against the accumulation of experimental errors.

Mitotic kinase oscillation governs the latching of cell cycle switches

In 1996, Kim Nasmyth¹ proposed that the eukaryotic cell cycle is an alternating sequence of transitions from G₁ to S-G₂-M and back again. These two phases correlate to high activity of cyclin-dependent kinases (CDKs) that trigger S-G₂-M events and CDK antagonists that stabilize G₁ phase. We associated these “alternative phases” with the coexistence of two stable steady states of the biochemical reactions among CDKs and their antagonists.^2,3 Transitions between these steady states (G₁-to-S and M-to-G₁) are driven by “helper” proteins. The fact that the transitions are irreversible is guaranteed by a “latching” property of the molecular switches, as we have argued in previous publications.^4,5 Here, we show that if the latch is broken, then the biochemical reactions can swing back-and-forth across the transitions; either G₁-S-G₁-S … (periodic DNA replication without mitosis or cell division) or M-(G₁)-M-(G₁) … (periodic Cdc14 release, without fully exiting mitosis). Using mathematical modeling of the molecular control circuit in budding yeast, we provide a fresh account of aberrant cell cycles in mutant strains: endoreplication in the clb1-5Δ strain⁶ and periodic release and resequestration of Cdc14 (an “exit” phosphatase) in the CLB2kdΔ strain.^7,8 In our opinion, these “endocycles” are not autonomous oscillatory modules that must be entrained by the CDK oscillator^6,7 but rather inadvertent and deleterious oscillations that are normally suppressed by the CDK latching-gate mechanism.⁸

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Mitotic kinases enforce once-per-cycle occurrence of cell cycle events

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Many cell cycle events can occur periodically in the absence of mitosis

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Are they autonomous oscillators or accidental consequences of faulty regulation?

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Regulatory mutations allow negative feedback loops to sustain oscillations

Combined multiple transcriptional repression mechanisms generate ultrasensitivity and oscillations

Transcriptional repression can occur via various mechanisms, such as blocking, sequestration and displacement. For instance, the repressors can hold the activators to prevent binding with DNA or can bind to the DNA-bound activators to block their transcriptional activity. Although the transcription can be completely suppressed with a single mechanism, multiple repression mechanisms are used together to inhibit transcriptional activators in many systems, such as circadian clocks and NF-κB oscillators. This raises the question of what advantages arise if seemingly redundant repression mechanisms are combined. Here, by deriving equations describing the multiple repression mechanisms, we find that their combination can synergistically generate a sharply ultrasensitive transcription response and thus strong oscillations. This rationalizes why the multiple repression mechanisms are used together in various biological oscillators. The critical role of such combined transcriptional repression for strong oscillations is further supported by our analysis of formerly identified mutations disrupting the transcriptional repression of the mammalian circadian clock. The hitherto unrecognized source of the ultrasensitivity, the combined transcriptional repressions, can lead to robust synthetic oscillators with a previously unachievable simple design.

Condensation of the Drosophila nerve cord is oscillatory and depends on coordinated mechanical interactions

During development, organs reach precise shapes and sizes. Organ morphology is not always obtained through growth; a classic counterexample is the condensation of the nervous system during Drosophila embryogenesis. The mechanics underlying such condensation remain poorly understood. Here, we characterize the condensation of the embryonic ventral nerve cord (VNC) at both subcellular and tissue scales. This analysis reveals that condensation is not a unidirectional continuous process but instead occurs through oscillatory contractions. The VNC mechanical properties spatially and temporally vary, and forces along its longitudinal axis are spatially heterogeneous. We demonstrate that the process of VNC condensation is dependent on the coordinated mechanical activities of neurons and glia. These outcomes are consistent with a viscoelastic model of condensation, which incorporates time delays and effective frictional interactions. In summary, we have defined the progressive mechanics driving VNC condensation, providing insights into how a highly viscous tissue can autonomously change shape and size.

Nonmodular oscillator and switch based on RNA decay drive regeneration of multimodal gene expression

Periodic gene expression dynamics are key to cell and organism physiology. Studies of oscillatory expression have focused on networks with intuitive regulatory negative feedback loops, leaving unknown whether other common biochemical reactions can produce oscillations. Oscillation and noise have been proposed to support mammalian progenitor cells’ capacity to restore heterogenous, multimodal expression from extreme subpopulations, but underlying networks and specific roles of noise remained elusive. We use mass-action-based models to show that regulated RNA degradation involving as few as two RNA species—applicable to nearly half of human protein-coding genes—can generate sustained oscillations without explicit feedback. Diverging oscillation periods synergize with noise to robustly restore cell populations’ bimodal expression on timescales of days. The global bifurcation organizing this divergence relies on an oscillator and bistable switch which cannot be decomposed into two structural modules. Our work reveals surprisingly rich dynamics of post-transcriptional reactions and a potentially widespread mechanism underlying development, tissue regeneration, and cancer cell heterogeneity.

History Dependence in a Chemical Reaction Network Enables Dynamic Switching

This work describes an enzymatic autocatalytic network capable of dynamic switching under out-of-equilibrium conditions. The network, wherein a molecular fuel (trypsinogen) and an inhibitor (soybean trypsin inhibitor) compete for a catalyst (trypsin), is kept from reaching equilibria using a continuous flow stirred tank reactor. A so-called 'linear inhibition sweep' is developed (i.e., a molecular analogue of linear sweep voltammetry) to intentionally perturb the competition between autocatalysis and inhibition, and used to demonstrate that a simple molecular system, comprising only three components, is already capable of a variety of essential neuromorphic behaviors (hysteresis, synchronization, resonance, and adaptation). This research provides the first steps in the development of a strategy that uses the principles in systems chemistry to transform chemical reaction networks into platforms capable of neural network computing.

Temporal control of the integrated stress response by a stochastic molecular switch

Stress granules (SGs) are formed in the cytosol as an acute response to environmental cues and activation of the integrated stress response (ISR), a central signaling pathway controlling protein synthesis. Using chronic virus infection as stress model, we previously uncovered a unique temporal control of the ISR resulting in recurrent phases of SG assembly and disassembly. Here, we elucidate the molecular network generating this fluctuating stress response by integrating quantitative experiments with mathematical modeling and find that the ISR operates as a stochastic switch. Key elements controlling this switch are the cooperative activation of the stress-sensing kinase PKR, the ultrasensitive response of SG formation to the phosphorylation of the translation initiation factor eIF2α, and negative feedback via GADD34, a stress-induced subunit of protein phosphatase 1. We identify GADD34 messenger RNA levels as the molecular memory of the ISR that plays a central role in cell adaptation to acute and chronic stress.

Mathematical analysis of robustness of oscillations in models of the mammalian circadian clock

Circadian rhythms in a wide range of organisms are mediated by molecular mechanisms based on transcription-translation feedback. In this paper, we use bifurcation theory to explore mathematical models of genetic oscillators, based on Kim & Forger’s interpretation of the circadian clock in mammals. At the core of their models is a negative feedback loop whereby PER proteins (PER1 and PER2) bind to and inhibit their transcriptional activator, BMAL1. For oscillations to occur, the dissociation constant of the PER:BMAL1 complex, K^d, must be ≤ 0.04 nM, which is orders of magnitude smaller than a reasonable expectation of 1–10 nM for this protein complex. We relax this constraint by two modifications to Kim & Forger’s ‘single negative feedback’ (SNF) model: first, by introducing a multistep reaction chain for posttranscriptional modifications of Per mRNA and posttranslational phosphorylations of PER, and second, by replacing the first-order rate law for degradation of PER in the nucleus by a Michaelis-Menten rate law. These modifications increase the maximum allowable K^d to ~2 nM. In a third modification, we consider an alternative rate law for gene transcription to resolve an unrealistically large rate of Per2 transcription at very low concentrations of BMAL1. Additionally, we studied extensions of the SNF model to include a second negative feedback loop (involving REV-ERB) and a supplementary positive feedback loop (involving ROR). Contrary to Kim & Forger’s observations of these extended models, we find that, with our modifications, the supplementary positive feedback loop makes the oscillations more robust than observed in the models with one or two negative feedback loops. However, all three models are similarly robust when accounting for circadian rhythms (~24 h period) with K^d ≥ 1 nM. Our results provide testable predictions for future experimental studies.

The circadian rhythm aligns bodily functions to the day/night cycle and is important for our health. The rhythm originates from an intracellular molecular clock mechanism that mediates rhythmic gene expression. It is long understood that transcriptional negative feedback with sufficient time delay is key to generating circadian oscillations. However, some of the most widely cited mathematical models for the circadian clock suffer from problems of parameter ‘fragilities’. That is, sustained oscillations are possible only for physically unrealistic parameter values. A recent model by Kim & Forger nicely incorporates the inhibitory binding of PER proteins to their transcription activator BMAL1, but oscillations in the Kim-Forger model require a binding affinity between PER and BMAL1 that is orders of magnitude larger than observed binding affinities of protein complexes. To rectify this problem, we make several physiologically credible modifications to the Kim-Forger model, which allow oscillations to occur with more realistic binding affinities. The modified model is further extended to explore the potential roles of supplementary feedback loops in the mammalian clock mechanism. Ultimately, accurate models of the circadian clock will provide better predictive tools for chronotherapy and chrono-pharmacology studies.

Dynamical behaviors of quorum sensing network mediated by combinatorial perturbation

The dynamical behaviors of the quorum sensing (QS) system are closely related to the release drugs and control the PH value in microorganisms and plants. However, the effect of the main molecules AiiA, LuxI, H$ _2 $O$ _2 $, and time delayed individual and combinatorial perturbation on the QS system dynamics and the above-mentioned biological phenomena is still unclear, which are seen as a key consideration in our paper. This paper formulates a QS computational model by incorporating these several substances. First, for the protein production time delay, a critical value is given by Hopf bifurcation theory. It is found that a larger time delay can lead to a larger amplitude and a longer period. This indicates that the length of time for protein synthesis has a regulatory effect on the release of drugs from the bacterial population. Second, hen the concentrations of AiiA, LuxI, and H$ _2 $O$ _2 $ is modulated individually, the QS system undergoes periodic oscillation and bistable state. Meanwhile, oscillatory and bistable regions can be significantly affected by simultaneously perturbing any two parameters related to AiiA, LuxI, and H$ _2 $O$ _2 $. This means that the individual or simultaneous changes of the three intrinsic molecular concentrations can effectively control the drugs release and the PH value in microorganisms and plants. Finally, the sensitivity relationship between the critical value of the delay and AiiA, LuxI, H$ _2 $O$ _2 $ parameters is analyzed.

A variable refractory period increases collective performance in noisy environments

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

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

The multifunctional adaptor protein HIP-55 couples Smad7 to accelerate TGF-β type I receptor degradation

Transforming growth factor β (TGF-β) is a multifunctional polypeptide that plays critical roles in regulating a broad range of cellular functions and physiological processes. TGF-β signalling dysfunction contributes to many disorders, such as cardiovascular diseases, cancer and immunological diseases. The homoeostasis of negative feedback regulation is critical for signal robustness, duration and specificity, which precisely control physiological and pathophysiological processes. However, the underlying mechanism by which the negative regulation of TGF-β signalling is integrated and coordinated is still unclear. Here, we reveal that haematopoietic progenitor kinase-interacting protein of 55 kDa (HIP-55) was upregulated upon TGF-β stimulation, while the loss of HIP-55 caused TGF-β signalling overactivation and the abnormal accumulation of downstream extracellular matrix (ECM) genes. HIP-55 interacts with Smad7 and competes with Smad7/Axin complex formation to inhibit the Axin-mediated degradation of Smad7. HIP-55 further couples Smad7 to TβRI but not TβRII, driving TβRI degradation. Altogether, our findings demonstrate a new mechanism by which the effector and negative feedback functions of HIP-55 are coupled and may provide novel strategies for the treatment of TGF-β signalling-related human diseases.

Synthetic biology tools for engineering Goodwin oscillation in Trypanosoma brucei brucei

Kinetoplastid protozoa possess properties that are highly divergent from the mammalian, yeast and bacterial cells more commonly used in synthetic biology and represent a tantalisingly untapped source of bioengineering potential. Trypanosoma brucei brucei (T. b. brucei), an established model organism for studying the Kinetoplastida, is non-pathogenic to humans and provides an interesting test case for establishing synthetic biology in this phylogenetic class. To demonstrate further the tractability of Kinetoplastida to synthetic biology, we sought to construct and demonstrate a Goodwin oscillator, the simplest oscillatory gene network, in T. b. brucei for the first time. We report one completed iteration of the archetypal synthetic biology Design-Build-Test-Learn (DBTL) cycle; firstly, using Ab initio mathematical modelling of the behaviour a theoretical, oscillatory, trypanosomal synthetic gene network (SGN) to inform the design of a plasmid encoding that network. Once assembled, the plasmid was then used to generate a stable transfectant T. b. brucei cell line. To test the performance of the oscillatory SGN, a novel experimental setup was established to capture images of the fluorescent signal from motion-restricted live cells. Data captured were consistent with oscillatory behaviour of the SGN, with cellular fluorescence observed to oscillate with a period of 50 min, with varying amplitude and linear growth trend. This first DBTL cycle establishes a foundation for future cycles in which the SGN design and experimental monitoring setup can be further refined.

Intermittent ERK oscillations downstream of FGF in mouse embryonic stem cells

Signal transduction networks generate characteristic dynamic activities to process extracellular signals and guide cell fate decisions such as to divide or differentiate. The differentiation of pluripotent cells is controlled by FGF/ERK signaling. However, only a few studies have addressed the dynamic activity of the FGF/ERK signaling network in pluripotent cells at high time resolution. Here, we use live cell sensors in wild-type and Fgf4-mutant mouse embryonic stem cells to measure dynamic ERK activity in single cells, for defined ligand concentrations and differentiation states. These sensors reveal pulses of ERK activity. Pulsing patterns are heterogeneous between individual cells. Consecutive pulse sequences occur more frequently than expected from simple stochastic models. Sequences become more prevalent with higher ligand concentration, but are rarer in more differentiated cells. Our results suggest that FGF/ERK signaling operates in the vicinity of a transition point between oscillatory and non-oscillatory dynamics in embryonic stem cells. The resulting heterogeneous dynamic signaling activities add a new dimension to cellular heterogeneity that may be linked to divergent fate decisions in stem cell cultures.

Controlling Chaos in Van Der Pol Dynamics Using Signal-Encoded Deep Learning

Controlling nonlinear dynamics is a long-standing problem in engineering. Harnessing known physical information to accelerate or constrain stochastic learning pursues a new paradigm of scientific machine learning. By linearizing nonlinear systems, traditional control methods cannot learn nonlinear features from chaotic data for use in control. Here, we introduce Physics-Informed Deep Operator Control (PIDOC), and by encoding the control signal and initial position into the losses of a physics-informed neural network (PINN), the nonlinear system is forced to exhibit the desired trajectory given the control signal. PIDOC receives signals as physics commands and learns from the chaotic data output from the nonlinear van der Pol system, where the output of the PINN is the control. Applied to a benchmark problem, PIDOC successfully implements control with a higher stochasticity for higher-order terms. PIDOC has also been proven to be capable of converging to different desired trajectories based on case studies. Initial positions slightly affect the control accuracy at the beginning stage yet do not change the overall control quality. For highly nonlinear systems, PIDOC is not able to execute control with a high accuracy compared with the benchmark problem. The depth and width of the neural network structure do not greatly change the convergence of PIDOC based on case studies of van der Pol systems with low and high nonlinearities. Surprisingly, enlarging the control signal does not help to improve the control quality. The proposed framework can potentially be applied to many nonlinear systems for nonlinear controls.

From the Belousov-Zhabotinsky reaction to biochemical clocks, traveling waves and cell cycle regulation

In the last 20 years, a growing army of systems biologists has employed quantitative experimental methods and theoretical tools of data analysis and mathematical modeling to unravel the molecular details of biological control systems with novel studies of biochemical clocks, cellular decision-making, and signaling networks in time and space. Few people know that one of the roots of this new paradigm in cell biology can be traced to a serendipitous discovery by an obscure Russian biochemist, Boris Belousov, who was studying the oxidation of citric acid. The story is told here from an historical perspective, tracing its meandering path through glycolytic oscillations, cAMP signaling, and frog egg development. The connections among these diverse themes are drawn out by simple mathematical models (nonlinear differential equations) that share common structures and properties.

Radical pairs can explain magnetic field and lithium effects on the circadian clock

Drosophila's circadian clock can be perturbed by magnetic fields, as well as by lithium administration. Cryptochromes are critical for the circadian clock. Further, the radical pairs in cryptochrome also can explain magnetoreception in animals. Based on a simple radical pair mechanism model of the animal magnetic compass, we show that both magnetic fields and lithium can influence the spin dynamics of the naturally occurring radical pairs and hence modulate the circadian clock's rhythms. Using a simple chemical oscillator model for the circadian clock, we show that the spin dynamics influence a rate in the chemical oscillator model, which translates into a change in the circadian period. Our model can reproduce the results of two independent experiments, magnetic field and lithium effects on the circadian clock. Our model predicts that stronger magnetic fields would shorten the clock's period. We also predict that lithium influences the clock in an isotope-dependent manner. Furthermore, our model also predicts that magnetic fields and hyperfine interactions modulate oxidative stress. The findings of this work suggest that the quantum nature of radical pairs might play roles in the brain, as another piece of evidence in addition to recent results on xenon anesthesia and lithium effects on hyperactivity.

Mathematical Modeling in Circadian Rhythmicity

Circadian clocks are autonomous systems able to oscillate in a self-sustained manner in the absence of external cues, although such Zeitgebers are typically present. At the cellular level, the molecular clockwork consists of a complex network of interlocked feedback loops. This chapter discusses self-sustained circadian oscillators in the context of nonlinear dynamics theory. We suggest basic steps that can help in constructing a mathematical model and introduce how self-sustained generations can be modeled using ordinary differential equations. Moreover, we discuss how coupled oscillators synchronize among themselves or entrain to periodic signals. The development of mathematical models over the last years has helped to understand such complex network systems and to highlight the basic building blocks in which oscillating systems are built upon. We argue that, through theoretical predictions, the use of simple models can guide experimental research and is thus suitable to model biological systems qualitatively.

Mathematical Modelling in Plant Synthetic Biology

Mathematical modelling techniques are integral to current research in plant synthetic biology. Modelling approaches can provide mechanistic understanding of a system, allowing predictions of behaviour and thus providing a tool to help design and analyse biological circuits. In this chapter, we provide an overview of mathematical modelling methods and their significance for plant synthetic biology. Starting with the basics of dynamics, we describe the process of constructing a model over both temporal and spatial scales and highlight crucial approaches, such as stochastic modelling and model-based design. Next, we focus on the model parameters and the techniques required in parameter analysis. We then describe the process of selecting a model based on tests and criteria and proceed to methods that allow closer analysis of the system's behaviour. Finally, we highlight the importance of uncertainty in modelling approaches and how to deal with a lack of knowledge, noisy data, and biological variability; all aspects that play a crucial role in the cooperation between the experimental and modelling components. Overall, this chapter aims to illustrate the importance of mathematical modelling in plant synthetic biology, providing an introduction for those researchers who are working with or working on modelling techniques.

Teaching computational systems biology with an eye on quantitative systems pharmacology at the undergraduate level: Why do it, who would take it, and what should we teach?

Computational systems biology (CSB) is a field that emerged primarily as the product of research activities. As such, it grew in several directions in a distributed and uncoordinated manner making the area appealing and fascinating. The idea of not having to follow a specific path but instead creating one fueled innovation. As the field matured, several interdisciplinary graduate programs emerged attempting to educate future generations of computational systems biologists. These educational initiatives coordinated the dissemination of information across student populations that had already decided to specialize in this field. However, we are now entering an era where CSB, having established itself as a valuable research discipline, is attempting the next major step: Entering undergraduate curricula. As interesting as this endeavor may sound, it has several difficulties, mainly because the field is not uniformly defined. In this manuscript, we argue that this diversity is a significant advantage and that several incarnations of an undergraduate-level CSB biology course could, and should, be developed tailored to programmatic needs. In this manuscript, we share our experiences creating a course as part of a Biomedical Engineering program.

Discovering adaptation-capable biological network structures using control-theoretic approaches

Constructing biological networks capable of performing specific biological functionalities has been of sustained interest in synthetic biology. Adaptation is one such ubiquitous functional property, which enables every living organism to sense a change in its surroundings and return to its operating condition prior to the disturbance. In this paper, we present a generic systems theory-driven method for designing adaptive protein networks. First, we translate the necessary qualitative conditions for adaptation to mathematical constraints using the language of systems theory, which we then map back as ‘design requirements’ for the underlying networks. We go on to prove that a protein network with different input–output nodes (proteins) needs to be at least of third-order in order to provide adaptation. Next, we show that the necessary design principles obtained for a three-node network in adaptation consist of negative feedback or a feed-forward realization. We argue that presence of a particular class of negative feedback or feed-forward realization is necessary for a network of any size to provide adaptation. Further, we claim that the necessary structural conditions derived in this work are the strictest among the ones hitherto existed in the literature. Finally, we prove that the capability of producing adaptation is retained for the admissible motifs even when the output node is connected with a downstream system in a feedback fashion. This result explains how complex biological networks achieve robustness while keeping the core motifs unchanged in the context of a particular functionality. We corroborate our theoretical results with detailed and thorough numerical simulations. Overall, our results present a generic, systematic and robust framework for designing various kinds of biological networks.

Biological systems display a remarkable diversity of functionalities, many of which can be conceived as the response of a large network composed of small interconnecting modules. Unravelling the connection pattern, i.e. design principles, behind important biological functionalities is one of the most challenging problems in systems biology. One such phenomenon is perfect adaptation, which merits special attention owing to its universal presence ranging from chemotaxis in bacterial cells to calcium homeostasis in mammalian cells. The present work focuses on finding the design principles for perfect adaptation in the presence of a stair-case type disturbance. To this end, the current work proposes a systems-theoretic approach to deduce precise mathematical (hence structural) conditions that comply with the key performance parameters for adaptation. The approach is agnostic to the particularities of the reaction kinetics, underlining the dominant role of the topological structure on the response of the network. Notably, the design principles obtained in this work serve as the most strict necessary structural conditions for a network of any size to provide perfect adaptation.

Boolean modelling in plant biology

Signalling and genetic networks underlie most biological processes and are often complex, containing many highly connected components. Modelling these networks can provide insight into mechanisms but is challenging given that rate parameters are often not well defined. Boolean modelling, in which components can only take on a binary value with connections encoded by logic equations, is able to circumvent some of these challenges, and has emerged as a viable tool to probe these complex networks. In this review, we will give an overview of Boolean modelling, with a specific emphasis on its use in plant biology. We review how Boolean modelling can be used to describe biological networks and then discuss examples of its applications in plant genetics and plant signalling.

Diverse role of decoys on emergence and precision of oscillations in a biomolecular clock

Biomolecular clocks are key drivers of oscillatory dynamics in diverse biological processes including cell-cycle regulation, circadian rhythms, and pattern formation during development. A minimal clock implementation is based on the classical Goodwin oscillator, in which a repressor protein inhibits its own synthesis via time-delayed negative feedback. Clock motifs, however, do not exist in isolation; its components are open to interacting with the complex environment inside cells. For example, there are ubiquitous high-affinity binding sites along the genome, known as decoys, where transcription factors such as repressor proteins can potentially interact. This binding affects the availability of transcription factors and has often been ignored in theoretical studies. How does such genomic decoy binding impact the clock's robustness and precision? To address this question, we systematically analyze deterministic and stochastic models of the Goodwin oscillator in the presence of reversible binding of the repressor to a finite number of decoy sites. Our analysis reveals that the relative stability of decoy-bound repressors compared to the free repressor plays distinct roles on the emergence and precision of oscillations. Interestingly, active degradation of the bound repressor can induce sustained oscillations that are otherwise absent without decoys. In contrast, decoy abundances can kill oscillation dynamics if the bound repressor is protected from degradation. Taking into account low copy-number fluctuations in clock components, we show that the degradation of the bound repressors enhances precision by attenuating noise in both the amplitude and period of oscillations. Overall, these results highlight the versatile role of otherwise hidden decoys in shaping the stochastic dynamics of biological clocks and emphasize the importance of synthetic decoys in designing robust clocks.

The finding and researching algorithm for potentially oscillating enzymatic systems

Many processes in living organisms are subject to periodic oscillations at different hierarchical levels of their organization: from molecular-genetic to population and ecological. Oscillatory processes are responsible for cell cycles in both prokaryotes and eukaryotes, for circadian rhythms, for synchronous coupling of respiration with cardiac contractions, etc. Fluctuations in the numbers of organisms in natural populations can be caused by the populations' own properties, their age structure, and ecological relationships with other species. Along with experimental approaches, mathematical and computer modeling is widely used to study oscillating biological systems. This paper presents classical mathematical models that describe oscillatory behavior in biological systems. Methods for the search for oscillatory molecular-genetic systems are presented by the example of their special case - oscillatory enzymatic systems. Factors influencing the cyclic dynamics in living systems, typical not only of the molecular-genetic level, but of higher levels of organization as well, are considered. Application of different ways to describe gene networks for modeling oscillatory molecular-genetic systems is considered, where the most important factor for the emergence of cyclic behavior is the presence of feedback. Techniques for finding potentially oscillatory enzymatic systems are presented. Using the method described in the article, we present and analyze, in a step-by-step manner, first the structural models (graphs) of gene networks and then the reconstruction of the mathematical models and computational experiments with them. Structural models are ideally suited for the tasks of an automatic search for potential oscillating contours (linked subgraphs), whose structure can correspond to the mathematical model of the molecular-genetic system that demonstrates oscillatory behavior in dynamics. At the same time, it is the numerical study of mathematical models for the selected contours that makes it possible to confirm the presence of stable limit cycles in them. As an example of application of the technology, a network of 300 metabolic reactions of the bacterium Escherichia coli was analyzed using mathematical and computer modeling tools. In particular, oscillatory behavior was shown for a loop whose reactions are part of the tryptophan biosynthesis pathway.

Dual roles of SIRT1 in the BAX switch through the P53 module: A mathematical modeling study

SIRT1 is a multifunctional deacetylase that participates in a variety of cellular physiological processes to cope with stress. The anticancer protein P53 is an important target of SIRT1. It has been found that SIRT1 is involved in apoptosis by regulating the activity and intracellular location of P53. Moreover, P53-dependent apoptosis is inseparable from the BCL-2 protein family. Among the members of this family, BAX’s switching dynamics may play a key role in apoptosis. Therefore, a challenging question arises: what effect does SIRT1 have on the BAX switch? To answer this question, we built a small-scale protein network model. Through computer simulation, the properties of SIRT1 that on the one hand promote and on the other inhibit apoptosis are revealed. We found that the opening time of the BAX switch will be delayed in the case of either SIRT1 excess or deficiency. Similarly, the stimulus threshold required for apoptosis will also increase in the above two scenarios. Thereby, we proposed that SIRT1 has an optimal content at which the probability of apoptosis is greatest. In addition, P53 oscillation requires the concentration of SIRT1 to be higher than the optimal value. This work may be helpful both experimentally and clinically.

Towards organism-level systems biology by next-generation genetics and whole-organ cell profiling

The system-level identification and analysis of molecular and cellular networks in mammals can be accelerated by "next-generation" genetics, which is defined as genetics that can achieve desired genetic makeup in a single generation without any animal crossing. We recently established a highly efficient procedure for producing knock-out (KO) mice using the "Triple-CRISPR" method, which targets a single gene by triple gRNAs in the CRISPR/Cas9 system. This procedure achieved an almost perfect KO efficiency (96-100%). We also established a highly efficient procedure, the "ES-mouse" method, for producing knock-in (KI) mice within a single generation. In this method, ES cells were treated with three inhibitors to keep their potency and then injected into 8-cell-stage embryos. These procedures dramatically shortened the time required to produce KO or KI mice from years down to about 3 months. The produced KO and KI mice can also be systematically profiled at a single-cell resolution by the "whole-organ cell profiling," which was realized by tissue-clearing methods, such as CUBIC, and an advanced light-sheet microscopy. The review describes the establishment and application of these technologies above in analyzing the three states (NREM sleep, REM sleep, and awake) of mammalian brains. It also discusses the role of calcium and muscarinic receptors in these states as well as the current challenges and future opportunities in the next-generation mammalian genetics and whole-organ cell profiling for organism-level systems biology.

The Goodwin Oscillator and its Legacy

In the 1960's Brian Goodwin published a couple of mathematical models showing how feedback inhibition can lead to oscillations and discussed possible implications of this behaviour for the physiology of the cell. He also presented key ideas about the rich dynamics that may result from the coupling between such biochemical oscillators. Goodwin's work motivated a series of theoretical investigations aiming at identifying minimal mechanisms to generate limit cycle oscillations and deciphering design principles of biological oscillators. The three-variable Goodwin model (adapted by Griffith) can be seen as a core model for a large class of biological systems, ranging from ultradian to circadian clocks. We summarize here main ideas and results brought by Goodwin and review a couple of modeling works directly or indirectly inspired by Goodwin's findings.

Substrate induced generation of transient self-assembled catalytic systems

Living matter is sustained under non-equilibrium conditions via continuous expense of energy which is coordinated by complex organized events. Spatiotemporal control over exquisite functions arises from chemical complexity under non-equilibrium conditions. For instance, extant biology often uses substrate binding events to access temporally stable protein conformations which show acceleration of catalytic rates to subsequently degrade the substrate. Furthermore, thermodynamically activated but kinetically stable esters (GTP) induce the change of conformation of cytoskeleton proteins (microtubules) which leads to rapid polymerization and triggers an augmentation of catalytic rates to subsequently degrade the ester. Importantly, high-energy assemblies composed of non-activated building blocks (GDP-tubulin) are accessed utilizing the energy dissipated from the catalytic conversion of GTP to GDP from the assembled state. Notably, some experimental studies with simple self-assembled systems have elegantly mimicked the phenomena of substrate induced transient generation of catalytic conformations. Through this review, we endeavour to highlight those select studies which have used simple building blocks to demonstrate substrate induced self-assemblies that subsequently show rate acceleration to convert the substrate into waste. The concept of substrate induced self-assembly of building blocks and rate acceleration from the assembled state has the potential to play a predominant role in the preparation of non-equilibrium systems. The design strategies covered in this review can inspire the possibilities of accessing high energy self-assembled structures that are seen in living systems.

This review highlights the studies which show substrate induced generation of transient catalytic moieties. Examples have been discussed with keeping an eye on the design strategies for development of non-equilibrium high energy assemblies as seen in Nature.

Auto-Regulation of Transcription and Translation: Oscillations, Excitability and Intermittency

Several members of the Hes/Her family, conserved targets of the Notch signalling pathway, encode transcriptional repressors that dimerise, bind DNA and self-repress. Such autoinhibition of transcription can yield homeostasis and, in the presence of delays that account for processes such as transcription, splicing and transport, oscillations. Whilst previous models of autoinhibition of transcription have tended to treat processes such as translation as being unregulated (and hence linear), here we develop and explore a mathematical model that considers autoinhibition of transcription together with nonlinear regulation of translation. It is demonstrated that such a model can yield, in the absence of delays, nonlinear dynamical behaviours such as excitability, homeostasis, oscillations and intermittency. These results indicate that regulation of translation as well as transcription allows for a much richer range of behaviours than is possible with autoregulation of transcription alone. A number of experiments are suggested that would that allow for the signature of autoregulation of translation as well as transcription to be experimentally detected in a Notch signalling system.

Stochastic pH Oscillations in a Model of the Urea-Urease Reaction Confined to Lipid Vesicles

The urea-urease clock reaction is a pH switch from acid to basic that can turn into a pH oscillator if it occurs inside a suitable open reactor. We numerically study the confinement of the reaction to lipid vesicles, which permit the exchange with an external reservoir by differential transport, enabling the recovery of the pH level and yielding a constant supply of urea molecules. For microscopically small vesicles, the discreteness of the number of molecules requires a stochastic treatment of the reaction dynamics. Our analysis shows that intrinsic noise induces a significant statistical variation of the oscillation period, which increases as the vesicles become smaller. The mean period, however, is found to be remarkably robust for vesicle sizes down to approximately 200 nm, but the periodicity of the rhythm is gradually destroyed for smaller vesicles. The observed oscillations are explained as a canard-like limit cycle that differs from the wide class of conventional feedback oscillators.

A frequency-amplitude coordinator and its optimal energy consumption for biological oscillators

Biorhythm including neuron firing and protein-mRNA interaction are fundamental activities with diffusive effect. Their well-balanced spatiotemporal dynamics are beneficial for healthy sustainability. Therefore, calibrating both anomalous frequency and amplitude of biorhythm prevents physiological dysfunctions or diseases. However, many works were devoted to modulate frequency exclusively whereas amplitude is usually ignored, although both quantities are equally significant for coordinating biological functions and outputs. Especially, a feasible method coordinating the two quantities concurrently and precisely is still lacking. Here, for the first time, we propose a universal approach to design a frequency-amplitude coordinator rigorously via dynamical systems tools. We consider both spatial and temporal information. With a single well-designed coordinator, they can be calibrated to desired levels simultaneously and precisely. The practical usefulness and efficacy of our method are demonstrated in representative neuronal and gene regulatory models. We further reveal its fundamental mechanism and optimal energy consumption providing inspiration for biorhythm regulation in future.

Chemical engines: driving systems away from equilibrium through catalyst reaction cycles

Biological systems exhibit a range of complex functions at the micro- and nanoscales under non-equilibrium conditions (for example, transportation and motility, temporal control, information processing and so on). Chemists also employ out-of-equilibrium systems, for example in kinetic selection during catalysis, self-replication, dissipative self-assembly and synthetic molecular machinery, and in the form of chemical oscillators. Key to non-equilibrium behaviour are the mechanisms through which systems are able to extract energy from the chemical reactants ('fuel') that drive such processes. In this Perspective we relate different examples of such powering mechanisms using a common conceptual framework. We discuss how reaction cycles can be coupled to other dynamic processes through positive (acceleration) or negative (inhibition) catalysis to provide the thermodynamic impetus for diverse non-equilibrium behaviour, in effect acting as a 'chemical engine'. We explore the way in which the energy released from reaction cycles is harnessed through kinetic selection in a series of what have sometimes been considered somewhat disparate fields (systems chemistry, molecular machinery, dissipative assembly and chemical oscillators), highlight common mechanistic principles and the potential for the synchronization of chemical reaction cycles, and identify future challenges for the invention and application of non-equilibrium systems. Explicit recognition of the use of fuelling reactions to power structural change in catalysts may stimulate the investigation of known catalytic cycles as potential elements for chemical engines, a currently unexplored area of catalysis research.

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

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