Entrainment of the mammalian cell cycle by the circadian clock: modeling two coupled cellular rhythms

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.

Time-of-day effects of cancer drugs revealed by high-throughput deep phenotyping

The circadian clock, a fundamental biological regulator, governs essential cellular processes in health and disease. Circadian-based therapeutic strategies are increasingly gaining recognition as promising avenues. Aligning drug administration with the circadian rhythm can enhance treatment efficacy and minimize side effects. Yet, uncovering the optimal treatment timings remains challenging, limiting their widespread adoption. In this work, we introduce a high-throughput approach integrating live-imaging and data analysis techniques to deep-phenotype cancer cell models, evaluating their circadian rhythms, growth, and drug responses. We devise a streamlined process for profiling drug sensitivities across different times of the day, identifying optimal treatment windows and responsive cell types and drug combinations. Finally, we implement multiple computational tools to uncover cellular and genetic factors shaping time-of-day drug sensitivity. Our versatile approach is adaptable to various biological models, facilitating its broad application and relevance. Ultimately, this research leverages circadian rhythms to optimize anti-cancer drug treatments, promising improved outcomes and transformative treatment strategies.

Collective effects of cell cleavage dynamics

A conserved process of early embryonic development in metazoans is the reductive cell divisions following oocyte fertilization, termed cell cleavages. Cell cleavage cycles usually start synchronously, lengthen differentially between the embryonic cells becoming asynchronous, and cease before major morphogenetic events, such as germ layer formation and gastrulation. Despite exhibiting species-specific characteristics, the regulation of cell cleavage dynamics comes down to common controllers acting mostly at the single cell/nucleus level, such as nucleus-to-cytoplasmic ratio and zygotic genome activation. Remarkably, recent work has linked cell cleavage dynamics to the emergence of collective behavior during embryogenesis, including pattern formation and changes in embryo-scale mechanics, raising the question how single-cell controllers coordinate embryo-scale processes. In this review, we summarize studies across species where an association between cell cleavages and collective behavior was made, discuss the underlying mechanisms, and propose that cell-to-cell variability in cell cleavage dynamics can serve as a mechanism of long-range coordination in developing embryos.

Modeling the Circadian Control of the Cell Cycle and Its Consequences for Cancer Chronotherapy

The mammalian cell cycle is governed by a network of cyclin/Cdk complexes which signal the progression into the successive phases of the cell division cycle. Once coupled to the circadian clock, this network produces oscillations with a 24 h period such that the progression into each phase of the cell cycle is synchronized to the day-night cycle. Here, we use a computational model for the circadian clock control of the cell cycle to investigate the entrainment in a population of cells characterized by some variability in the kinetic parameters. Our numerical simulations showed that successful entrainment and synchronization are only possible with a sufficient circadian amplitude and an autonomous period close to 24 h. Cellular heterogeneity, however, introduces some variability in the entrainment phase of the cells. Many cancer cells have a disrupted clock or compromised clock control. In these conditions, the cell cycle runs independently of the circadian clock, leading to a lack of synchronization of cancer cells. When the coupling is weak, entrainment is largely impacted, but cells maintain a tendency to divide at specific times of day. These differential entrainment features between healthy and cancer cells can be exploited to optimize the timing of anti-cancer drug administration in order to minimize their toxicity and to maximize their efficacy. We then used our model to simulate such chronotherapeutic treatments and to predict the optimal timing for anti-cancer drugs targeting specific phases of the cell cycle. Although qualitative, the model highlights the need to better characterize cellular heterogeneity and synchronization in cell populations as well as their consequences for circadian entrainment in order to design successful chronopharmacological protocols.

Advances in circadian clock regulation of reproduction

The mammalian circadian clock is an endogenously regulated oscillator that is synchronized with solar time and cycle within a 24-h period. The circadian clock exists not only in the suprachiasmatic nucleus (SCN) of the hypothalamus, a central pacemaker of the circadian clock system, but also in numerous peripheral tissues known as peripheral circadian oscillators. The SCN and peripheral circadian oscillators mutually orchestrate the diurnal rhythms of various physiological and behavioral processes in a hierarchical manner. In the past two decades, peripheral circadian oscillators have been identified and their function has been determined in the mammalian reproductive system and its related endocrine glands, including the hypothalamus, pituitary gland, ovaries, testes, uterus, mammary glands, and prostate gland. Increasing evidence indicates that both the SCN and peripheral circadian oscillators play discrete roles in coordinating reproductive processes and optimizing fertility in mammals. The present study reviews recent evidence on circadian clock regulation of reproductive function in the hypothalamic-pituitary-gonadal axis and reproductive system. Additionally, we elucidate the effects of chronodisruption (as a result of, for example, shift work, jet lag, disrupted eating patterns, and sleep disorders) on mammalian reproductive performance from multiple aspects. Finally, we propose potential behavioral changes or pharmaceutical strategies for the prevention and treatment of reproductive disorders from the perspective of chronomedicine. Conclusively, this review will outline recent evidence on circadian clock regulation of reproduction, providing novel perspectives on the role of the circadian clock in maintaining normal reproductive functions and in diseases that negatively affect fertility.

Understanding the mechanisms of HPV-related carcinogenesis: Implications for cell cycle dynamics

The role of human papillomavirus (HPV) as a causative agent for epithelial cancers is well-known, but many open questions remain regarding the downstream gene regulatory effects of viral proteins E6 and E7 on the cell cycle. Here, we extend a cell cycle model originally presented by Gérard and Goldbeter (2009) in order to capture the effects of E6 and E7 on key actors in the cell cycle. Results suggest that E6 is sufficient to reverse p53-induced quiescence, while E7 is sufficient to reverse p16INK4a-induced quiescence; both E6 and E7 are necessary when p53 and p16INK4a are both active. Moreover, E7 appears to play a role as a "growth factor substitute", inducing cell division in the absence of growth factor. Low levels of E7 may permit regular cell division, but the results suggest that higher levels of E7 dysregulate the cell cycle in ways that may destabilize the cellular genome. The mechanisms explored here provide opportunities for developing new treatment targets that take advantage of the cell cycle regulatory system to prevent HPV-related cancer effects.

Machine learning alternative to systems biology should not solely depend on data

In recent years, artificial intelligence (AI)/machine learning has emerged as a plausible alternative to systems biology for the elucidation of biological phenomena and in attaining specified design objective in synthetic biology. Although considered highly disruptive with numerous notable successes so far, we seek to bring attention to both the fundamental and practical pitfalls of their usage, especially in illuminating emergent behaviors from chaotic or stochastic systems in biology. Without deliberating on their suitability and the required data qualities and pre-processing approaches beforehand, the research and development community could experience similar 'AI winters' that had plagued other fields. Instead, we anticipate the integration or combination of the two approaches, where appropriate, moving forward.

Cell size homeostasis under the circadian regulation of cell division in cyanobacteria

Bacterial cells maintain their characteristic cell size over many generations. Several rod-shaped bacteria, such as Escherichia coli and the cyanobacteria Synechococcus elongatus, divide after adding a constant length to their length at birth. Through this division control known as the adder mechanism, perturbation in cell length due to physiological fluctuation decays over generations at a rate of 2^-1 per cell division. However, previous experiments have shown that the circadian clock in cyanobacteria reduces cell division frequency at a specific time of day under constant light. This circadian gating should modulate the division control by the adder mechanism, but its significance remains unknown. Here we address how the circadian gating affects cell length, doubling time, and cell length stability in cyanobacteria by using mathematical models. We show that a cell subject to circadian gating grows for a long time, and gives birth to elongated daughter cells. These elongated daughter cells grow faster than the previous generation, as elongation speed is proportional to cell length and divide in a short time before the next gating. Hence, the distributions of doubling time and cell length become bimodal, as observed in experimental data. Interestingly, the average doubling time over the population of cells is independent of gating because the extension of doubling time by gating is compensated by its reduction in the subsequent generation. On the other hand, average cell length is increased by gating, suggesting that the circadian clock controls cell length. We then show that the decay rate of perturbation in cell length depends on the ratio of delay in division by the gating τ_G to the average doubling time τ₀ as [Formula: see text] . We estimated τ_G≈2.5, τ₀≈13.6 hours, and τ_G/τ₀≈0.18 from experimental data, indicating that a long doubling time in cyanobacteria maintains the decay rate similar to that of the adder mechanism. Thus, our analysis suggests that the acquisition of the circadian clock during evolution did not impose a constraint on cell size homeostasis in cyanobacteria.

Bilateral Feedback in Oscillator Model Is Required to Explain the Coupling Dynamics of Hes1 with the Cell Cycle

Biological processes are governed by the expression of proteins, and for some proteins, their level of expression can fluctuate periodically over time (i.e., they oscillate). Many oscillatory proteins (e.g., cell cycle proteins and those from the HES family of transcription factors) are connected in complex ways, often within large networks. This complexity can be elucidated by developing intuitive mathematical models that describe the underlying critical aspects of the relationships between these processes. Here, we provide a mathematical explanation of a recently discovered biological phenomenon: the phasic position of the gene Hes1’s oscillatory expression at the beginning of the cell cycle of an individual human breast cancer stem cell can have a predictive value on how long that cell will take to complete a cell cycle. We use a two-component model of coupled oscillators to represent Hes1 and the cell cycle in the same cell with minimal assumptions. Inputting only the initial phase angles, we show that this model is capable of predicting the dynamic mitosis to mitosis behaviour of Hes1 and predicting cell cycle length patterns as found in real-world experimental data. Moreover, we discover that bidirectional coupling between Hes1 and the cell cycle is critical within the system for the data to be reproduced and that nonfixed asymmetry in the interactions between the oscillators is required. The phase dynamics we present here capture the complex interplay between Hes1 and the cell cycle, helping to explain nongenetic cell cycle variability, which has critical implications in cancer treatment contexts.

A yeast cell cycle model integrating stress, signaling, and physiology

The cell division cycle in eukaryotic cells is a series of highly coordinated molecular interactions that ensure that cell growth, duplication of genetic material, and actual cell division are precisely orchestrated to give rise to two viable progeny cells. Moreover, the cell cycle machinery is responsible for incorporating information about external cues or internal processes that the cell must keep track of to ensure a coordinated, timely progression of all related processes. This is most pronounced in multicellular organisms, but also a cardinal feature in model organisms such as baker's yeast. The complex and integrative behavior is difficult to grasp and requires mathematical modeling to fully understand the quantitative interplay of the single components within the entire system. Here, we present a self-oscillating mathematical model of the yeast cell cycle that comprises all major cyclins and their main regulators. Furthermore, it accounts for the regulation of the cell cycle machinery by a series of external stimuli such as mating pheromones and changes in osmotic pressure or nutrient quality. We demonstrate how the external perturbations modify the dynamics of cell cycle components and how the cell cycle resumes after adaptation to or relief from stress.

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’.

Time-keeping and decision-making in living cells: Part I

To survive and reproduce, a cell must process information from its environment and its own internal state and respond accordingly, in terms of metabolic activity, gene expression, movement, growth, division and differentiation. These signal–response decisions are made by complex networks of interacting genes and proteins, which function as biochemical switches and clocks, and other recognizable information-processing circuitry. This theme issue of Interface Focus (in two parts) brings together articles on time-keeping and decision-making in living cells—work that uses precise mathematical modelling of underlying molecular regulatory networks to understand important features of cell physiology. Part I focuses on time-keeping: mechanisms and dynamics of biological oscillators and modes of synchronization and entrainment of oscillators, with special attention to circadian clocks.

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.

Circadian and Immunity Cycle Talk in Cancer Destination: From Biological Aspects to In Silico Analysis

Simple Summary

The circadian cycle is a natural cycle of the body repeated every 24 h, based on a day and night rhythm, and it affects many body processes. The present article reviews the importance and role of the circadian cycle in cancer and its association with the immune system and immunotherapy drugs at the cellular and molecular levels. It also examines the genes and cellular pathways involved in both circadian and immune systems. It offers possible computational solutions to increase the effectiveness of cancer treatment concerning the circadian cycle.

Abstract

Cancer is the leading cause of death and a major problem to increasing life expectancy worldwide. In recent years, various approaches such as surgery, chemotherapy, radiation, targeted therapies, and the newest pillar, immunotherapy, have been developed to treat cancer. Among key factors impacting the effectiveness of treatment, the administration of drugs based on the circadian rhythm in a person and within individuals can significantly elevate drug efficacy, reduce adverse effects, and prevent drug resistance. Circadian clocks also affect various physiological processes such as the sleep cycle, body temperature cycle, digestive and cardiovascular processes, and endocrine and immune systems. In recent years, to achieve precision patterns for drug administration using computational methods, the interaction of the effects of drugs and their cellular pathways has been considered more seriously. Integrated data-derived pathological images and genomics, transcriptomics, and proteomics analyses have provided an understanding of the molecular basis of cancer and dramatically revealed interactions between circadian and immunity cycles. Here, we describe crosstalk between the circadian cycle signaling pathway and immunity cycle in cancer and discuss how tumor microenvironment affects the influence on treatment process based on individuals’ genetic differences. Moreover, we highlight recent advances in computational modeling that pave the way for personalized immune chronotherapy.

Circadian clocks and their role in lactation competence

Circadian rhythms are 24 h cycles of behavior, physiology and gene expression that function to synchronize processes across the body and coordinate physiology with the external environment. Circadian clocks are central to maintaining homeostasis and regulating coordinated changes in physiology in response to internal and external cues. Orchestrated changes occur in maternal physiology during the periparturient period to support the growth of the fetus and the energetic and nutritional demands of lactation. Discoveries in our lab made over a decade ago led us to hypothesize that the circadian timing system functions to regulate metabolic and mammary specific changes that occur to support a successful lactation. Findings of studies that ensued are summarized, and point to the importance of circadian clocks in the regulation of lactation competence. Disruption of the circadian timing system can negatively affect mammary gland development and differentiation, alter maternal metabolism and impair milk production.

The Expression and Function of Circadian Rhythm Genes in Hepatocellular Carcinoma

Hepatocellular carcinoma (HCC) is among the most common and lethal form of cancer worldwide. However, its diagnosis and treatment are still dissatisfactory, due to limitations in the understanding of its pathogenic mechanism. Therefore, it is important to elucidate the molecular mechanisms and identify novel therapeutic targets for HCC. Circadian rhythm-related genes control a variety of biological processes. These genes play pivotal roles in the initiation and progression of HCC and are potential diagnostic markers and therapeutic targets. This review gives an update on the research progress of circadian rhythms, their effects on the initiation, progression, and prognosis of HCC, in a bid to provide new insights for the research and treatment of HCC.

Improving cancer treatments via dynamical biophysical models

Despite significant advances in oncological research, cancer nowadays remains one of the main causes of mortality and morbidity worldwide. New treatment techniques, as a rule, have limited efficacy, target only a narrow range of oncological diseases, and have limited availability to the general public due their high cost. An important goal in oncology is thus the modification of the types of antitumor therapy and their combinations, that are already introduced into clinical practice, with the goal of increasing the overall treatment efficacy. One option to achieve this goal is optimization of the schedules of drugs administration or performing other medical actions. Several factors complicate such tasks: the adverse effects of treatments on healthy cell populations, which must be kept tolerable; the emergence of drug resistance due to the intrinsic plasticity of heterogeneous cancer cell populations; the interplay between different types of therapies administered simultaneously. Mathematical modeling, in which a tumor and its microenvironment are considered as a single complex system, can address this complexity and can indicate potentially effective protocols, that would require experimental verification. In this review, we consider classical methods, current trends and future prospects in the field of mathematical modeling of tumor growth and treatment. In particular, methods of treatment optimization are discussed with several examples of specific problems related to different types of treatment.

Core circadian clock transcription factor BMAL1 regulates mammary epithelial cell growth, differentiation, and milk component synthesis

The role the mammary epithelial circadian clock plays in gland development and lactation is unknown. We hypothesized that mammary epithelial clocks function to regulate mammogenesis and lactogenesis, and propose the core clock transcription factor BMAL1:CLOCK regulates genes that control mammary epithelial development and milk synthesis. Our objective was to identify transcriptional targets of BMAL1 in undifferentiated (UNDIFF) and lactogen differentiated (DIFF) mammary epithelial cells (HC11) using ChIP-seq. Ensembl gene IDs with the nearest transcriptional start site to ChIP-seq peaks were explored as potential targets, and represented 846 protein coding genes common to UNDIFF and DIFF cells and 2773 unique to DIFF samples. Genes with overlapping peaks between samples (1343) enriched cell-cell adhesion, membrane transporters and lipid metabolism categories. To functionally verify targets, an HC11 line with Bmal1 gene knocked out (BMAL1-KO) using CRISPR-CAS was created. BMAL1-KO cultures had lower cell densities over an eight-day growth curve, which was associated with increased (p<0.05) levels of reactive oxygen species and lower expression of superoxide dismutase 3 (Sod3). RT-qPCR analysis also found lower expression of the putative targets, prolactin receptor (Prlr), Ppara, and beta-casein (Csn2). Findings support our hypothesis and highlight potential importance of clock in mammary development and substrate transport.

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.

Circadian disruption in lung cancer

Despite major developments in lung cancer investigations and the progress of innovative oncology treatments in recent decades, lung cancer continues to be the predominant cause of cancer-related mortality globally, with over a million deaths each year. This highlights the urgent need to develop a deeper understanding of the current state of cancer care. At the environmental and cellular levels, circadian rhythms are closely associated with living organisms. In humans, the suprachiasmatic nucleus is the principal circadian pacemaker. Circadian gene feedback loops regulate the clock, connecting peripheral tissue metabolism, cell proliferation, DNA repair, and cell death to energy homeostasis, physical activity, and neurohormonal regulation at the organismal level. Endogenous circadian homeostasis has been frequently disturbed in modern civilizations, resulting in a higher risk of many disorders, including lung cancer. Despite major developments in lung cancer investigations and the progress of innovative oncology treatments in recent decades, lung cancer continues to be the predominant cause of cancer-related mortality globally, with over a million deaths each year. This highlights the urgent need to develop a deeper understanding of the current state of cancer care. At the environmental and cellular levels, circadian rhythms are closely associated with living organisms. In humans, the suprachiasmatic nucleus is the principal circadian pacemaker. Circadian gene feedback loops regulate the clock, connecting peripheral tissue metabolism, cell proliferation, DNA repair, and cell death to energy homeostasis, physical activity, and neurohormonal regulation at the organismal level. Endogenous circadian homeostasis has been frequently disturbed in modern civilizations, resulting in a higher risk of many disorders, including lung cancer. The mammalian circadian clock controls metabolism and cell division, and disruption of these processes may lead to cancer pathogenesis. Furthermore, circadian disturbance has recently been identified as a self-regulating cancer risk factor and is listed as a carcinogen. The theory that both somatic and systemic disturbances of circadian rhythms are related to a higher risk of lung cancer development and poor prognosis is addressed in this study. The chronotherapy principles hold much more promise for enhancing the lung cancer care options currently available. Developing a better understanding of the molecular interactions that control the physiological equilibrium between both the circadian rhythm and the cycle of cell division could significantly influence the development of novel treatments for lung cancer and other diseases.

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

The fundamental mechanisms that control and regulate biological organisms exhibit a surprising level of complexity. Oscillators are perhaps the simplest motifs that produce time-varying dynamics and are ubiquitous in biological systems. It is also known that such biological oscillators interact with each other-for instance, circadian oscillators affect the cell cycle, and somitogenesis clock proteins in adjacent cells affect each other in developing embryos. Therefore, it is vital to understand the effects that can emerge from non-linear interaction between oscillations. Here, we show how oscillations typically arise in biology and take the reader on a tour through the great variety in dynamics that can emerge even from a single pair of coupled oscillators. We explain how chaotic dynamics can emerge and outline the methods of detecting this in experimental time traces. Finally, we discuss the potential role of such complex dynamical features in biological systems.

A design principle for posttranslational chaotic oscillators

Chaos behavior has been observed in various cellular and molecular processes. Here, we modeled reversible phosphorylation dynamics to elucidate a design principle for autonomous chaos generation that may arise from generic enzymatic reactions. A comprehensive parameter search demonstrated that the reaction system composed of a set of kinases and phosphatases and two substrates with two modification sites exhibits chaos behavior. All reactions are described according to the Michaelis-Menten reaction scheme without exotic functions being applied to enzymes and substrates. Clustering analysis of parameter sets that can generate chaos behavior revealed the existence of motif structures. These chaos motifs allow the two-substrate species to interact via enzyme availability and constrain the two substrates' dynamic changes in phosphorylation status so that they occur at different timescales. This chaos motif structure is found in several enzymatic reactions, suggesting that chaos behavior may underlie cellular autonomy in a variety of biochemical systems.

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Two substrates with reversible two-site phosphorylation can exhibit chaos behavior

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The chaos does not require autocatalysis or allosteric regulation of enzymes

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The chaos is a result of the coupling of two substrates via enzyme availability

Trajectory-based energy landscapes of gene regulatory networks

Multistability and natural biological variability can result in significant heterogeneity within a cell population, leading to challenges in understanding and modulating cell behavior. Energy landscapes can offer qualitatively intuitive visualizations of cell phenotype and facilitate a more quantitative understanding of cellular dynamics, but current methods for landscape generation are mathematically involved and often require specific system properties (e.g., ergodicity or independent gene/protein probability distributions) that do not always hold. Here, we present a simple kinetic Monte Carlo-based method for landscape generation from a system of ordinary differential equations using only simulation trajectories initialized throughout the phase space of interest. The resulting landscape produces three quantitative features relevant to understanding cell behavior: stability (reflected by the depth or potential of landscape valleys), velocity (representing average directional movement on the landscape), and variance in velocity (indicative of landscape positions with heterogeneous movements). We applied this method to a genetic toggle switch, a core decision-making network in binary cellular responses, to elucidate effects of biologically relevant intrinsic and extrinsic cues. Intrinsic noise, such as stochasticity in transcription-translation and differences in cell cycle position, manifests through changes in valley width and position, reflecting increased population heterogeneity and more probabilistic cell fate transitions. The landscapes also capture the effect of an external inducer, revealing a quantitative correlation between the rate of cell fate transition and the energy barrier above a threshold inducer concentration determined by the permissivity of the valley. Further, in tracking dynamically changing landscapes under time-varying external cues, we unexpectedly found that an oscillatory inducer input can modulate cell fate heterogeneity and lead to periodic cell fate transitions entrained to the input frequency, depending on the intrinsic degradation rate of the switch. The landscape generation approach outlined herein is generalizable to other network topologies and may provide new quantitative insights into their dynamics.

The Role of Omega-3 Polyunsaturated Fatty Acids from Different Sources in Bone Development

N-3 polyunsaturated fatty acids (PUFAs) are essential nutrients that must be obtained from the diet. We have previously showed that endogenous n-3 PUFAs contribute to skeletal development and bone quality in fat-1 mice. Unlike other mammals, these transgenic mice, carry the n-3 desaturase gene and thus can convert n-6 to n-3 PUFAs endogenously. Since this model does not mimic dietary exposure to n-3 PUFAs, diets rich in fish and flaxseed oils were used to further elucidate the role of n-3 PUFAs in bone development. Our investigation reveals that dietary n-3 PUFAs decrease fat accumulation in the liver, lower serum fat levels, and alter fatty acid (FA) content in liver and serum. Bone analyses show that n-3 PUFAs improve mechanical properties, which were measured using a three-point bending test, but exert complex effects on bone structure that vary according to its source. In a micro-CT analysis, we found that the flaxseed oil diet improves trabecular bone micro-architecture, whereas the fish oil diet promotes higher bone mineral density (BMD) with no effect on trabecular bone. The transcriptome characterization of bone by RNA-seq identified regulatory mechanisms of n-3 PUFAs via modulation of the cell cycle and peripheral circadian rhythm genes. These results extend our knowledge and provide insights into the molecular mechanisms of bone remodeling regulation induced by different sources of dietary n-3 PUFAs.

The Lineage Before Time: Circadian and Nonclassical Clock Influences on Development

Diverse factors including metabolism, chromatin remodeling, and mitotic kinetics influence development at the cellular level. These factors are well known to interact with the circadian transcriptional-translational feedback loop (TTFL) after its emergence. What is only recently becoming clear, however, is how metabolism, mitosis, and epigenetics may become organized in a coordinated cyclical precursor signaling module in pluripotent cells prior to the onset of TTFL cycling. We propose that both the precursor module and the TTFL module constrain cellular identity when they are active during development, and that the emergence of these modules themselves is a key lineage marker. Here we review the component pathways underlying these ideas; how proliferation, specification, and differentiation decisions in both developmental and adult stem cell populations are or are not regulated by the classical TTFL; and emerging evidence that we propose implies a primordial clock that precedes the classical TTFL and influences early developmental decisions.

Circadian clock effects on cellular proliferation: Insights from theory and experiments

Oscillations of the cellular circadian clock have emerged as an important regulator of many physiological processes, both in health and in disease. One such process, cellular proliferation, is being increasingly recognized to be affected by the circadian clock. Here, we review how a combination of experimental and theoretical work has furthered our understanding of the way circadian clocks couple to the cell cycle and play a role in tissue homeostasis and cancer. Finally, we discuss recently introduced methods for modeling coupling of clocks based on techniques from survival analysis and machine learning and highlight their potential importance for future studies.

Spotlight on Circadian Genes and Colorectal Cancer Crosstalk

Mammalian physiology is regulated by circadian clock through oscillating feedback loops controlling cellular processes and behaviors. Recent findings have led to an interesting connection between circadian disruption and colorectal cancer progression and incidence through controlling the hallmarks of cancer, namely cell cycle, cell metabolism and cell death. Deeper understanding of the circadian mechanisms that define the colorectal cancer pathophysiology is the need of the hour to define a chronotherapy for improving colorectal cancer patient survival. This review identifies the key areas in which circadian genes interact with cellular pathways to modify the outcome with respect to colorectal cancer incidence and progression.

Cell cycle period control through modulation of clock inputs

In this work, we study period control of the mammalian cell cycle via coupling with the cellular clock. For this, we make use of the oscillators’ synchronization dynamics and investigate methods of slowing down the cell cycle with the use of clock inputs. Clock control of the cell cycle is well established via identified molecular mechanisms, such as the CLOCK:BMAL1-mediated induction of the wee1 gene, resulting in the WEE1 kinase that represses the active form of mitosis promoting factor (MPF), the essential cell cycle component. To investigate the coupling dynamics of these systems, we use previously developed models of the clock and cell cycle oscillators and center our studies on unidirectional clock [Formula: see text] cell cycle coupling. Moreover, we propose an hypothesis of a Growth Factor (GF)-responsive clock, involving a pathway of the non-essential cell cycle complex cyclin D/CDK4. We observe a variety of rational ratios of clock to cell cycle period, such as: 1:1, 3:2, 4:3, and 5:4. Finally, our protocols of period control are successful in effectively slowing down the cell cycle by the use of clock modulating inputs, some of which correspond to existing drugs.

Nonequilibrium Thermodynamics in Cell Biology: Extending Equilibrium Formalism to Cover Living Systems

We discuss new developments in the nonequilibrium dynamics and thermodynamics of living systems, giving a few examples to demonstrate the importance of nonequilibrium thermodynamics for understanding biological dynamics and functions. We study single-molecule enzyme dynamics, in which the nonequilibrium thermodynamic and dynamic driving forces of chemical potential and flux are crucial for the emergence of non-Michaelis-Menten kinetics. We explore single-gene expression dynamics, in which nonequilibrium dissipation can suppress fluctuations. We investigate the cell cycle and identify the nutrition supply as the energy input that sustains the stability, speed, and coherence of cell cycle oscillation, from which the different vital phases of the cell cycle emerge. We examine neural decision-making processes and find the trade-offs among speed, accuracy, and thermodynamic costs that are important for neural function. Lastly, we consider the thermodynamic cost for specificity in cellular signaling and adaptation.

Analytical distributions for detailed models of stochastic gene expression in eukaryotic cells

The stochasticity of gene expression presents significant challenges to the modeling of genetic networks. A two-state model describing promoter switching, transcription, and messenger RNA (mRNA) decay is the standard model of stochastic mRNA dynamics in eukaryotic cells. Here, we extend this model to include mRNA maturation, cell division, gene replication, dosage compensation, and growth-dependent transcription. We derive expressions for the time-dependent distributions of nascent mRNA and mature mRNA numbers, provided two assumptions hold: 1) nascent mRNA dynamics are much faster than those of mature mRNA; and 2) gene-inactivation events occur far more frequently than gene-activation events. We confirm that thousands of eukaryotic genes satisfy these assumptions by using data from yeast, mouse, and human cells. We use the expressions to perform a sensitivity analysis of the coefficient of variation of mRNA fluctuations averaged over the cell cycle, for a large number of genes in mouse embryonic stem cells, identifying degradation and gene-activation rates as the most sensitive parameters. Furthermore, it is shown that, despite the model's complexity, the time-dependent distributions predicted by our model are generally well approximated by the negative binomial distribution. Finally, we extend our model to include translation, protein decay, and auto-regulatory feedback, and derive expressions for the approximate time-dependent protein-number distributions, assuming slow protein decay. Our expressions enable us to study how complex biological processes contribute to the fluctuations of gene products in eukaryotic cells, as well as allowing a detailed quantitative comparison with experimental data via maximum-likelihood methods.

Control of synchronization ratios in clock/cell cycle coupling by growth factors and glucocorticoids

The cell cycle and the circadian clock are essential cyclic cellular processes often synchronous in healthy cells. In this work, we use previously developed mathematical models of the mammalian cell cycle and circadian cellular clock in order to investigate their dynamical interactions. Firstly, we study unidirectional cell cycle → clock coupling by proposing a mechanism of mitosis promoting factor (MPF)-controlled REV-ERBα degradation. Secondly, we analyse a bidirectional coupling configuration, where we add the CLOCK : BMAL1-mediated MPF repression via the WEE1 kinase to the first system. Our simulations reproduce ratios of clock to cell cycle period in agreement with experimental observations and give predictions of the system's synchronization state response to a variety of control parameters. Specifically, growth factors accelerate the coupled oscillators and dexamethasone (Dex) drives the system from a 1 : 1 to a 3 : 2 synchronization state. Furthermore, simulations of a Dex pulse reveal that certain time regions of pulse application drive the system from 1 : 1 to 3 : 2 synchronization while others have no effect, revealing the existence of a responsive and an irresponsive system's phase, a result we contextualize with observations on the segregation of Dex-treated cells into two populations.

Low-dimensional Dynamics of Two Coupled Biological Oscillators

The circadian clock and the cell cycle are two biological oscillatory processes that coexist within individual cells. These two oscillators were found to interact, which can lead to their synchronization. Here, we develop a method to identify a low-dimensional stochastic model of the coupled system directly from time-lapse imaging in single cells. In particular, we infer the coupling and non-linear dynamics of the two oscillators from thousands of mouse and human single-cell fluorescence microscopy traces. This coupling predicts multiple phase-locked states showing different degrees of robustness against molecular fluctuations inherent to cellular-scale biological oscillators. For the 1:1 state, the predicted phase-shifts upon period perturbations were validated experimentally. Moreover, the phase-locked states are temperature-independent and evolutionarily conserved from mouse to human, hinting at a common underlying dynamical mechanism. Finally, we detect a signature of the coupled dynamics in a physiological context, explaining why tissues with different proliferation states exhibited shifted circadian clock phases.

Transcription-based circadian mechanism controls the duration of molecular clock states in response to signaling inputs

The molecular oscillator of the mammalian circadian clock consists in a dynamical network of genes and proteins whose main regulatory mechanisms occur at the transcriptional level. From a dynamical point of view, the mechanisms leading to an oscillatory solution with an orderly protein peak expression and a clear day/night phase distinction remain unclear. Our goal is to identify the essential interactions needed to generate phase opposition between the activating CLOCK:BMAL1 and the repressing PER:CRY complexes and to better distinguish these two main clock molecular phases relating to rest/activity and fast/feeding cycles. To do this, we develop a transcription-based mathematical model centered on linear combinations of the clock controlled elements (CCEs): E-box, R-box and D-box. Each CCE is responsive to activators and repressors. After model calibration with single-cell data, we explore entrainment and period tuning via interplay with metabolism. Variation of the PER degradation rate γ_p, relating to the tau mutation, results in asymmetric changes in the duration of the different clock molecular phases. Time spent at the state of high PER/PER:CRY decreases with γ_p, while time spent at the state of high BMAL1 and CRY1, both proteins with activity in promoting insulin sensitivity, remains constant. This result suggests a possible mechanism behind the altered metabolism of tau mutation animals. Furthermore, we expose the clock system to two regulatory inputs, one relating to the fast/feeding cycle and the other to the light-dependent synchronization signaling. We observe the phase difference between these signals to also affect the relative duration of molecular clock states. Simulated circadian misalignment, known to correlate with insulin resistance, leads to decreased duration of BMAL1 expression. Our results reveal a possible mechanism for clock-controlled metabolic homeostasis, whereby the circadian clock controls the relative duration of different molecular (and metabolic) states in response to signaling inputs.

Robust synchronization of the cell cycle and the circadian clock through bidirectional coupling

The cell cycle and the circadian clock represent major cellular rhythms, which appear to be coupled. Thus the circadian factor BMAL1 controls the level of cell cycle proteins such as Cyclin E and WEE1, the latter of which inhibits the kinase CDK1 that governs the G2/M transition. In reverse the cell cycle impinges on the circadian clock through direct control by CDK1 of REV-ERBα, which negatively regulates BMAL1. These observations provide evidence for bidirectional coupling of the cell cycle and the circadian clock. By merging detailed models for the two networks in mammalian cells, we previously showed that unidirectional coupling to the circadian clock can entrain the cell cycle to 24 or 48 h, depending on the cell cycle autonomous period, while complex oscillations occur when entrainment fails. Here we show that the reverse unidirectional coupling via phosphorylation of REV-ERBα or via mitotic inhibition of transcription, both controlled by CDK1, can elicit entrainment of the circadian clock by the cell cycle. We then determine the effect of bidirectional coupling of the cell cycle and circadian clock as a function of their relative coupling strengths. In contrast to unidirectional coupling, bidirectional coupling markedly reduces the likelihood of complex oscillations. While the two rhythms oscillate independently as long as both couplings are weak, one rhythm entrains the other if one of the couplings dominates. If the couplings in both directions become stronger and of comparable magnitude, the two rhythms synchronize, generally at an intermediate period within the range defined by the two autonomous periods prior to coupling. More surprisingly, synchronization may also occur at a period slightly below or above this range, while in some conditions the synchronization period can even be much longer. Two or even three modes of synchronization may sometimes coexist, yielding examples of birhythmicity or trirhythmicity. Because synchronization readily occurs in the form of simple periodic oscillations over a wide range of coupling strengths and in the presence of multiple connections between the two oscillatory networks, the results indicate that bidirectional coupling favours the robust synchronization of the cell cycle and the circadian clock.

Allostatic adaptation and personalized physiological trade-offs in the circadian regulation of the HPA axis: A mathematical modeling approach

The hypothalamic-pituitary-adrenal (HPA) axis orchestrates the physiological response to unpredictable acute stressors. Moreover, the HPA axis exhibits prominent circadian activity and synchronizes peripheral circadian clocks to daily environmental cycles, thereby promoting homeostasis. Persistent disruption of homeostatic glucocorticoid circadian rhythmicity due to chronic stress exposure is correlated with the incidence of various pathological conditions including depression, diabetes and cancer. Allostatic habituation of the HPA axis, such that glucocorticoid levels retain homeostatic levels upon chronic exposure to stress, can therefore confer fitness advantages by preventing the sustained dysregulation of glucocorticoid-responsive signaling pathways. However, such allostatic adaptation results in a physiological cost (allostatic load) that might impair the homeostatic stress-responsive and synchronizing functions of the HPA axis. We use mathematical modeling to characterize specific chronic stress-induced allostatic adaptations in the HPA network. We predict the existence of multiple individualized regulatory strategies enabling the maintenance of homeostatic glucocorticoid rhythms, while allowing for flexible HPA response characteristics. We show that this regulatory variability produces a trade-off between the stress-responsive and time-keeping properties of the HPA axis. Finally, allostatic regulatory adaptations are predicted to cause a time-of-day dependent sensitization of the acute stress response and impair the entrainability of the HPA axis.

Cancer and the Circadian Clock

The circadian clock is a master regulator of mammalian physiology, regulating daily oscillations of crucial biological processes and behaviors. Notably, circadian disruption has recently been identified as an independent risk factor for cancer and classified as a carcinogen. As such, it is imperative to discern the underpinning mechanisms by which circadian disruption alters cancer risk. Emergent data, reviewed herein, demonstrate that circadian regulatory functions play critical roles in several hallmarks of cancer, including control of cell proliferation, cell death, DNA repair, and metabolic alteration. Developing a deeper understanding of circadian-cancer regulation cross-talk holds promise for developing new strategies for cancer interception, prevention, and management.

Exploring the underlying mechanisms of the coupling between cell differentiation and cell cycle

Differentiation and replication are the two major fates of the cells. They are the fundamental processes for completing the cellular functions. Although the underlying biological processes have been considerably explored for each of these processes and significant progresses have been made, global quantification and physical understanding are still challenging, especially for the relationship among them. In this study, we developed a theoretical framework for both the cell cycle and cell differentiation by exploring the associated global dynamics and their underlying relationship. We found that the dynamics of the cell cycle and cell differentiation is governed by both the landscape gradient and rotational curl flux. While landscape attracts the system down to the stable state basins, the curl flux drives the stable oscillation flow. We uncovered the irregular sombrero-shaped landscapes of the cell cycle at different developmental stages. We studied how the cells develop from undifferentiated cells to differentiated cells and how the cell cycle proceeds at different developmental stages. We investigated how the cell differentiation can influence the cell cycle where more progressive differentiation can lead to the changes of the cell cycle oscillations. In contrast, we can also quantitatively illustrate how the cell cycle can influence the cell differentiation where cell cycle regulation can lead to the changes of the differentiation processes. Through the landscape and flux analysis, we uncovered the key regulatory elements controlling the progression of the cell differentiation and cell cycle. This can help to design an effective strategy for drug discovery against associated diseases.

Research on circadian clock genes in non-small-cell lung carcinoma

Circadian clock genes have become a hot topic in cancer research in recent years, and more and more studies are showing that clock genes are involved in regulating cell proliferation cycle and apoptosis of malignant tumors, neuroendocrine and immune function, and other processes. Lung cancer is a malignant tumor with increasing incidence worldwide. The pathogenesis of lung cancer is extremely complicated and includes genetic factors, living environment, and smoking, and the occurrence of lung cancer is related to the regulation of many oncogenes and tumor suppressor genes. But there are few studies on clock genes in lung cancer. Studies on clock genes may help to better understand the mechanism of lung cancer development for an improved treatment. The expressions of all 14 kinds of clock genes in adenocarcinoma (ADC) and squamous cell carcinoma (SCC), two main kinds of non-small-cell lung cancer (NSCLC), were studied based on integration and analysis of data from The Cancer Genome Atlas (TCGA) to show the association between clock gene expression and prognosis of cancer patients. Analysis of TCGA data indicated that overexpression of Cry2, BMAL1, and RORA with underexpression of Timeless and NPAS2 was associated with a favorable prognosis of ADC, and the expression of NPAS2 was associated with the time of patient survival. Additionally, the expression of Cry2 was related to TNM stage. In SCC, high expression of DEC1 was correlated with poor overall survival in patients and the expression of Timeless was associated with the time of patient survival. In NSCLC, circadian clock genes constitute cancer circadian rhythm by interacting with each other, showing that asynchrony with normal tissues, which collectively controlling the occurrence and development of NSCLC.

Multi-rhythmicity generated by coupling two cellular rhythms

The cell cycle and the circadian clock represent two major cellular rhythms, which are coupled because the circadian clock governs the synthesis of several proteins of the network that drives the mammalian cell cycle. Analysis of a detailed model for these coupled cellular rhythms previously showed that the cell cycle can be entrained at the circadian period of 24 h, or at a period of 48 h, depending on the autonomous period of the cell cycle and on the coupling strength. We show by means of numerical simulations that multiple stable periodic regimes, i.e. multi-rhythmicity, may originate from the coupling of the two cellular rhythms. In these conditions, the cell cycle can evolve to any one of two (birhythmicity) or three stable periodic regimes (trirhythmicity). When applied at the right phase, transient perturbations of appropriate duration and magnitude can induce the switch between the different oscillatory states. Such switching is characterized by final state sensitivity, which originates from the complex structure of the attraction basins. By providing a novel instance of multi-rhythmicity in a realistic model for the coupling of two major cellular rhythms, the results throw light on the conditions in which multiple stable periodic regimes may coexist in biological systems.

The positive circadian regulators CLOCK and BMAL1 control G2/M cell cycle transition through Cyclin B1

We previously identified a tight bidirectional phase coupling between the circadian clock and the cell cycle. To understand the role of the CLOCK/BMAL1 complex, representing the main positive regulator of the circadian oscillator, we knocked down Bmal1 or Clock in NIH3T33C mouse fibroblasts (carrying fluorescent reporters for clock and cell cycle phase) and analyzed timing of cell division in individual cells and cell populations. Inactivation of Bmal1 resulted in a loss of circadian rhythmicity and a lengthening of the cell cycle, originating from delayed G2/M transition. Subsequent molecular analysis revealed reduced levels of Cyclin B1, an important G2/M regulator, upon suppression of Bmal1 gene expression. In complete agreement with these experimental observations, simulation of Bmal1 knockdown in a computational model for coupled mammalian circadian clock and cell cycle oscillators (now incorporating Cyclin B1 induction by BMAL1) revealed a lengthening of the cell cycle. Similar data were obtained upon knockdown of Clock gene expression. In conclusion, the CLOCK/BMAL1 complex controls cell cycle progression at the level of G2/M transition through regulation of Cyclin B1 expression.

Deciphering the Dynamics of Interlocked Feedback Loops in a Model of the Mammalian Circadian Clock

Mathematical models of fundamental biological processes play an important role in consolidating theory and experiments, especially if they are systematically developed, thoroughly characterized, and well tested by experimental data. In this work, we report a detailed bifurcation analysis of a mathematical model of the mammalian circadian clock network developed by Relógio et al., noteworthy for its consistency with available data. Using one- and two-parameter bifurcation diagrams, we explore how oscillations in the model depend on the expression levels of its constituent genes and the activities of their encoded proteins. These bifurcation diagrams allow us to decipher the dynamics of interlocked feedback loops by parametric variation of genes and proteins in the model. Among other results, we find that REV-ERB, a member of a subfamily of orphan nuclear receptors, plays a critical role in the intertwined dynamics of Relógio's model. The bifurcation diagrams reported here can be used for predicting how the core clock network responds-in terms of period, amplitude and phases of oscillations-to different perturbations.

Circadian Rhythm of Glucocorticoid Administration Entrains Clock Genes in Immune Cells: A DREAM Trial Ancillary Study

CONTEXT

Adrenal insufficiency (AI) requires lifelong glucocorticoid (GC) replacement. Conventional therapies do not mimic the endogenous cortisol circadian rhythm. Clock genes are essential components of the machinery controlling circadian functions and are influenced by GCs. However, clock gene expression has never been investigated in patients with AI.

OBJECTIVE

To evaluate the effect of the timing of GC administration on circadian gene expression in peripheral blood mononuclear cells (PBMCs) of patients from the Dual Release Hydrocortisone vs Conventional Glucocorticoid Replacement in Hypocortisolism (DREAM) trial.

DESIGN

Outcome assessor-blinded, randomized, active comparator clinical trial.

PARTICIPANTS AND INTERVENTION

Eighty-nine patients with AI were randomly assigned to continue their multiple daily GC doses or switch to an equivalent dose of once-daily modified-release hydrocortisone and were compared with 25 healthy controls; 65 patients with AI and 18 controls consented to gene expression analysis.

RESULTS

Compared with healthy controls, 19 of the 68 genes were found modulated in patients with AI at baseline, 18 of which were restored to control levels 12 weeks after therapy was switched: ARNTL [BMAL] (P = 0.024), CLOCK (P = 0.016), AANAT (P = 0.021), CREB1 (P = 0.010), CREB3 (P = 0.037), MAT2A (P = 0.013); PRKAR1A, PRKAR2A, and PRKCB (all P < 0.010) and PER3, TIMELESS, CAMK2D, MAPK1, SP1, WEE1, CSNK1A1, ONP3, and PRF1 (all P < 0.001). Changes in WEE1, PRF1, and PER3 expression correlated with glycated hemoglobin, inflammatory monocytes, and CD16+ natural killer cells.

CONCLUSIONS

Patients with AI on standard therapy exhibit a dysregulation of circadian genes in PBMCs. The once-daily administration reconditions peripheral tissue gene expression to levels close to controls, paralleling the clinical outcomes of the DREAM trial (NCT02277587).

Dissipative structures in biological systems: bistability, oscillations, spatial patterns and waves

The goal of this review article is to assess how relevant is the concept of dissipative structure for understanding the dynamical bases of non-equilibrium self-organization in biological systems, and to see where it has been applied in the five decades since it was initially proposed by Ilya Prigogine. Dissipative structures can be classified into four types, which will be considered, in turn, and illustrated by biological examples: (i) multistability, in the form of bistability and tristability, which involve the coexistence of two or three stable steady states, or in the form of birhythmicity, which involves the coexistence between two stable rhythms; (ii) temporal dissipative structures in the form of sustained oscillations, illustrated by biological rhythms; (iii) spatial dissipative structures, known as Turing patterns; and (iv) spatio-temporal structures in the form of propagating waves. Rhythms occur with widely different periods at all levels of biological organization, from neural, cardiac and metabolic oscillations to circadian clocks and the cell cycle; they play key roles in physiology and in many disorders. New rhythms are being uncovered while artificial ones are produced by synthetic biology. Rhythms provide the richest source of examples of dissipative structures in biological systems. Bistability has been observed experimentally, but has primarily been investigated in theoretical models in an increasingly wide range of biological contexts, from the genetic to the cell and animal population levels, both in physiological conditions and in disease. Bistable transitions have been implicated in the progression between the different phases of the cell cycle and, more generally, in the process of cell fate specification in the developing embryo. Turing patterns are exemplified by the formation of some periodic structures in the course of development and by skin stripe patterns in animals. Spatio-temporal patterns in the form of propagating waves are observed within cells as well as in intercellular communication. This review illustrates how dissipative structures of all sorts abound in biological systems.This article is part of the theme issue 'Dissipative structures in matter out of equilibrium: from chemistry, photonics and biology (part 1)'.

Analysis Tools for Interconnected Boolean Networks With Biological Applications

Boolean networks with asynchronous updates are a class of logical models particularly well adapted to describe the dynamics of biological networks with uncertain measures. The state space of these models can be described by an asynchronous state transition graph, which represents all the possible exits from every single state, and gives a global image of all the possible trajectories of the system. In addition, the asynchronous state transition graph can be associated with an absorbing Markov chain, further providing a semi-quantitative framework where it becomes possible to compute probabilities for the different trajectories. For large networks, however, such direct analyses become computationally untractable, given the exponential dimension of the graph. Exploiting the general modularity of biological systems, we have introduced the novel concept of asymptotic graph, computed as an interconnection of several asynchronous transition graphs and recovering all asymptotic behaviors of a large interconnected system from the behavior of its smaller modules. From a modeling point of view, the interconnection of networks is very useful to address for instance the interplay between known biological modules and to test different hypotheses on the nature of their mutual regulatory links. This paper develops two new features of this general methodology: a quantitative dimension is added to the asymptotic graph, through the computation of relative probabilities for each final attractor and a companion cross-graph is introduced to complement the method on a theoretical point of view.

Modeling the interactions of sense and antisense Period transcripts in the mammalian circadian clock network

In recent years, it has become increasingly apparent that antisense transcription plays an important role in the regulation of gene expression. The circadian clock is no exception: an antisense transcript of the mammalian core-clock gene PERIOD2 (PER2), which we shall refer to as Per2AS RNA, oscillates with a circadian period and a nearly 12 h phase shift from the peak expression of Per2 mRNA. In this paper, we ask whether Per2AS plays a regulatory role in the mammalian circadian clock by studying in silico the potential effects of interactions between Per2 and Per2AS RNAs on circadian rhythms. Based on the antiphasic expression pattern, we consider two hypotheses about how Per2 and Per2AS mutually interfere with each other's expression. In our pre-transcriptional model, the transcription of Per2AS RNA from the non-coding strand represses the transcription of Per2 mRNA from the coding strand and vice versa. In our post-transcriptional model, Per2 and Per2AS transcripts form a double-stranded RNA duplex, which is rapidly degraded. To study these two possible mechanisms, we have added terms describing our alternative hypotheses to a published mathematical model of the molecular regulatory network of the mammalian circadian clock. Our pre-transcriptional model predicts that transcriptional interference between Per2 and Per2AS can generate alternative modes of circadian oscillations, which we characterize in terms of the amplitude and phase of oscillation of core clock genes. In our post-transcriptional model, Per2/Per2AS duplex formation dampens the circadian rhythm. In a model that combines pre- and post-transcriptional controls, the period, amplitude and phase of circadian proteins exhibit non-monotonic dependencies on the rate of expression of Per2AS. All three models provide potential explanations of the observed antiphasic, circadian oscillations of Per2 and Per2AS RNAs. They make discordant predictions that can be tested experimentally in order to distinguish among these alternative hypotheses.

Exploring the Underlying Mechanisms of the Xenopus laevis Embryonic Cell Cycle

The cell cycle is an indispensable process in proliferation and development. Despite significant efforts, global quantification and physical understanding are still challenging. In this study, we explored the mechanisms of the Xenopus laevis embryonic cell cycle by quantifying the underlying landscape and flux. We uncovered the Mexican hat landscape of the Xenopus laevis embryonic cell cycle with several local basins and barriers on the oscillation path. The local basins characterize the different phases of the Xenopus laevis embryonic cell cycle, and the local barriers represent the checkpoints. The checkpoint mechanism of the cell cycle is revealed by the landscape basins and barriers. While landscape shape determines the stabilities of the states on the oscillation path, the curl flux force determines the stability of the cell cycle flow. Replication is fundamental for biology of living cells. We quantify the input energy (through the entropy production) as the thermodynamic requirement for initiation and sustainability of single cell life (cell cycle). Furthermore, we also quantify curl flux originated from the input energy as the dynamical requirement for the emergence of a new stable phase (cell cycle). This can provide a new quantitative insight for the origin of single cell life. In fact, the curl flux originated from the energy input or nutrition supply determines the speed and guarantees the progression of the cell cycle. The speed of the cell cycle is a hallmark of cancer. We characterized the quality of the cell cycle by the coherence time and found it is supported by the flux and energy cost. We are also able to quantify the degree of time irreversibility by the cross correlation function forward and backward in time from the stochastic traces in the simulation or experiments, providing a way for the quantification of the time irreversibility and the flux. Through global sensitivity analysis upon landscape and flux, we can identify the key elements for controlling the cell cycle speed. This can help to design an effective strategy for drug discovery against cancer.

Bioinformatic Analysis of Circadian Expression of Oncogenes and Tumor Suppressor Genes

Background

Circadian rhythms are physiological and behavioral cycles with a period of approximately 24 hours that control various functions including gene expression. Circadian disruption is associated with a variety of diseases, especially cancer. Although some of the oncogenes and tumor suppressor genes (TSGs) are known as clock-controlled genes (CCGs), the analysis and annotation of circadian expression of most human oncogenes and TSGs are still lacking. This study aims to investigate the circadian expression of a list of human oncogenes and TSGs.

Methods

A bioinformatic analysis was conducted on a gene library comprising 120 genes to investigate the circadian expression of human oncogenes and TSGs. To achieve this purpose, the genotranscriptomic data were retrieved from COSMIC and analyzed by R statistical software. Furthermore, the acquired data were analyzed at the transcriptomic and proteomic levels using several publicly available databases. Also, the significance of all analyses was confirmed statistically.

Results

Altogether, our results indicated that 7 human oncogenes/TSGs may be expressed and function in a circadian manner. These oncogenes/TSGs showed a circadian expression pattern at CircaDB database and associated with at least one of the circadian genes/CCGs based on both genotranscriptomic and correlation analyses.

Conclusions

Although 4 of 7 finally outputted genes have been previously reported to be clock controlled, heretofore there is no report about the circadian expression of 3 other genes. Considering the importance of oncogenes/TSGs in the initiation and progression of cancer, further studies are suggested for the identification of exact circadian expression patterns of these 3 human oncogenes/TSGs.