Design principles of biochemical oscillators

Oscillatory Dynamics and Regulatory Mechanisms of the p53-Per2 Network in DNA-Damaged Cells

Circadian rhythm disruptions are linked to increased cancer risk and unfavorable prognosis in patients with cancer, highlighting the critical role of the interplay between the circadian rhythm factor Per2 and the tumor suppressor p53. This brief presents, for the first time, a mathematical model to capture the dynamics of the p53-Per2 network in DNA-damaged cells. The model accurately describes the different stages of the process from unstressed cells to cellular repair and finally to apoptosis as the degree of DNA damage increases. Furthermore, it is found that increasing the inhibition of Per2 by p53 leads to the phase advance of Per2 oscillations, whereas by modulating the inhibition of Mdm2 by Per2, an independent amplitude modulation of active p53 can be achieved, with the range of modulation increasing with the strength of the inhibition. Moreover, the effects of time delays inherent in the transcription, translation, and nuclear translocation of Per2 on the circadian rhythm of DNA-damaged cells are quantitatively investigated by theoretical analyses. It is found that time delays can induce stable oscillations through a supercritical Hopf bifurcation, thereby maintaining the circadian function of DNA-damaged cells and enhancing their DNA-damage repair capacity. This study proposes new insights into cancer prevention and treatment strategies.

Computational modeling of plant root development: the art and the science

Plant root development, like any developmental process, arises from the interplay between processes like gene expression, cell-cell signaling, cell growth and division, and tissue mechanics, which unfold over a wide range of temporal and spatial scales. Computational models are uniquely suited to integrate these different processes and spatio-temporal scales to investigate how their interplay determines developmental outcomes and have become part of mainstream plant developmental research. Still, for non-modeling experts, it often remains unclear how models are built, why a particular modeling approach was chosen, and how to interpret and value model outcomes. This review attempts to explain the science behind the art of model building, illustrating the simplifications that are often made to keep models simple to understand and when these are and are not justified. Similarly, it discusses when it is safe to ignore certain processes like growth or tissue mechanics and when it is not. Additionally, this review discusses a range of major breakthrough modeling articles. Their approaches are linked to classical concepts and models in developmental biology like the French flag positional information gradient of Lewis Wolpert and the repetitive patterning mechanism proposed by Turing, in addition to highlighting the lessons they taught us on plant root development.

Exploring nonlinear phenomena in animal vocalizations through oscillator theory

Animal vocalizations comprise a rich array of complex sounds that exhibit nonlinear phenomena (NLP), which have fascinated researchers for decades. From the melodic songs of birds to the clicks and whistles of dolphins, many species have been found to produce nonlinear vocalizations, offering a valuable perspective on the mechanisms underlying sound production and potential adaptive functions. By leveraging on the principles of oscillator theory and nonlinear dynamics, animal vocalizations, which are based on coupled oscillators, can be described and conveniently classified. We review the basic ingredients for self-sustained oscillations and how different NLP can emerge. We discuss important terms in the context of oscillator theory: attractor types, phase space, bifurcations and Arnold tongue diagrams. Through a comparative analysis of observed NLP and bifurcation diagrams, our study reviews how the tools of nonlinear dynamics can provide insights into the intricate complexity of animal vocalizations, as well as into the evolutionary pressures and adaptive strategies that have shaped the diverse communication systems of the animal kingdom.This article is part of the theme issue, 'Nonlinear phenomena in vertebrate vocalizations: mechanisms and communicative functions'.

Stochastic analysis of human ovarian aging and menopause timing

Menopause marks a critically important biological event that ends a woman's fertility. It is a result of ovarian aging and depletion of ovarian reserve. Although many aspects of these processes are now well understood, the overall dynamic picture remains unclear. Here, we present a novel theoretical framework to analyze human ovarian aging dynamics and menopause timing. Our method is based on stochastic analysis of underlying processes stimulated by observing follicles sequentially transitioning between different stages during ovulation. This allows us to obtain a fully quantitative description of ovarian aging and menopause timing consistent with available experimental observations. Our model accurately predicts the average age of menopause across geographically diverse human populations. Theoretical analysis suggests a universal relation between the initial follicle reserve, the depletion rates, and the threshold that triggers menopause. In addition, it is found that the distributions of menopause times are quite narrow, and it is proposed that this might be a result of a precise regulation due to the synchronization of transitions between different stages of follicles. Our theoretical approach not only quantitatively explains the dynamics of human ovarian aging and menopause timing but also provides important insights into individual variability in ovarian aging. It can be used as a powerful tool for predicting menopause timing and investigating complex processes of reproductive aging.

Energy-based analysis of biochemical oscillators using bond graphs and linear control theory

The bond graph approach has been recognized as a useful conceptual basis for understanding the behaviour of living entities modelled as a system with hierarchical interacting parts exchanging energy. One such behaviour is oscillation, which underpins many essential biological functions. In this paper, energy-based modelling of biochemical systems using the bond graph approach is combined with classical feedback control theory to give a novel approach to the analysis, and potentially synthesis, of biochemical oscillators. It is shown that oscillation is dependent on the interplay between active and passive feedback and this interplay is formalized using classical frequency-response analysis of feedback systems. In particular, the phase margin is suggested as a simple scalar indicator of the presence or absence of oscillations; it is shown how this indicator can be used to investigate the effect of both the structure and parameters of biochemical system on oscillation. It follows that the combination of classical feedback control theory and the bond graph approach to systems biology gives a novel analysis and design methodology for biochemical oscillators. The approach is illustrated using an introductory example similar to the Goodwin oscillator, the Sel'kov model of glycolytic oscillations and the repressilator.

Minimal catalytic dissipative assemblies via cooperation of an amino acid, a nucleobase precursor and a cofactor

Functions arising from cooperation between protobiopolymers have fueled the chemical emergence of living matter, which requires a continuous supply of energy to exist in a far-from-equilibrium state. Non-equilibrium conditions imparted by available energy sources have played critical roles in the appearance of complex co-assembled architectures, which exploit the properties of different classes of biopolymers. Such co-assemblies formed from mixtures of nitrogenous heterocycles as protonucleobases and peptide precursors might have acted as early versions of catalytic machinery, capable of sustaining chemical reaction networks. Herein, we show the generation of catalytic non-equilibrium networks from a mixture of a nitrogenous heterocycle, an amino acid and a cofactor driven by an aromatic substrate. The cooperation, a result of supramolecular interactions between different components, rendered the assemblies capable of activating the cofactor towards oxidative degradation of the substrate, which resulted in autonomous disassembly (negative feedback). Furthermore, utilising promiscuous hydrolytic capability, the transient co-assemblies could metabolise a precursor to generate additional amounts of the substrate, enhancing the lifetime (positive feedback) of the assemblies.

Veratridine-Induced Oscillations in Nav 1.7 but Not Nav 1.5 Sodium Channels Are Revealed by Membrane Potential Sensitive Dye

Voltage-gated sodium channels (Navs) are critical for membrane potential depolarisation in cells, with especially important roles in neuronal and cardiomyocyte membranes. Their malfunction results in a range of disorders, and they are the target of many widely used drugs. A rapid yet accurate functional assay is therefore desirable both to probe for novel active compounds and to better understand the many different Nav isoforms. Here, we use fluorescence to monitor Nav function: cells expressing either the cardiac Nav 1.5 or pain-associated Nav 1.7 were loaded with fluorescent membrane potential sensitive dye and then stimulated with veratridine. Cells expressing Nav 1.5 show a concentration-dependent slow rise and then a plateau in fluorescence. In contrast, cells expressing Nav 1.7 show a more rapid rise and then unexpected oscillatory behavior. Inhibition by flecainide and mexiletine demonstrates that these oscillations are Nav-dependent. Thus, we show that this fluorescent membrane potential dye can provide useful functional data and that we can readily distinguish between these two Nav isoforms because of the behavior of cells expressing them when activated by veratridine. We consider these distinct behaviors may be due to different interactions of veratridine with the different Nav isoforms, although more studies are needed to understand the mechanism underlying the oscillations.

Dynamics of phase tumbling and the reentrainment of circadian oscillators

Circadian clocks are comprised of networks of cellular oscillators that synchronize to produce endogenous daily rhythms in gene expression and protein abundance. These clocks have evolved to align the physiology and behavior of organisms to the 24-h environmental cycles arising from Earth's rotation. Rapid travel across time zones causes misalignment between an organism's circadian rhythms and its environment, leading to sleep problems and other jet lag symptoms until the circadian system entrains to the external cycles of the new time zone. Experimental and modeling work has shown that phase tumbling, defined as desynchronizing networks of circadian oscillators prior to an abrupt phase shift of the light-dark cycle, can speed up the process of reentrainment. Here, we use a mathematical model of circadian oscillators and 2-D entrainment maps to analyze the conditions under which phase tumbling has a positive, neutral, or negative effect on reentrainment time. We find that whether or not phase tumbling is beneficial depends on the size of the external phase shift and the location of the perturbed oscillator with respect to the fixed points and invariant manifolds of the entrainment map.

Ion current oscillation of polyelectrolyte modified micropipettes

Here, we report for the first time that ion current oscillation (ICO) with periodic amplitude and frequency can autonomously occur at polyimidazole brush (PvimB) modified pipettes in an asymmetric solution with a pH gradient (e.g. pH 6.0/pH 8.0). Experimental results demonstrated that under a strong bias voltage, the proton responsive PvimB-modified pipettes exhibited significant current switching behavior under negative bias voltages, which contributed to the periodic oscillating ion current under constant biases. Based on this dynamic, the frequency and amplitude of the ICO phenomenon were regulated by adjusting the pH gradient in the asymmetric solution. ICOs under different bias voltages were further explored to show the voltage-dependent nature of this phenomenon. This observation of ICO phenomena offers a new strategy for designing iontronic devices with dynamic conductivity changes induced by surface chemical interactions within spatial confinements.

Identify structures underlying out-of-equilibrium reaction networks with random graph analysis

Network measures have proven very successful in identifying structural patterns in complex systems (e.g., a living cell, a neural network, the Internet). How such measures can be applied to understand the rational and experimental design of chemical reaction networks (CRNs) is unknown. Here, we develop a procedure to model CRNs as a mathematical graph on which network measures and a random graph analysis can be applied. We used an enzymatic CRN (for which a mass-action model was previously developed) to show that the procedure provides insights into its network structure and properties. Temporal analyses, in particular, revealed when feedback interactions emerge in such a network, indicating that CRNs comprise various reactions that are being added and removed over time. We envision that the procedure, including the temporal network analysis method, could be broadly applied in chemistry to characterize the network properties of many other CRNs, promising data-driven analysis of future molecular systems of ever greater complexity.

Mechanistic origins of temperature scaling in the early embryonic cell cycle

Temperature profoundly impacts organismal physiology and ecological dynamics, particularly affecting ectothermic species and making them especially vulnerable to climate changes. Although complex physiological processes usually involve dozens of enzymes, empirically it is found that the rates of these processes often obey the Arrhenius equation, which was originally proposed for individual chemical reactions. Here we have examined the temperature scaling of the early embryonic cell cycle, with the goal of understanding why the Arrhenius equation approximately holds and why it breaks down at temperature extremes. Using experimental data from Xenopus laevis, Xenopus tropicalis, and Danio rerio, plus published data from Caenorhabditis elegans, Caenorhabditis briggsae, and Drosophila melanogaster, we find that the apparent activation energies (E a values) for the early embryonic cell cycle for diverse ectotherms are all similar, 75 ± 7 kJ/mol (mean ± std.dev., n = 6), which corresponds to aQ 10 value at 20°C of 2.8 ± 0.2 (mean ± std.dev., n = 6). Using computational models, we find that the approximate Arrhenius scaling and the deviations from it at high and low temperatures can be accounted for by biphasic temperature scaling in critical individual components of the cell cycle oscillator circuit, by imbalances in theE a values for different partially rate-determining enzymes, or by a combination of both. Experimental studies of cycling Xenopus extracts indicate that both of these mechanisms contribute to the general scaling of temperature, and in vitro studies of individual cell cycle regulators confirm that there is in fact a substantial imbalance in theirE a values. These findings provide mechanistic insights into the dynamic interplay between temperature and complex biochemical processes, and into why biological systems fail at extreme temperatures.

Dynamics of spindle assembly and position checkpoints: Integrating molecular mechanisms with computational models

Mitotic checkpoints orchestrate cell division through intricate molecular networks that ensure genomic stability. While experimental research has uncovered key aspects of checkpoint function, the complexity of protein interactions and spatial dynamics necessitates computational modeling for a deeper, system-level understanding. This review explores mathematical frameworks-from ordinary differential equations to stochastic simulations, which reveal checkpoint dynamics across multiple scales, encompassing models ranging from simple protein interactions to whole-system simulations with thousands of parameters. These approaches have elucidated fundamental properties, including bistable switches driving spindle assembly checkpoint (SAC) activation, spatial organization principles underlying spindle position checkpoint (SPOC) signaling, and critical system-level features ensuring checkpoint robustness. This study evaluates diverse modeling approaches, from rule-based models to chemical organization theory, highlighting their successful application in predicting protein localization patterns and checkpoint response dynamics validated through live-cell imaging. Contemporary challenges persist in integrating spatial and temporal scales, refining parameter estimation, and enhancing spatial modeling fidelity. However, recent advances in single-molecule imaging, data-driven algorithms, and machine learning techniques, particularly deep learning for parameter optimization, present transformative opportunities for improving model accuracy and predictive power. By bridging molecular mechanisms with system-level behaviors through validated computational frameworks, this review offers a comprehensive perspective on the mathematical modeling of cell cycle control, with practical implications for cancer research and therapeutic development.

Global niche partitioning of purine and pyrimidine cross-feeding among ocean microbes

Cross-feeding involves microbes consuming exudates of other surrounding microbes, mediating elemental cycling. Characterizing the diversity of cross-feeding pathways in ocean microbes illuminates evolutionary forces driving self-organization of ocean ecosystems. Here, we uncover a purine and pyrimidine cross-feeding network in globally abundant groups. The cyanobacterium Prochlorococcus exudes both compound classes, which metabolic reconstructions suggest follows synchronous daily genome replication. Co-occurring heterotrophs differentiate into purine- and pyrimidine-using generalists or specialists that use compounds for different purposes. The most abundant heterotroph, SAR11, is a specialist that uses purines as sources of energy, carbon, and/or nitrogen, with subgroups differentiating along ocean-scale gradients in the supply of energy and nitrogen, in turn producing putative cryptic nitrogen cycles that link many microbes. Last, in an SAR11 subgroup that dominates where Prochlorococcus is abundant, adenine additions to cultures inhibit DNA synthesis, poising cells for replication. We argue that this subgroup uses inferred daily adenine pulses from Prochlorococcus to synchronize to the daily photosynthate supply from surrounding phytoplankton.

Regulatory Effects of Cooperativity and Signal Profile on Adaptive Dynamics in Incoherent Feedforward Loop Networks

Cellular adaptation to external signals is essential for biological functions, and it is an important field of interest in systems biology. This study examines the impact of cooperativity on the adaptation response of the Incoherent Feedforward Loop (IFFL) network motif to various signal profiles. Through comprehensive simulations, we studied how the IFFL motif responds to constant and pulse-type signals under varying levels of cooperativity. The results of our study demonstrate that positive cooperativity generally enhances the system's ability to adapt to different signal profiles. Nevertheless, given specific signal profiles, higher levels of cooperativity may decrease the system's adaptability. On the other hand, the adaptive response breaks down for negative cooperativity. For constant signals, increased positive cooperativity leads to a response with higher amplitude, and it accelerates the response time but delays the return time required to settle back down to the pre-stimulus state. Upon signal cessation, high positive cooperativity not only slows the system's response and return times but, in some cases, can lead to a complete temporary halt in response. For the pulse-like signal, cooperativity increases the maximum amplitude of the oscillatory response. These insights highlight the delicate balance between cooperativity and signal profile in cellular adaptation mechanisms involving the IFFL network motif.

Modeling ncRNA-Mediated Circuits in Cell Fate Decision: From Systems Biology to Synthetic Biology

Noncoding RNAs (ncRNAs) play critical roles in essential cell fate decisions. However, the exact molecular mechanisms underlying ncRNA-mediated bistable switches remain elusive and controversial. In recent years, systematic mathematical and quantitative experimental analyses have made significant contributions to elucidating the molecular mechanisms of controlling ncRNA-mediated cell fate decision processes. In this chapter, we review and summarize the general framework of mathematical modeling of ncRNA in a pedagogical way and the application of this general framework to real biological processes. We discuss the emerging properties resulting from the reciprocal regulation between mRNA, miRNA, and competing endogenous mRNA (ceRNA). We also explore the efforts within the synthetic biology approach to understand the fundamental design principles underlying cell fate decisions. Both the positive feedback loops between ncRNAs and transcription factors and the emerging properties from the miRNA-mRNA reciprocal regulation enable bistable switches to direct cell fate decisions.

Mathematical Modeling for Oscillations Driven by Noncoding RNAs

In this chapter, we first survey strategies for the mathematical modeling of gene regulatory networks for capturing physiologically important dynamics in cells such as oscillations. We focus on models based on ordinary differential equations with various forms of nonlinear functions that describe gene regulations. We next use a small system of a microRNA and its mRNA target to illustrate a recently discovered oscillator driven by noncoding RNAs. This oscillator has unique features that distinguish it from conventional biological oscillators, including the absence of an imposed negative feedback loop and the divergence of the periods. The latter property may serve crucial biological functions for restoring heterogeneity of cell populations on the timescale of days. We describe general requirements for obtaining the limit cycle oscillations in terms of underlying biochemical reactions and kinetic rate constants. We discuss future directions stemming from this minimal, noncoding RNA-based model for gene expression oscillation.

Benchmark for quantitative characterization of circadian clock cycles

Understanding circadian clock mechanisms is fundamental in order to counteract the harmful effects of clock malfunctioning and associated diseases. Biochemical, genetic and systems biology approaches have provided invaluable information on the mechanisms of the circadian clock, from which many mathematical models have been developed to understand the dynamics and quantitative properties of the circadian oscillator. To better analyze and compare quantitatively all these circadian cycles, we propose a method based on a previously proposed circadian cycle segmentation into stages. We notably identify a sequence of eight stages that characterize the progress of the circadian cycle. Next, we apply our approach to an experimental dataset and to five different models, all built with ordinary differential equations. Our method permits to assess the agreement of mathematical model cycles with biological properties or to detect some inconsistencies. As another application of our method, we provide insights on how this segmentation into stages can help to analyze the effect of a clock gene loss of function on the dynamic of a genetic oscillator. The strength of our method is to provide a benchmark for characterization, comparison and improvement of new mathematical models of circadian oscillators in a wide variety of model systems.

Emergent robust oscillatory dynamics in the interlocked feedback-feedforward loops

One of the challenges that beset modelling complex biological networks is to relate networks to function to dynamics. A further challenge is deciphering the cellular function and dynamics that can change drastically when the network edge is tinkered with by adding or removing it. To illustrate this, the authors took a well-studied three-variable Goodwin oscillatory motif with only a negative feedback loop. To this motif, an edge was added that results in an emergent structure consisting of new feedforward and feedback loops while retaining Goodwin's original negative feedback loop. To relate emergent structure to oscillatory dynamics, the authors took all the combinations of edge signs in the interlocked motif. Bifurcation analysis reveals that all the structural combinations can be grouped into two categories based on their unique dynamics. These two groups also exhibit unique amplitude-frequency (amp-freq) plots. These two categories are attributed to the emergence of interlocked motifs with specific edge signs. To support the ideas, a well-studied plant circadian model of Arabidopsis thaliana was taken to illustrate the importance of interlocked motifs in fine-tuning amplitude and frequency in circadian oscillators. The authors briefly discuss its implications for central oscillators' adaptation to different environmental cues.

Oscillatory autophagy induction is enabled by an updated AMPK-ULK1 regulatory wiring

Autophagy-dependent survival relies on a crucial oscillatory response during cellular stress. Although oscillatory behaviour is typically associated with processes like the cell cycle or circadian rhythm, emerging experimental and theoretical evidence suggests that such periodic dynamics may explain conflicting experimental results in autophagy research. In this study, we demonstrate that oscillatory behaviour in the regulation of the non-selective, stress-induced macroautophagy arises from a series of interlinked negative and positive feedback loops within the mTORC1-AMPK-ULK1 regulatory triangle. While many of these interactions have been known for decades, recent discoveries have revealed how mTORC1, AMPK, and ULK1 are truly interconnected. Although these new findings initially appeared contradictory to established models, additional experiments and our systems biology analysis clarify the updated regulatory structure. Through computational modelling of the autophagy oscillatory response, we show how this regulatory network governs autophagy induction. Our results not only reconcile previous conflicting experimental observations but also offer insights for refining autophagy regulation and advancing understanding of its mechanisms of action.

Temperature-dependent fold-switching mechanism of the circadian clock protein KaiB

The oscillator of the cyanobacterial circadian clock relies on the ability of the KaiB protein to switch reversibly between a stable ground-state fold (gsKaiB) and an unstable fold-switched fold (fsKaiB). Rare fold-switching events by KaiB provide a critical delay in the negative feedback loop of this posttranslational oscillator. In this study, we experimentally and computationally investigate the temperature dependence of fold switching and its mechanism. We demonstrate that the stability of gsKaiB increases with temperature compared to fsKaiB and that the Q10 value for the gsKaiB → fsKaiB transition is nearly three times smaller than that for the reverse transition in a construct optimized for NMR studies. Simulations and native-state hydrogen-deuterium exchange NMR experiments suggest that fold switching can involve both partially and completely unfolded intermediates. The simulations predict that the transition state for fold switching coincides with isomerization of conserved prolines in the most rapidly exchanging region, and we confirm experimentally that proline isomerization is a rate-limiting step for fold switching. We explore the implications of our results for temperature compensation, a hallmark of circadian clocks, through a kinetic model.

Oscillatory dynamics of Rac1 activity in Dictyostelium discoideum amoebae

Small GTPases of the Rho family play a central role in the regulation of cell motility by controlling the remodeling of the actin cytoskeleton. In the amoeboid cells of Dictyostelium discoideum, the active form of the Rho GTPase Rac1 regulates actin polymerases at the leading edge and actin filament bundling proteins at the posterior cortex of polarized cells. We monitored the spatiotemporal dynamics of Rac1 and its effector DGAP1 in vegetative amoebae using specific fluorescent probes. We observed that plasma membrane domains enriched in active Rac1 not only exhibited stable polarization, but also showed rotations and oscillations, whereas DGAP1 was depleted from these regions. To simulate the observed dynamics of the two proteins, we developed a mass-conserving reaction-diffusion model based on the circulation of Rac1 between the membrane and the cytoplasm coupled with its activation by GEFs, deactivation by GAPs and interaction with DGAP1. Our theoretical model accurately reproduced the experimentally observed dynamic patterns, including the predominant anti-correlation between active Rac1 and DGAP1. Significantly, the model predicted a new colocalization regime of these two proteins in polarized cells, which we confirmed experimentally. In summary, our results improve the understanding of Rac1 dynamics and reveal how the occurrence and transitions between different regimes depend on biochemical reaction rates, protein levels and cell size. This study not only expands our knowledge of the behavior of Rac1 GTPases in D. discoideum amoebae but also demonstrates how specific modes of interaction between Rac1 and its effector DGAP1 lead to their counterintuitively anti-correlated dynamics.

Feedback driven autonomous cycles of assembly and disassembly from minimal building blocks

The construction of complex systems by simple chemicals that can display emergent network dynamics might contribute to our understanding of complex behavior from simple organic reactions. Here we design single amino acid/dipeptide-based systems that exhibit multiple periodic changes of (dis)assembly under non-equilibrium conditions in closed system, importantly in the absence of evolved biocatalysts. The two-component based building block exploits pH driven non-covalent assembly and time-delayed accelerated catalysis from self-assembled state to install orthogonal feedback loops with a single batch of reactants. Mathematical modelling of the reaction network establishes that the oscillations are transient for this network structure and helps to predict the relative contribution of the feedback loop to the ability of the system to exhibit such transient oscillation. Such autonomous systems with purely synthetic molecules are the starting point that can enable the design of active materials with emergent properties.

Extrinsic fluctuations in the p53 cycle

Fluctuations are inherent to biological systems, arising from the stochastic nature of molecular interactions, and influence various aspects of system behavior, stability, and robustness. These fluctuations can be categorized as intrinsic, stemming from the system's inherent structure and dynamics, and extrinsic, arising from external factors, such as temperature variations. Understanding the interplay between these fluctuations is crucial for obtaining a comprehensive understanding of biological phenomena. However, studying these effects poses significant computational challenges. In this study, we used an underexplored methodology to analyze the effect of extrinsic fluctuations in stochastic systems using ordinary differential equations instead of solving the master equation with stochastic parameters. By incorporating temperature fluctuations into reaction rates, we explored the impact of extrinsic factors on system dynamics. We constructed a master equation and calculated the equations for the dynamics of the first two moments, offering computational efficiency compared with directly solving the chemical master equation. We applied this approach to analyze a biological oscillator, focusing on the p53 model and its response to temperature-induced extrinsic fluctuations. Our findings underscore the impact of extrinsic fluctuations on the nature of oscillations in biological systems, with alterations in oscillatory behavior depending on the characteristics of extrinsic fluctuations. We observed an increased oscillation amplitude and frequency of the p53 concentration cycle. This study provides valuable insights into the effects of extrinsic fluctuations on biological oscillations and highlights the importance of considering them in more complex systems to prevent unwanted scenarios related to health issues.

Newton's cradle: Cell cycle regulation by two mutually inhibitory oscillators

The cell division cycle is a fundamental physiological process displaying a great degree of plasticity during the course of multicellular development. This plasticity is evident in the transition from rapid and stringently-timed divisions of the early embryo to subsequent size-controlled mitotic cycles. Later in development, cells may pause and restart proliferation in response to myriads of internal or external signals, or permanently exit the cell cycle following terminal differentiation or senescence. Beyond this, cells can undergo modified cell division variants, such as endoreplication, which increases their ploidy, or meiosis, which reduces their ploidy. This wealth of behaviours has led to numerous conceptual analogies intended as frameworks for understanding the proliferative program. Here, we aim to unify these mechanisms under one dynamical paradigm. To this end, we take a control theoretical approach to frame the cell cycle as a pair of arrestable and mutually-inhibiting, doubly amplified, negative feedback oscillators controlling chromosome replication and segregation events, respectively. Under appropriate conditions, this framework can reproduce fixed-period oscillations, checkpoint arrests of variable duration, and endocycles. Subsequently, we use phase plane and bifurcation analysis to explain the dynamical basis of these properties. Then, using a physiologically realistic, biochemical model, we show that the very same regulatory structure underpins the diverse functions of the cell cycle control network. We conclude that Newton's cradle may be a suitable mechanical analogy of how the cell cycle is regulated.

Cell Cycle Complexity: Exploring the Structure of Persistent Subsystems in 414 Models

Background: The regulation of cellular proliferation and genomic integrity is controlled by complex surveillance mechanisms known as cell cycle checkpoints. Disruptions in these checkpoints can lead to developmental defects and tumorigenesis. Methods: To better understand these mechanisms, computational modeling has been employed, resulting in a dataset of 414 mathematical models in the BioModels database. These models vary significantly in detail and simulated processes, necessitating a robust analytical approach. Results: In this study, we apply the chemical organization theory (COT) to these models to gain insights into their dynamic behaviors. COT, which handles both ordinary and partial differential equations (ODEs and PDEs), is utilized to analyze the compartmentalized structures of these models. COT's framework allows for the examination of persistent subsystems within these models, even when detailed kinetic parameters are unavailable. By computing and analyzing the lattice of organizations, we can compare and rank models based on their structural features and dynamic behavior. Conclusions: Our application of the COT reveals that models with compartmentalized organizations exhibit distinctive structural features that facilitate the understanding of phenomena such as periodicity in the cell cycle. This approach provides valuable insights into the dynamics of cell cycle control mechanisms, refining existing models and potentially guiding future research in this area.

Depth-Dependent Contributions of Various Vascular Zones to Cerebral Autoregulation and Functional Hyperemia: An In-Silico Analysis

Autoregulation and neurogliavascular coupling are key mechanisms that modulate myogenic tone (MT) in vessels to regulate cerebral blood flow (CBF) during resting state and periods of increased neural activity, respectively. To determine relative contributions of distinct vascular zones across different cortical depths in CBF regulation, we developed a simplified yet detailed and computationally efficient model of the mouse cerebrovasculature. The model integrates multiple simplifications and generalizations regarding vascular morphology, the hierarchical organization of mural cells, and potentiation/inhibition of MT in vessels. Our analysis showed that autoregulation is the result of the synergy between these factors, but achieving an optimal balance across all cortical depths and throughout the autoregulation range is a complex task. This complexity explains the non-uniformity observed experimentally in capillary blood flow at different cortical depths. In silico simulations of cerebral autoregulation support the idea that the cerebral vasculature does not maintain a plateau of blood flow throughout the autoregulatory range and consists of both flat and sloped phases. We learned that small-diameter vessels with large contractility, such as penetrating arterioles and precapillary arterioles, have major control over intravascular pressure at the entry points of capillaries and play a significant role in CBF regulation. However, temporal alterations in capillary diameter contribute moderately to cerebral autoregulation and minimally to functional hyperemia. In addition, hemodynamic analysis shows that while hemodynamics within capillaries remain relatively stable across all cortical depths throughout the entire autoregulation range, significant variability in hemodynamics can be observed within the first few branch orders of precapillary arterioles or transitional zone vessels. The computationally efficient cerebrovasculature model, proposed in this study, provides a novel framework for analyzing dynamics of the CBF regulation where hemodynamic and vasodynamic interactions are the foundation on which more sophisticated models can be developed.

Multiscale Approach for Tuning Communication among Chemical Oscillators Confined in Biomimetic Microcompartments

Inspired by the biological world, new cross-border disciplines and technologies have emerged. Relevant examples include systems chemistry, which offers a bottom-up approach toward chemical complexity, and bio/chemical information and communication technology (bio/chemical ICT), which explores the conditions for propagating signals among individual microreactors separated by selectively permeable membranes. To fabricate specific arrays of microreactors, microfluidics has been demonstrated as an excellent method. In particular, droplet-based microfluidics is a powerful tool for encapsulating biological entities and chemical reagents in artificial microenvironments, mostly water-in-oil microdroplets. In these systems, the interfaces are liquid-liquid, and their physicochemical properties are key factors for tuning the coupling between molecular diffusion. Simple and double emulsions, where aqueous domains are in equilibrium with oil domains through boundary layers of amphiphilic molecules, are organized assemblies with high interfacial-area-to-volume ratios. These membranes can be engineered to obtain different surface charges, single- or multilayer stacking, and a variable degree of defects in molecular packing. Emulsions find application in many fields, including the food industry, pharmaceutics, and cosmetics. Furthermore, micro- and nanoemulsions can be used to model the propagation of chemical species through long distances, which is not only vital for cell signaling but also significant in molecular computing. Here we present in-depth research on the faceted world of solutions confined in restricted environments. In particular, we focused on the multiscale aspects of structure and dynamics from molecular to micro and macro levels. The Belousov-Zhabotinsky chemical reaction, known for its robustness and well-documented oscillatory behavior, was selected to represent a generic signal emitter/receiver confined within microenvironments separated by liquid-liquid interfaces. In this pulse generator, the temporal and spatial progressions are governed by periodic fluctuations in the concentration of chemical species, which act as activatory or inhibitory messengers over long distances. When organized into "colonies" or arrays, these micro-oscillators exhibit emergent dynamical behaviors at the population level. These behaviors can be finely tuned by manipulating the geometrical distribution of the oscillators and the properties of the interfaces at the nanoscale. By carefully selecting the membrane composition, it is possible to drive the system toward either in-phase, antiphase, or mixed synchronization regimes among individual oscillators, depending on messenger molecules. This relatively simple lab-scale model replicates some of the communication strategies commonly found in biological systems, particularly those based on the passive diffusion of chemical and electrical signals. It can help shed light on fundamental life processes and inspire new implementations in molecular computing and smart materials.

Effects of growth feedback on adaptive gene circuits: A dynamical understanding

The successful integration of engineered gene circuits into host cells remains a significant challenge in synthetic biology due to circuit-host interactions, such as growth feedback, where the circuit influences cell growth and vice versa. Understanding the dynamics of circuit failures and identifying topologies resilient to growth feedback are crucial for both fundamental and applied research. Utilizing transcriptional regulation circuits with adaptation as a paradigm, we systematically study more than four hundred topological structures and uncover various categories of failures. Three dynamical mechanisms of circuit failures are identified: continuous deformation of the response curve, strengthened or induced oscillations, and sudden switching to coexisting attractors. Our extensive computations also uncover a scaling law between a circuit robustness measure and the strength of growth feedback. Despite the negative effects of growth feedback on the majority of circuit topologies, we identify several circuits that maintain optimal performance as designed, a feature important for applications.

Arp2/3-dependent endocytosis ensures Cdc42 oscillations by removing Pak1-mediated negative feedback

The GTPase Cdc42 regulates polarized growth in most eukaryotes. In the bipolar yeast Schizosaccharomyces pombe, Cdc42 activation cycles periodically at sites of polarized growth. These periodic cycles are caused by alternating positive feedback and time-delayed negative feedback loops. At each polarized end, negative feedback is established when active Cdc42 recruits the Pak1 kinase to prevent further Cdc42 activation. It is unclear how Cdc42 activation returns to each end after Pak1-dependent negative feedback. We find that disrupting branched actin-mediated endocytosis disables Cdc42 reactivation at the cell ends. Using experimental and mathematical approaches, we show that endocytosis-dependent Pak1 removal from the cell ends allows the Cdc42 activator Scd1 to return to that end to enable reactivation of Cdc42. Moreover, we show that Pak1 elicits its own removal via activation of endocytosis. These findings provide a deeper insight into the self-organization of Cdc42 regulation and reveal previously unknown feedback with endocytosis in the establishment of cell polarity.

Repurposing a Catalytic Cycle for Transient Self-Assembly

Life operates out of equilibrium to enable various sophisticated behaviors. Synthetic chemists have strived to mimic biological nonequilibrium systems in such fields as autonomous molecular machines and dissipative self-assembly. Central to these efforts has been the development of new chemical reaction cycles, which drive systems out of equilibrium by conversion of chemical fuel into waste species. However, the construction of reaction cycles has been challenging due to the difficulty of finding compatible reactions that constitute a cycle. Here, we realize an alternative approach by repurposing a known catalytic cycle as a chemical reaction cycle for driving dissipative self-assembly. This approach can overcome the compatibility problem because all steps involved in a catalytic cycle are already known to proceed concurrently under the same conditions. Our repurposing approach is applicable to diverse combinations of catalytic cycles and systems to drive out of equilibrium, which will substantially broaden the scope of out-of-equilibrium systems.

Learning metabolic dynamics from irregular observations by Bidirectional Time-Series State Transfer Network

Modeling microbial metabolic dynamics is important for the rational optimization of both biosynthetic systems and industrial processes to facilitate green and efficient biomanufacturing. Classical approaches utilize explicit equation systems to represent metabolic networks, enabling the quantification of pathway fluxes to identify metabolic bottlenecks. However, these white-box models, despite their diverse applications, have limitations in simulating metabolic dynamics and are intrinsically inaccurate for industrial strains that lack information on network structures and kinetic parameters. On the other hand, black-box models do not rely on prior mechanistic knowledge of strains but are built upon observed time-series trajectories of biosynthetic systems in action. In practice, these observations are typically irregular, with discontinuously observed time points across multiple independent batches, each time point potentially containing missing measurements. Learning from such irregular data remains challenging for existing approaches. To address this issue, we present the Bidirectional Time-Series State Transfer Network (BTSTN) for modeling metabolic dynamics directly from irregular observations. Using evaluation data sets derived from both ideal dynamic systems and a real-world fermentation process, we demonstrate that BTSTN accurately reconstructs dynamic behaviors and predicts future trajectories. This approach exhibits enhanced robustness against missing measurements and noise, as compared to the state-of-the-art methods.IMPORTANCEIndustrial biosynthetic systems often involve strains with unclear genetic backgrounds, posing challenges in modeling their distinct metabolic dynamics. In such scenarios, white-box models, which commonly rely on inferred networks, are thereby of limited applicability and accuracy. In contrast, black-box models, such as statistical models and neural networks, are directly fitted or learned from observed time-series trajectories of biosynthetic systems in action. These methods typically assume regular observations without missing time points or measurements. If the observations are irregular, a pre-processing step becomes necessary to obtain a fully filled data set for subsequent model training, which, at the same time, inevitably introduces errors into the resulting models. BTSTN is a novel approach that natively learns from irregular observations. This distinctive feature makes it a unique addition to the current arsenal of technologies modeling metabolic dynamics.

Synthetic gene circuit evolution: Insights and opportunities at the mid-scale

Directed evolution focuses on optimizing single genetic components for predefined engineering goals by artificial mutagenesis and selection. In contrast, experimental evolution studies the adaptation of entire genomes in serially propagated cell populations, to provide an experimental basis for evolutionary theory. There is a relatively unexplored gap at the middle ground between these two techniques, to evolve in vivo entire synthetic gene circuits with nontrivial dynamic function instead of single parts or whole genomes. We discuss the requirements for such mid-scale evolution, with hypothetical examples for evolving synthetic gene circuits by appropriate selection and targeted shuffling of a seed set of genetic components in vivo. Implementing similar methods should aid the rapid generation, functionalization, and optimization of synthetic gene circuits in various organisms and environments, accelerating both the development of biomedical and technological applications and the understanding of principles guiding regulatory network evolution.

Bifurcations in coupled amyloid-β aggregation-inflammation systems

A complex interplay between various processes underlies the neuropathology of Alzheimer's disease (AD) and its progressive course. Several lines of evidence point to the coupling between Aβ aggregation and neuroinflammation and its role in maintaining brain homeostasis during the long prodromal phase of AD. Little is however known about how this protective mechanism fails and as a result, an irreversible and progressive transition to clinical AD occurs. Here, we introduce a minimal model of a coupled system of Aβ aggregation and inflammation, numerically simulate its dynamical behavior, and analyze its bifurcation properties. The introduced model represents the following events: generation of Aβ monomers, aggregation of Aβ monomers into oligomers and fibrils, induction of inflammation by Aβ aggregates, and clearance of various Aβ species. Crucially, the rates of Aβ generation and clearance are modulated by inflammation level following a Hill-type response function. Despite its relative simplicity, the model exhibits enormously rich dynamics ranging from overdamped kinetics to sustained oscillations. We then specify the region of inflammation- and coupling-related parameters space where a transition to oscillatory dynamics occurs and demonstrate how changes in Aβ aggregation parameters could shift this oscillatory region in parameter space. Our results reveal the propensity of coupled Aβ aggregation-inflammation systems to oscillatory dynamics and propose prolonged sustained oscillations and their consequent immune system exhaustion as a potential mechanism underlying the transition to a more progressive phase of amyloid pathology in AD. The implications of our results in regard to early diagnosis of AD and anti-AD drug development are discussed.

Cycling and spiral-wave modes in an active cyclic Potts model

We studied the nonequilibrium dynamics of a cycling three-state Potts model using simulations and theory. This model can be tuned from thermal-equilibrium to far-from-equilibrium conditions. At low cycling energy, the homogeneous dominant state cycles via nucleation and growth, while spiral waves are formed at high energy. For large systems, a discontinuous transition occurs from these cyclic homogeneous phases to spiral waves, while the opposite transition is absent. Conversely, these two modes can coexist for small systems. The waves can be reproduced by a continuum theory, and the transition can be understood from the competition between nucleation and growth.

A Chemical Reaction Network Drives Complex Population Dynamics in Oscillating Self-Reproducing Vesicles

We report chemically fueled oscillations of vesicles. The population cycling of vesicles is driven by their self-reproduction and collapse within a biphasic reaction network involving the interplay of molecular and supramolecular events. We studied the oscillations on the molecular and supramolecular scales and tracked vesicle populations in time by interferometric scattering microscopy and dynamic light scattering. Complex supramolecular events were observed during oscillations─including vesicle reproduction, growth, and decomposition─and differences in the number, size, and mass of aggregates can often be observed within and between pulses. This system's dynamic behavior is reminiscent of a reproductive cycle in living cells.

Choreographing Oscillatory Hydrodynamics with DNA-Coated Gold Nanoparticles

Periodic responses to nonperiodic energy inputs, such as oscillations, are hallmarks of living systems. Nanoparticle-based systems have largely remained unexplored in the generation of oscillatory features. Here, we demonstrate a nanosystem featuring hierarchical response to light, where thermoplasmonic effects and reversible DNA-hybridization generate thermal convective forces and ultimately, oscillatory hydrodynamic flows. The slow aggregation of gold nanoparticles (AuNPs) serves as a positive feedback, while fast photothermal disassembly acts as negative feedback. These asymmetric feedback loops, combined with thermal hysteresis for time-delay, are essential ingredients for orchestrating an oscillating response.

A Partial Tumor Irradiation Approach for Complex Bulky Disease

A large proportion of cancer patients present with unresectable bulky disease at baseline or following treatment failure. The data available in the literature suggest that the vast majority of these patients do not benefit from available standard therapies. Therefore the clinical outcomes are poor; patients are desperate and usually relegated to palliative or best supportive care as the only options. Large tumor masses are usually hypoxic, resistant to radiation and systemic therapy, with extensive regional infiltration of the surrounding critical organs, the presence of which makes it impossible to deliver a radical dose of radiation. Promising data in terms of improved therapeutic ratio where such complex tumors are concerned can be seen with the use of new emerging unconventional radiotherapy techniques known as spatially fractionated radiotherapies (SFRT). One of them is PATHY, or PArtial Tumor irradiation targeting HYpoxic segment, which is characterized by a very short treatment course offering a large spectrum of therapeutic benefits in terms of the symptom relief, quality of life, local tumor control, neoadjuvant and immunomodulatory effects.

Biological Rhythms Generated by a Single Activator-Repressor Loop with Inhomogeneity and Diffusion

Common models of circadian rhythms are typically constructed as compartmental reactions of well-mixed biochemicals, incorporating a negative-feedback loop consisting of several intermediate reaction steps essentially required to produce oscillations. Spatial transport of each reactant is often represented as an extra compartmental reaction step. Contrary to this traditional understanding, in this Letter we demonstrate that a single activation-repression biochemical reaction pair is sufficient to generate sustained oscillations if the sites of both reactions are spatially separated and molecular transport is mediated by diffusion. Our proposed scenario represents the simplest configuration in terms of the participating chemical reactions and offers a conceptual basis for understanding biological oscillations and inspiring in vitro assays aimed at constructing minimal clocks.

Enhancing model identification with SINDy via nullcline reconstruction

Many dynamical systems exhibit oscillatory behavior that can be modeled with differential equations. Recently, these equations have increasingly been derived through data-driven methods, including the transparent technique known as Sparse Identification of Nonlinear Dynamics (SINDy). This paper illustrates the importance of accurately determining the system's limit cycle position in phase space for identifying sparse and effective models. We introduce a method for identifying the limit cycle position and the system's nullclines by applying SINDy to datasets adjusted with various offsets. This approach is evaluated using three criteria: model complexity, coefficient of determination, and generalization error. We applied this method to several models: the oscillatory FitzHugh-Nagumo model, a more complex model consisting of two coupled cubic differential equations with a single stable state, and a multistable model of glycolytic oscillations. Our results confirm that incorporating detailed information about the limit cycle in phase space enhances the accuracy of model identification in oscillatory systems.

Design patterns for the construction of computational biological models

Computational biological models have proven to be an invaluable tool for understanding and predicting the behaviour of many biological systems. While it may not be too challenging for experienced researchers to construct such models from scratch, it is not a straightforward task for early stage researchers. Design patterns are well-known techniques widely applied in software engineering as they provide a set of typical solutions to common problems in software design. In this paper, we collect and discuss common patterns that are usually used during the construction and execution of computational biological models. We adopt Petri nets as a modelling language to provide a visual illustration of each pattern; however, the ideas presented in this paper can also be implemented using other modelling formalisms. We provide two case studies for illustration purposes and show how these models can be built up from the presented smaller modules. We hope that the ideas discussed in this paper will help many researchers in building their own future models.

Temperature-Dependent Fold-Switching Mechanism of the Circadian Clock Protein KaiB

The oscillator of the cyanobacterial circadian clock relies on the ability of the KaiB protein to switch reversibly between a stable ground-state fold (gsKaiB) and an unstable fold-switched fold (fsKaiB). Rare fold-switching events by KaiB provide a critical delay in the negative feedback loop of this post-translational oscillator. In this study, we experimentally and computationally investigate the temperature dependence of fold switching and its mechanism. We demonstrate that the stability of gsKaiB increases with temperature compared to fsKaiB and that the Q10 value for the gsKaiB → fsKaiB transition is nearly three times smaller than that for the reverse transition. Simulations and native-state hydrogen-deuterium exchange NMR experiments suggest that fold switching can involve both subglobally and near-globally unfolded intermediates. The simulations predict that the transition state for fold switching coincides with isomerization of conserved prolines in the most rapidly exchanging region, and we confirm experimentally that proline isomerization is a rate-limiting step for fold switching. We explore the implications of our results for temperature compensation, a hallmark of circadian clocks, through a kinetic model.

Change in RhoGAP and RhoGEF availability drives transitions in cortical patterning and excitability in Drosophila

Actin cortex patterning and dynamics are critical for cell shape changes. These dynamics undergo transitions during development, often accompanying changes in collective cell behavior. Although mechanisms have been established for individual cells' dynamic behaviors, the mechanisms and specific molecules that result in developmental transitions in vivo are still poorly understood. Here, we took advantage of two developmental systems in Drosophila melanogaster to identify conditions that altered cortical patterning and dynamics. We identified a Rho guanine nucleotide exchange factor (RhoGEF) and Rho GTPase activating protein (RhoGAP) pair required for actomyosin waves in egg chambers. Specifically, depletion of the RhoGEF, Ect2, or the RhoGAP, RhoGAP15B, disrupted actomyosin wave induction, and both proteins relocalized from the nucleus to the cortex preceding wave formation. Furthermore, we found that overexpression of a different RhoGEF and RhoGAP pair, RhoGEF2 and Cumberland GAP (C-GAP), resulted in actomyosin waves in the early embryo, during which RhoA activation precedes actomyosin assembly by ∼4 s. We found that C-GAP was recruited to actomyosin waves, and disrupting F-actin polymerization altered the spatial organization of both RhoA signaling and the cytoskeleton in waves. In addition, disrupting F-actin dynamics increased wave period and width, consistent with a possible role for F-actin in promoting delayed negative feedback. Overall, we showed a mechanism involved in inducing actomyosin waves that is essential for oocyte development and is general to other cell types, such as epithelial and syncytial cells.

Cell coupling compensates for changes in single-cell Her6 dynamics and provides phenotypic robustness

This paper investigates the effect of altering the protein expression dynamics of the bHLH transcription factor Her6 at the single-cell level in the embryonic zebrafish telencephalon. Using a homozygote endogenous Her6:Venus reporter and 4D single-cell tracking, we show that Her6 oscillates in neural telencephalic progenitors and that the fusion of protein destabilisation (PEST) domain alters its expression dynamics, causing most cells to downregulate Her6 prematurely. However, counterintuitively, oscillatory cells increase, with some expressing Her6 at high levels, resulting in increased heterogeneity of Her6 expression in the population. These tissue-level changes appear to be an emergent property of coupling between single-cells, as revealed by experimentally disrupting Notch signalling and by computationally modelling alterations in Her6 protein stability. Despite the profound differences in the single-cell Her6 dynamics, the size of the telencephalon is only transiently altered and differentiation markers do not exhibit significant differences early on; however, a small increase is observed at later developmental stages. Our study suggests that cell coupling provides a compensation strategy, whereby an almost normal phenotype is maintained even though single-cell gene expression dynamics are abnormal, granting phenotypic robustness.

Mechanochemical dynamics of collective cells and hierarchical topological defects in multicellular lumens

Collective cell dynamics is essential for tissue morphogenesis and various biological functions. However, it remains incompletely understood how mechanical forces and chemical signaling are integrated to direct collective cell behaviors underlying tissue morphogenesis. Here, we propose a three-dimensional (3D) mechanochemical theory accounting for biochemical reaction-diffusion and cellular mechanotransduction to investigate the dynamics of multicellular lumens. We show that the interplay between biochemical signaling and mechanics can trigger either pitchfork or Hopf bifurcation to induce diverse static mechanochemical patterns or generate oscillations with multiple modes both involving marked mechanical deformations in lumens. We uncover the crucial role of mechanochemical feedback in emerging morphodynamics and identify the evolution and morphogenetic functions of hierarchical topological defects including cell-level hexatic defects and tissue-level orientational defects. Our theory captures the common mechanochemical traits of collective dynamics observed in experiments and could provide a mechanistic context for understanding morphological symmetry breaking in 3D lumen-like tissues.

A mathematical model for the role of dopamine-D2 self-regulation in the production of ultradian rhythms

Many self-motivated and goal-directed behaviours display highly flexible, approximately 4 hour ultradian (shorter than a day) oscillations. Despite lacking direct correspondence to physical cycles in the environment, these ultradian rhythms may be involved in optimizing functional interactions with the environment and reflect intrinsic neural dynamics. Current evidence supports a role of mesostriatal dopamine (DA) in the expression and propagation of ultradian rhythmicity, however, the biochemical processes underpinning these oscillations remain to be identified. Here, we use a mathematical model to investigate D2 autoreceptor-dependent DA self-regulation as the source of ultradian behavioural rhythms. DA concentration at the midbrain-striatal synapses is governed through a dual-negative feedback-loop structure, which naturally gives rise to rhythmicity. This model shows the propensity of striatal DA to produce an ultradian oscillation characterized by a flexible period that is highly sensitive to parameter variations. Circadian (approximately 24 hour) regulation consolidates the ultradian oscillations and alters their response to the phase-dependent, rapid-resetting effect of a transient excitatory stimulus. Within a circadian framework, the ultradian rhythm orchestrates behavioural activity and enhances responsiveness to an external stimulus. This suggests a role for the circadian-ultradian timekeeping hierarchy in governing organized behaviour and shaping daily experience through coordinating the motivation to engage in recurring, albeit not highly predictable events, such as social interactions.

Network switches and their role in circadian clocks

Circadian rhythms are generated by complex interactions among genes and proteins. Self-sustained ∼24 h oscillations require negative feedback loops and sufficiently strong nonlinearities that are the product of molecular and network switches. Here, we review common mechanisms to obtain switch-like behavior, including cooperativity, antagonistic enzymes, multisite phosphorylation, positive feedback, and sequestration. We discuss how network switches play a crucial role as essential components in cellular circadian clocks, serving as integral parts of transcription-translation feedback loops that form the basis of circadian rhythm generation. The design principles of network switches and circadian clocks are illustrated by representative mathematical models that include bistable systems and negative feedback loops combined with Hill functions. This work underscores the importance of negative feedback loops and network switches as essential design principles for biological oscillations, emphasizing how an understanding of theoretical concepts can provide insights into the mechanisms generating biological rhythms.

From biological data to oscillator models using SINDy

Periodic changes in the concentration or activity of different molecules regulate vital cellular processes such as cell division and circadian rhythms. Developing mathematical models is essential to better understand the mechanisms underlying these oscillations. Recent data-driven methods like SINDy have fundamentally changed model identification, yet their application to experimental biological data remains limited. This study investigates SINDy's constraints by directly applying it to biological oscillatory data. We identify insufficient resolution, noise, dimensionality, and limited prior knowledge as primary limitations. Using various generic oscillator models of different complexity and/or dimensionality, we systematically analyze these factors. We then propose a comprehensive guide for inferring models from biological data, addressing these challenges step by step. Our approach is validated using glycolytic oscillation data from yeast.

Selenium catalysis enables negative feedback organic oscillators

The construction of materials regulated by chemical reaction networks requires regulatory motifs that can be stacked together into systems with desired properties. Multiple autocatalytic reactions producing thiols are known. However, negative feedback loop motifs are unavailable for thiol chemistry. Here, we develop a negative feedback loop based on the selenocarbonates. In this system, thiols induce the release of aromatic selenols that catalyze the oxidation of thiols by organic peroxides. This negative feedback loop has two important features. First, catalytic oxidation of thiols follows Michaelis-Menten-like kinetics, thus increasing nonlinearity for the negative feedback. Second, the strength of the negative feedback can be tuned by varying substituents in selenocarbonates. When combined with the autocatalytic production of thiols in a flow reactor, this negative feedback loop induces sustained oscillations. The availability of this negative feedback motif enables the future construction of oscillatory, homeostatic, adaptive, and other regulatory circuits in life-inspired systems and materials.

Artificial Homeostasis Systems Based on Feedback Reaction Networks: Design Principles and Future Promises

Feedback-controlled chemical reaction networks (FCRNs) are indispensable for various biological processes, such as cellular mechanisms, patterns, and signaling pathways. Through the intricate interplay of many feedback loops (FLs), FCRNs maintain a stable internal cellular environment. Currently, creating minimalistic synthetic cells is the long-term objective of systems chemistry, which is motivated by such natural integrity. The design, kinetic optimization, and analysis of FCRNs to exhibit functions akin to those of a cell still pose significant challenges. Indeed, reaching synthetic homeostasis is essential for engineering synthetic cell components. However, maintaining homeostasis in artificial systems against various agitations is a difficult task. Several biological events can provide us with guidelines for a conceptual understanding of homeostasis, which can be further applicable in designing artificial synthetic systems. In this regard, we organize our review with artificial homeostasis systems driven by FCRNs at different length scales, including homogeneous, compartmentalized, and soft material systems. First, we stretch a quick overview of FCRNs in different molecular and supramolecular systems, which are the essential toolbox for engineering different nonlinear functions and homeostatic systems. Moreover, the existing history of synthetic homeostasis in chemical and material systems and their advanced functions with self-correcting, and regulating properties are also emphasized.

A robust ultrasensitive transcriptional switch in noisy cellular environments

Ultrasensitive transcriptional switches enable sharp transitions between transcriptional on and off states and are essential for cells to respond to environmental cues with high fidelity. However, conventional switches, which rely on direct repressor-DNA binding, are extremely noise-sensitive, leading to unintended changes in gene expression. Here, through model simulations and analysis, we discovered that an alternative design combining three indirect transcriptional repression mechanisms, sequestration, blocking, and displacement, can generate a noise-resilient ultrasensitive switch. Although sequestration alone can generate an ultrasensitive switch, it remains sensitive to noise because the unintended transcriptional state induced by noise persists for long periods. However, by jointly utilizing blocking and displacement, these noise-induced transitions can be rapidly restored to the original transcriptional state. Because this transcriptional switch is effective in noisy cellular contexts, it goes beyond previous synthetic transcriptional switches, making it particularly valuable for robust synthetic system design. Our findings also provide insights into the evolution of robust ultrasensitive switches in cells. Specifically, the concurrent use of seemingly redundant indirect repression mechanisms in diverse biological systems appears to be a strategy to achieve noise-resilience of ultrasensitive switches.