A tale of two rhythms: Locked clocks and chaos in biology

External low energy electromagnetic fields affect heart dynamics: surrogate for system synchronization, chaos control and cancer patient's health

All cells in the human body, including cancer cells, possess specific electrical properties crucial for their functions. These properties are notably different between normal and cancerous cells. Cancer cells are characterized by autonomous oscillations and damped electromagnetic field (EMF) activation. Cancer reduces physiological variability, implying a systemic disconnection that desynchronizes bodily systems and their inherent random processes. The dynamics of heart rate, in this context, could reflect global physiological network instability in the sense of entrainment. Using a medical device that employs an active closed-loop system, such as administering specifically modulated EMF frequencies at targeted intervals and at low energies, we can evaluate the periodic oscillations of the heart. This procedure serves as a closed-loop control mechanism leading to a temporary alteration in plasma membrane ionic flow and the heart's periodic oscillation dynamics. The understanding of this phenomenon is supported by computer simulations of a mathematical model, which are validated by experimental data. Heart dynamics can be quantified using difference logistic equations, and it correlates with improved overall survival rates in cancer patients.

Mean-field approximation for networks with synchrony-driven adaptive coupling

Synaptic plasticity plays a fundamental role in neuronal dynamics, governing how connections between neurons evolve in response to experience. In this study, we extend a network model of θ-neuron oscillators to include a realistic form of adaptive plasticity. In place of the less tractable spike-timing-dependent plasticity, we employ recently validated phase-difference-dependent plasticity rules, which adjust coupling strengths based on the relative phases of θ-neuron oscillators. We explore two distinct implementations of this plasticity: pairwise updates to individual coupling strengths and global updates applied to the mean coupling strength. We derive a mean-field approximation and assess its accuracy by comparing it to θ-neuron simulations across various stability regimes. The synchrony of the system is quantified using the Kuramoto order parameter. Through bifurcation analysis and the calculation of maximal Lyapunov exponents, we uncover interesting phenomena such as bistability and chaotic dynamics via period-doubling and boundary crisis bifurcations. These behaviors emerge as a direct result of adaptive coupling and are absent in systems without such plasticity.

Emerging cancer therapies: targeting physiological networks and cellular bioelectrical differences with non-thermal systemic electromagnetic fields in the human body - a comprehensive review

A steadily increasing number of publications support the concept of physiological networks, and how cellular bioelectrical properties drive cell proliferation and cell synchronization. All cells, especially cancer cells, are known to possess characteristic electrical properties critical for physiological behavior, with major differences between normal and cancer cell counterparts. This opportunity can be explored as a novel treatment modality in Oncology. Cancer cells exhibit autonomous oscillations, deviating from normal rhythms. In this context, a shift from a static view of cellular processes is required for a better understanding of the dynamic connections between cellular metabolism, gene expression, cell signaling and membrane polarization as states in constant flux in realistic human models. In oncology, radiofrequency electromagnetic fields have produced sustained responses and improved quality of life in cancer patients with minimal side effects. This review aims to show how non-thermal systemic radiofrequency electromagnetic fields leads to promising therapeutic responses at cellular and tissue levels in humans, supporting this newly emerging cancer treatment modality with early favorable clinical experience specifically in advanced cancer.

Entrainment and multi-stability of the p53 oscillator in human cells

The tumor suppressor p53 responds to cellular stress and activates transcription programs critical for regulating cell fate. DNA damage triggers oscillations in p53 levels with a robust period. Guided by the theory of synchronization and entrainment, we developed a mathematical model and experimental system to test the ability of the p53 oscillator to entrain to external drug pulses of various periods and strengths. We found that the p53 oscillator can be locked and entrained to a wide range of entrainment modes. External periods far from p53's natural oscillations increased the heterogeneity between individual cells whereas stronger inputs reduced it. Single-cell measurements allowed deriving the phase response curves (PRCs) and multiple Arnold tongues of p53. In addition, multi-stability and non-linear behaviors were mathematically predicted and experimentally detected, including mode hopping, period doubling, and chaos. Our work revealed critical dynamical properties of the p53 oscillator and provided insights into understanding and controlling it. A record of this paper's transparent peer review process is included in the supplemental information.

From resonance to chaos by modulating spatiotemporal patterns through a synthetic optogenetic oscillator

Oscillations are a recurrent phenomenon in biological systems across scales, but deciphering their fundamental principles is very challenging. Here, we tackle this challenge by redesigning the wellcharacterised synthetic oscillator known as "repressilator" in Escherichia coli and controlling it using optogenetics, creating the "optoscillator". Bacterial colonies manifest oscillations as spatial ring patterns. When we apply periodic light pulses, the optoscillator behaves as a forced oscillator and we systematically investigate the properties of the rings under various light conditions. Combining experiments with mathematical modeling, we demonstrate that this simple oscillatory circuit can generate complex dynamics that are transformed into distinct spatial patterns. We report the observation of synchronisation, resonance, subharmonic resonance and period doubling. Furthermore, we present evidence of a chaotic regime. This work highlights the intricate spatiotemporal patterns accessible by synthetic oscillators and underscores the potential of our approach in revealing fundamental principles of biological oscillations.

Dynamical analysis of a periodically forced chaotic chemical oscillator

We present a comprehensive dynamical analysis of a chaotic chemical model referred to as the autocatalator, when subject to a periodic administration of one substrate. Our investigation encompasses the dynamical characterization of both unforced and forced systems utilizing isospikes and largest Lyapunov exponents-based parameter planes, bifurcation diagrams, and analysis of complex oscillations. Additionally, we present a phase diagram showing the effect of the period and amplitude of the forcing signal on the system's behavior. Furthermore, we show how the landscapes of parameter planes are altered in response to forcing application. This analysis contributes to a deeper understanding of the intricate dynamics induced by the periodic forcing of a chaotic system.

Coupling allows robust mammalian redox circadian rhythms despite heterogeneity and noise

Circadian clocks are endogenous oscillators present in almost all cells that drive daily rhythms in physiology and behavior. There are two mechanisms that have been proposed to explain how circadian rhythms are generated in mammalian cells: through a transcription-translation feedback loop (TTFL) and based on oxidation/reduction reactions, both of which are intrinsically stochastic and heterogeneous at the single cell level. In order to explore the emerging properties of stochastic and heterogeneous redox oscillators, we simplify a recently developed kinetic model of redox oscillations to an amplitude-phase oscillator with 'twist' (period-amplitude correlation) and subject to Gaussian noise. We show that noise and heterogeneity alone lead to fast desynchronization, and that coupling between noisy oscillators can establish robust and synchronized rhythms with amplitude expansions and tuning of the period due to twist. Coupling a network of redox oscillators to a simple model of the TTFL also contributes to synchronization, large amplitudes and fine-tuning of the period for appropriate interaction strengths. These results provide insights into how the circadian clock compensates randomness from intracellular sources and highlight the importance of noise, heterogeneity and coupling in the context of circadian oscillators.

Chaos in a bacterial stress response

Cellular responses to environmental changes are often highly heterogeneous and exhibit seemingly random dynamics. The astonishing insight of chaos theory is that such unpredictable patterns can, in principle, arise without the need for any random processes, i.e., purely deterministically without noise. However, while chaos is well understood in mathematics and physics, its role in cell biology remains unclear because the complexity and noisiness of biological systems make testing difficult. Here, we show that chaos explains the heterogeneous response of Escherichia coli cells to oxidative stress. We developed a theoretical model of the gene expression dynamics and demonstrate that chaotic behavior arises from rapid molecular feedbacks that are coupled with cell growth dynamics and cell-cell interactions. Based on theoretical predictions, we then designed single-cell experiments to show we can shift gene expression from periodic oscillations to chaos on demand. Our work suggests that chaotic gene regulation can be employed by cell populations to generate strong and variable responses to changing environments.

An improved rhythmicity analysis method using Gaussian Processes detects cell-density dependent circadian oscillations in stem cells

MOTIVATION

Detecting oscillations in time series remains a challenging problem even after decades of research. In chronobiology, rhythms (for instance in gene expression, eclosion, egg-laying, and feeding) tend to be low amplitude, display large variations amongst replicates, and often exhibit varying peak-to-peak distances (non-stationarity). Most currently available rhythm detection methods are not specifically designed to handle such datasets, and are also limited by their use of P-values in detecting oscillations.

RESULTS

We introduce a new method, ODeGP (Oscillation Detection using Gaussian Processes), which combines Gaussian Process regression and Bayesian inference to incorporate measurement errors, non-uniformly sampled data, and a recently developed non-stationary kernel to improve detection of oscillations. By using Bayes factors, ODeGP models both the null (non-rhythmic) and the alternative (rhythmic) hypotheses, thus providing an advantage over P-values. Using synthetic datasets, we first demonstrate that ODeGP almost always outperforms eight commonly used methods in detecting stationary as well as non-stationary symmetric oscillations. Next, by analyzing existing qPCR datasets, we demonstrate that our method is more sensitive compared to the existing methods at detecting weak and noisy oscillations. Finally, we generate new qPCR data on mouse embryonic stem cells. Surprisingly, we discover using ODeGP that increasing cell-density results in rapid generation of oscillations in the Bmal1 gene, thus highlighting our method's ability to discover unexpected and new patterns. In its current implementation, ODeGP is meant only for analyzing single or a few time-trajectories, not genome-wide datasets.

AVAILABILITY AND IMPLEMENTATION

ODeGP is available at https://github.com/Shaonlab/ODeGP.

E-box binding transcription factors in cancer

E-boxes are important regulatory elements in the eukaryotic genome. Transcription factors can bind to E-boxes through their basic helix-loop-helix or zinc finger domain to regulate gene transcription. E-box-binding transcription factors (EBTFs) are important regulators of development and essential for physiological activities of the cell. The fundamental role of EBTFs in cancer has been highlighted by studies on the canonical oncogene MYC, yet many EBTFs exhibit common features, implying the existence of shared molecular principles of how they are involved in tumorigenesis. A comprehensive analysis of TFs that share the basic function of binding to E-boxes has been lacking. Here, we review the structure of EBTFs, their common features in regulating transcription, their physiological functions, and their mutual regulation. We also discuss their converging functions in cancer biology, their potential to be targeted as a regulatory network, and recent progress in drug development targeting these factors in cancer therapy.

Coupled oscillator cooperativity as a control mechanism in chronobiology

Control of dynamical processes is vital for maintaining correct cell regulation and cell-fate decisions. Numerous regulatory networks show oscillatory behavior; however, our knowledge of how one oscillator behaves when stimulated by two or more external oscillatory signals is still missing. We explore this problem by constructing a synthetic oscillatory system in yeast and stimulate it with two external oscillatory signals. Letting model verification and prediction operate in a tight interplay with experimental observations, we find that stimulation with two external signals expands the plateau of entrainment and reduces the fluctuations of oscillations. Furthermore, by adjusting the phase differences of external signals, one can control the amplitude of oscillations, which is understood through the signal delay of the unperturbed oscillatory network. With this we reveal a direct amplitude dependency of downstream gene transcription. Taken together, these results suggest a new path to control oscillatory systems by coupled oscillator cooperativity.

An improved rhythmicity analysis method using Gaussian Processes detects cell-density dependent circadian oscillations in stem cells

Detecting oscillations in time series remains a challenging problem even after decades of research. In chronobiology, rhythms in time series (for instance gene expression, eclosion, egg-laying and feeding) datasets tend to be low amplitude, display large variations amongst replicates, and often exhibit varying peak-to-peak distances (non-stationarity). Most currently available rhythm detection methods are not specifically designed to handle such datasets. Here we introduce a new method, ODeGP ( O scillation De tection using G aussian P rocesses), which combines Gaussian Process (GP) regression with Bayesian inference to provide a flexible approach to the problem. Besides naturally incorporating measurement errors and non-uniformly sampled data, ODeGP uses a recently developed kernel to improve detection of non-stationary waveforms. An additional advantage is that by using Bayes factors instead of p-values, ODeGP models both the null (non-rhythmic) and the alternative (rhythmic) hypotheses. Using a variety of synthetic datasets we first demonstrate that ODeGP almost always outperforms eight commonly used methods in detecting stationary as well as non-stationary oscillations. Next, on analyzing existing qPCR datasets that exhibit low amplitude and noisy oscillations, we demonstrate that our method is more sensitive compared to the existing methods at detecting weak oscillations. Finally, we generate new qPCR time-series datasets on pluripotent mouse embryonic stem cells, which are expected to exhibit no oscillations of the core circadian clock genes. Surprisingly, we discover using ODeGP that increasing cell density can result in the rapid generation of oscillations in the Bmal1 gene, thus highlighting our method’s ability to discover unexpected patterns. In its current implementation, ODeGP (available as an R package) is meant only for analyzing single or a few time-trajectories, not genome-wide datasets.

Systematic analysis of negative and positive feedback loops for robustness and temperature compensation in circadian rhythms

Temperature compensation and robustness to biological noise are two key characteristics of the circadian clock. These features allow the circadian pacemaker to maintain a steady oscillation in a wide range of environmental conditions. The presence of a time-delayed negative feedback loop in the regulatory network generates autonomous circadian oscillations in eukaryotic systems. In comparison, the circadian clock of cyanobacteria is controlled by a strong positive feedback loop. Positive feedback loops with substrate depletion can also generate oscillations, inspiring other circadian clock models. What makes a circadian oscillatory network robust to extrinsic noise is unclear. We investigated four basic circadian oscillators with negative, positive, and combinations of positive and negative feedback loops to explore network features necessary for circadian clock resilience. We discovered that the negative feedback loop system performs the best in compensating temperature changes. We also show that a positive feedback loop can reduce extrinsic noise in periods of circadian oscillators, while intrinsic noise is reduced by negative feedback loops.

Jag1-Notch cis-interaction determines cell fate segregation in pancreatic development

The Notch ligands Jag1 and Dll1 guide differentiation of multipotent pancreatic progenitor cells (MPCs) into unipotent pro-acinar cells (PACs) and bipotent duct/endocrine progenitors (BPs). Ligand-mediated trans-activation of Notch receptors induces oscillating expression of the transcription factor Hes1, while ligand-receptor cis-interaction indirectly represses Hes1 activation. Despite Dll1 and Jag1 both displaying cis- and trans-interactions, the two mutants have different phenotypes for reasons not fully understood. Here, we present a mathematical model that recapitulates the spatiotemporal differentiation of MPCs into PACs and BPs. The model correctly captures cell fate changes in Notch pathway knockout mice and small molecule inhibitor studies, and a requirement for oscillatory Hes1 expression to maintain the multipotent state. Crucially, the model entails cell-autonomous attenuation of Notch signaling by Jag1-mediated cis-inhibition in MPC differentiation. The model sheds light on the underlying mechanisms, suggesting that cis-interaction is crucial for exiting the multipotent state, while trans-interaction is required for adopting the bipotent fate.

Enhanced DNA repair through droplet formation and p53 oscillations

Living organisms are constantly exposed to DNA damage, and optimal repair is therefore crucial. A characteristic hallmark of the response is the formation of sub-compartments around the site of damage, known as foci. Following multiple DNA breaks, the transcription factor p53 exhibits oscillations in its nuclear concentration, but how this dynamics can affect the repair remains unknown. Here, we formulate a theory for foci formation through droplet condensation and discover how oscillations in p53, with its specific periodicity and amplitude, optimize the repair process by preventing Ostwald ripening and distributing protein material in space and time. Based on the theory predictions, we reveal experimentally that the oscillatory dynamics of p53 does enhance the repair efficiency. These results connect the dynamical signaling of p53 with the microscopic repair process and create a new paradigm for the interplay of complex dynamics and phase transitions in biology.

OSCILLATORS IN THE MICROVASCULATURE - GLYCOCALYX AND BEYOND

The growing recognition of abundance of oscillating functions in biological systems has motivated this brief overview which narrows down on the microvasculature. Specifically, it encompasses self-sustained oscillations of blood flow, hematocrit and viscosity at bifurcations; their effects on the oscillations of endothelial glycocalyx, mechanotransduction and its termination to prime endothelial cells for the subsequent mechanical signaling event; oscillating affinity of hyaluronan-CD44 binding domain; spontaneous contractility of actomyosin complexes in the cortical actin web, its effects on the tension of the plasma membrane; reversible effects of sirtuin-1 on endothelial glycocalyx; and effects of plasma membrane tension on endo-and exocytosis. Some potential interactions between those oscillators - their coupling - are discussed together with their transition into chaotic movements. Future in-depth understanding of the oscillatory activities in the microvasculature could serve as a guide to its chronotherapy under pathological conditions.

Temperature Controls Onset and Period of NF-κB Oscillations and can Lead to Chaotic Dynamics

The transcription factor NF-κB plays a vital role in the control of the immune system, and following stimulation with TNF-α its nuclear concentration shows oscillatory behaviour. How environmental factors, in particular temperature, can control the oscillations and thereby affect gene stimulation is still remains to be resolved question. In this work, we reveal that the period of the oscillations decreases with increasing temperature. We investigate this using a mathematical model, and by applying results from statistical physics, we introduce temperature dependency to all rates, resulting in a remarkable correspondence between model and experiments. Our model predicts how temperature affects downstream protein production and find a crossover, where high affinity genes upregulates at high temperatures. Finally, we show how or that oscillatory temperatures can entrain NF-κB oscillations and lead to chaotic dynamics presenting a simple path to chaotic conditions in cellular biology.

The emergence of polyglot entrainment responses to periodic inputs in vicinities of Hopf bifurcations in slow-fast systems

Several distinct entrainment patterns can occur in the FitzHugh-Nagumo (FHN) model under external periodic forcing. Investigating the FHN model under different types of periodic forcing reveals the existence of multiple disconnected 1:1 entrainment segments for constant, low enough values of the input amplitude when the unforced system is in the vicinity of a Hopf bifurcation. This entrainment structure is termed polyglot to distinguish it from the single 1:1 entrainment region (monoglot) structure typically observed in Arnold tongue diagrams. The emergence of polyglot entrainment is then explained using phase-plane analysis and other dynamical system tools. Entrainment results are investigated for other slow-fast systems of neuronal, circadian, and glycolytic oscillations. Exploring these models, we found that polyglot entrainment structure (multiple 1:1 regions) is observed when the unforced system is in the vicinity of a Hopf bifurcation and the Hopf point is located near a knee of a cubic-like nullcline.

The Root Clock as a Signal Integrator System: Ensuring Balance for Survival

The root system is essential for the survival of terrestrial plants, plant development, and adaptation to changing environments. The development of the root system relies on post-embryonic organogenesis and more specifically on the formation and growth of lateral roots (LR). The spacing of LR along the main root is underpinned by a precise prepatterning mechanism called the Root Clock. In Arabidopsis, the primary output of this mechanism involves the generation of periodic gene expression oscillations in a zone close to the root tip called the Oscillation Zone (OZ). Because of these oscillations, pre-branch sites (PBS) are established in the positions from which LR will emerge, although the oscillations can also possibly regulate the root wavy pattern and growth. Furthermore, we show that the Root Clock is present in LR. In this review, we describe the recent advances unraveling the inner machinery of Root Clock as well as the new tools to track the Root Clock activity. Moreover, we discuss the basis of how Arabidopsis can balance the creation of a repetitive pattern while integrating both endogenous and exogenous signals to adapt to changing environmental conditions. These signals can work as entrainment signals, but in occasions they also affect the periodicity and amplitude of the oscillatory dynamics in gene expression. Finally, we identify similarities with the Segmentation Clock of vertebrates and postulate the existence of a determination front delimiting the end of the oscillations in gene expression and initiating LR organogenesis through the activation of PBS in an ARF7 dependent-manner.

Multi-synchronization and other patterns of multi-rhythmicity in oscillatory biological systems

While experimental and theoretical studies have established the prevalence of rhythmic behaviour at all levels of biological organization, less common is the coexistence between multiple oscillatory regimes (multi-rhythmicity), which has been predicted by a variety of models for biological oscillators. The phenomenon of multi-rhythmicity involves, most commonly, the coexistence between two (birhythmicity) or three (trirhythmicity) distinct regimes of self-sustained oscillations. Birhythmicity has been observed experimentally in a few chemical reactions and in biological examples pertaining to cardiac cell physiology, neurobiology, human voice patterns and ecology. The present study consists of two parts. We first review the mechanisms underlying multi-rhythmicity in models for biochemical and cellular oscillations in which the phenomenon was investigated over the years. In the second part, we focus on the coupling of the cell cycle and the circadian clock and show how an additional source of multi-rhythmicity arises from the bidirectional coupling of these two cellular oscillators. Upon bidirectional coupling, the two oscillatory networks generally synchronize in a unique manner characterized by a single, common period. In some conditions, however, the two oscillators may synchronize in two or three different ways characterized by distinct waveforms and periods. We refer to this type of multi-rhythmicity as ‘multi-synchronization’.

Principles, mechanisms and functions of entrainment in biological oscillators

Entrainment is a phenomenon in which two oscillators interact with each other, typically through physical or chemical means, to synchronize their oscillations. This phenomenon occurs in biology to coordinate processes from the molecular to organismal scale. Biological oscillators can be entrained within a single cell, between cells or to an external input. Using six illustrative examples of entrainable biological oscillators, we discuss the distinctions between entrainment and synchrony and explore features that contribute to a system's propensity to entrain. Entrainment can either enhance or reduce the heterogeneity of oscillations within a cell population, and we provide examples and mechanisms of each case. Finally, we discuss the known functions of entrainment and discuss potential functions from an evolutionary perspective.

Searching Deterministic Chaotic Properties in System-Wide Vulnerability Datasets

Cybersecurity is a never-ending battle against attackers, who try to identify and exploit misconfigurations and software vulnerabilities before being patched. In this ongoing conflict, it is important to analyse the properties of the vulnerability time series to understand when information systems are more vulnerable. We study computer systems’ software vulnerabilities and probe the relevant National Vulnerability Database (NVD) time-series properties. More specifically, we show through an extensive experimental study based on the National Institute of Standards and Technology (NIST) database that the relevant systems software time series present significant chaotic properties. Moreover, by defining some systems based on open and closed source software, we compare their chaotic properties resulting in statistical conclusions. The contribution of this novel study is focused on the prepossessing stage of vulnerabilities time series forecasting. The strong evidence of their chaotic properties as derived by this research effort could lead to a deeper analysis to provide additional tools to their forecasting process.

Principles underlying the complex dynamics of temperature entrainment by a circadian clock

Autonomously oscillating circadian clocks resonate with daily environmental (zeitgeber) rhythms to organize physiology around the solar day. Although entrainment properties and mechanisms have been studied widely and in great detail for light-dark cycles, entrainment to daily temperature rhythms remains poorly understood despite that they are potent zeitgebers. Here we investigate the entrainment of the chronobiological model organism Neurospora crassa, subject to thermocycles of different periods and fractions of warm versus cold phases, mimicking seasonal variations. Depending on the properties of these thermocycles, regularly entrained rhythms, period-doubling (frequency demultiplication) but also irregular aperiodic behavior occurs. We demonstrate that the complex nonlinear phenomena of experimentally observed entrainment dynamics can be understood by molecular mathematical modeling.

Autonomous clocks that regulate organelle biogenesis, cytoskeletal organization, and intracellular dynamics

How do cells perceive time? Do cells use temporal information to regulate the production/degradation of their enzymes, membranes, and organelles? Does controlling biological time influence cytoskeletal organization and cellular architecture in ways that confer evolutionary and physiological advantages? Potential answers to these fundamental questions of cell biology have historically revolved around the discussion of 'master' temporal programs, such as the principal cyclin-dependent kinase/cyclin cell division oscillator and the circadian clock. In this review, we provide an overview of the recent evidence supporting an emerging concept of 'autonomous clocks,' which under normal conditions can be entrained by the cell cycle and/or the circadian clock to run at their pace, but can also run independently to serve their functions if/when these major temporal programs are halted/abrupted. We begin the discussion by introducing recent developments in the study of such clocks and their roles at different scales and complexities. We then use current advances to elucidate the logic and molecular architecture of temporal networks that comprise autonomous clocks, providing important clues as to how these clocks may have evolved to run independently and, sometimes at the cost of redundancy, have strongly coupled to run under the full command of the cell cycle and/or the circadian clock. Next, we review a list of important recent findings that have shed new light onto potential hallmarks of autonomous clocks, suggestive of prospective theoretical and experimental approaches to further accelerate their discovery. Finally, we discuss their roles in health and disease, as well as possible therapeutic opportunities that targeting the autonomous clocks may offer.

A modular approach for modeling the cell cycle based on functional response curves

Modeling biochemical reactions by means of differential equations often results in systems with a large number of variables and parameters. As this might complicate the interpretation and generalization of the obtained results, it is often desirable to reduce the complexity of the model. One way to accomplish this is by replacing the detailed reaction mechanisms of certain modules in the model by a mathematical expression that qualitatively describes the dynamical behavior of these modules. Such an approach has been widely adopted for ultrasensitive responses, for which underlying reaction mechanisms are often replaced by a single Hill function. Also time delays are usually accounted for by using an explicit delay in delay differential equations. In contrast, however, S-shaped response curves, which by definition have multiple output values for certain input values and are often encountered in bistable systems, are not easily modeled in such an explicit way. Here, we extend the classical Hill function into a mathematical expression that can be used to describe both ultrasensitive and S-shaped responses. We show how three ubiquitous modules (ultrasensitive responses, S-shaped responses and time delays) can be combined in different configurations and explore the dynamics of these systems. As an example, we apply our strategy to set up a model of the cell cycle consisting of multiple bistable switches, which can incorporate events such as DNA damage and coupling to the circadian clock in a phenomenological way.

Bistability plays an important role in many biochemical processes and typically emerges from complex interaction patterns such as positive and double negative feedback loops. Here, we propose to theoretically study the effect of bistability in a larger interaction network. We explicitly incorporate a functional expression describing an S-shaped input-output curve in the model equations, without the need for considering the underlying biochemical events. This expression can be converted into a functional module for an ultrasensitive response, and a time delay is easily included as well. Exploiting the fact that several of these modules can easily be combined in larger networks, we construct a cell cycle model consisting of multiple bistable switches and show how this approach can account for a number of known properties of the cell cycle.