Observation of a chaotic multioscillatory metabolic attractor by real-time monitoring of a yeast continuous culture

The main sources of molecular organization in the cell. Atlas of self-organized and self-regulated dynamic biostructures

One of the most important goals of contemporary biology is to understand the principles of the molecular order underlying the complex dynamic architecture of cells. Here, we present an overview of the main driving forces involved in the cellular molecular complexity and in the emergent functional dynamic structures, spanning from the most basic molecular organization levels to the complex emergent integrative systemic behaviors. First, we address the molecular information processing which is essential in many complex fundamental mechanisms such as the epigenetic memory, alternative splicing, regulation of transcriptional system, and the adequate self-regulatory adaptation to the extracellular environment. Next, we approach the biochemical self-organization, which is central to understand the emergency of metabolic rhythms, circadian oscillations, and spatial traveling waves. Such a complex behavior is also fundamental to understand the temporal compartmentalization of the cellular metabolism and the dynamic regulation of many physiological activities. Numerous examples of biochemical self-organization are considered here, which show that practically all the main physiological processes in the cell exhibit this type of dynamic molecular organization. Finally, we focus on the biochemical self-assembly which, at a primary level of organization, is a basic but important mechanism for the order in the cell allowing biomolecules in a disorganized state to form complex aggregates necessary for a plethora of essential structures and physiological functions. In total, more than 500 references have been compiled in this review. Due to these main sources of order, systemic functional structures emerge in the cell, driving the metabolic functionality towards the biological complexity.

A reservoir of timescales emerges in recurrent circuits with heterogeneous neural assemblies

The temporal activity of many physical and biological systems, from complex networks to neural circuits, exhibits fluctuations simultaneously varying over a large range of timescales. Long-tailed distributions of intrinsic timescales have been observed across neurons simultaneously recorded within the same cortical circuit. The mechanisms leading to this striking temporal heterogeneity are yet unknown. Here, we show that neural circuits, endowed with heterogeneous neural assemblies of different sizes, naturally generate multiple timescales of activity spanning several orders of magnitude. We develop an analytical theory using rate networks, supported by simulations of spiking networks with cell-type specific connectivity, to explain how neural timescales depend on assembly size and show that our model can naturally explain the long-tailed timescale distribution observed in the awake primate cortex. When driving recurrent networks of heterogeneous neural assemblies by a time-dependent broadband input, we found that large and small assemblies preferentially entrain slow and fast spectral components of the input, respectively. Our results suggest that heterogeneous assemblies can provide a biologically plausible mechanism for neural circuits to demix complex temporal input signals by transforming temporal into spatial neural codes via frequency-selective neural assemblies.

A dynamically coherent pattern of rhythms that matches between distant species across the evolutionary scale

We address the temporal organization of circadian and ultradian rhythms, crucial for understanding biological timekeeping in behavior, physiology, metabolism, and alignment with geophysical time. Using a newly developed five-steps wavelet-based approach to analyze high-resolution time series of metabolism in yeast cultures and spontaneous movement, metabolism, and feeding behavior in mice, rats, and quails, we describe a dynamically coherent pattern of rhythms spanning over a broad range of temporal scales (hours to minutes). The dynamic pattern found shares key features among the four, evolutionary distant, species analyzed. Specifically, a branching appearance given by splitting periods from 24 h into 12 h, 8 h and below in mammalian and avian species, or from 14 h down to 0.07 h in yeast. Scale-free fluctuations with long-range correlations prevail below ~ 4 h. Synthetic time series modeling support a scenario of coexisting behavioral rhythms, with circadian and ultradian rhythms at the center of the emergent pattern observed.

Neural mechanisms underlying the temporal organization of naturalistic animal behavior

Naturalistic animal behavior exhibits a strikingly complex organization in the temporal domain, with variability arising from at least three sources: hierarchical, contextual, and stochastic. What neural mechanisms and computational principles underlie such intricate temporal features? In this review, we provide a critical assessment of the existing behavioral and neurophysiological evidence for these sources of temporal variability in naturalistic behavior. Recent research converges on an emergent mechanistic theory of temporal variability based on attractor neural networks and metastable dynamics, arising via coordinated interactions between mesoscopic neural circuits. We highlight the crucial role played by structural heterogeneities as well as noise from mesoscopic feedback loops in regulating flexible behavior. We assess the shortcomings and missing links in the current theoretical and experimental literature and propose new directions of investigation to fill these gaps.

An appreciation of the prescience of Don Gilbert (1930-2011): master of the theory and experimental unravelling of biochemical and cellular oscillatory dynamics

We review Don Gilbert's pioneering seminal contributions that both detailed the mathematical principles and the experimental demonstration of several of the key dynamic characteristics of life. Long before it became evident to the wider biochemical community, Gilbert proposed that cellular growth and replication necessitate autodynamic occurrence of cycles of oscillations that initiate, coordinate and terminate the processes of growth, during which all components are duplicated and become spatially re-organised in the progeny. Initiation and suppression of replication exhibit switch-like characteristics, that is, bifurcations in the values of parameters that separate static and autodynamic behaviour. His limit cycle solutions present models developed in a series of papers reported between 1974 and 1984, and these showed that most or even all of the major facets of the cell division cycle could be accommodated. That the cell division cycle may be timed by a multiple of shorter period (ultradian) rhythms, gave further credence to the central importance of oscillatory phenomena and homeodynamics as evident on multiple time scales (seconds to hours). Further application of the concepts inherent in limit cycle operation as hypothesised by Gilbert more than 50 years ago are now validated as being applicable to oscillatory transcript, metabolite and enzyme levels, cellular differentiation, senescence, cancerous states and cell death. Now, we reiterate especially for students and young colleagues, that these early achievements were even more exceptional, as his own lifetime's work on modelling was continued with experimental work in parallel with his predictions of the major current enterprises of biological research.

Quantitative Studies for Cell-Division Cycle Control

The cell-division cycle (CDC) is driven by cyclin-dependent kinases (CDKs). Mathematical models based on molecular networks, as revealed by molecular and genetic studies, have reproduced the oscillatory behavior of CDK activity. Thus, one basic system for representing the CDC is a biochemical oscillator (CDK oscillator). However, genetically clonal cells divide with marked variability in their total duration of a single CDC round, exhibiting non-Gaussian statistical distributions. Therefore, the CDK oscillator model does not account for the statistical nature of cell-cycle control. Herein, we review quantitative studies of the statistical properties of the CDC. Over the past 70 years, studies have shown that the CDC is driven by a cluster of molecular oscillators. The CDK oscillator is coupled to transcriptional and mitochondrial metabolic oscillators, which cause deterministic chaotic dynamics for the CDC. Recent studies in animal embryos have raised the possibility that the dynamics of molecular oscillators underlying CDC control are affected by allometric volume scaling among the cellular compartments. Considering these studies, we discuss the idea that a cluster of molecular oscillators embedded in different cellular compartments coordinates cellular physiology and geometry for successful cell divisions.

Electromagnetic Fields, Genomic Instability and Cancer: A Systems Biological View

This review discusses the use of systems biology in understanding the biological effects of electromagnetic fields, with particular focus on induction of genomic instability and cancer. We introduce basic concepts of the dynamical systems theory such as the state space and attractors and the use of these concepts in understanding the behavior of complex biological systems. We then discuss genomic instability in the framework of the dynamical systems theory, and describe the hypothesis that environmentally induced genomic instability corresponds to abnormal attractor states; large enough environmental perturbations can force the biological system to leave normal evolutionarily optimized attractors (corresponding to normal cell phenotypes) and migrate to less stable variant attractors. We discuss experimental approaches that can be coupled with theoretical systems biology such as testable predictions, derived from the theory and experimental methods, that can be used for measuring the state of the complex biological system. We also review potentially informative studies and make recommendations for further studies.

Temporal metabolic partitioning of the yeast and protist cellular networks: the cell is a global scale-invariant (fractal or self-similar) multioscillator

Britton Chance, electronics expert when a teenager, became an enthusiastic student of biological oscillations, passing on this enthusiasm to many students and colleagues, including one of us (DL). This historical essay traces BC's influence through the accumulated work of DL to DL's many collaborators. The overall temporal organization of mass-energy, information, and signaling networks in yeast in self-synchronized continuous cultures represents, until now, the most characterized example of in vivo elucidation of time structure. Continuous online monitoring of dissolved gases by direct measurement (membrane-inlet mass spectrometry, together with NAD(P)H and flavin fluorescence) gives strain-specific dynamic information from timescales of minutes to hours as does two-photon imaging. The predominantly oscillatory behavior of network components becomes evident, with spontaneously synchronized cellular respiration cycles between discrete periods of increased oxygen consumption (oxidative phase) and decreased oxygen consumption (reductive phase). This temperature-compensated ultradian clock provides coordination, linking temporally partitioned functions by direct feedback loops between the energetic and redox state of the cell and its growing ultrastructure. Multioscillatory outputs in dissolved gases with 13 h, 40 min, and 4 min periods gave statistical self-similarity in power spectral and relative dispersional analyses: i.e., complex nonlinear (chaotic) behavior and a functional scale-free (fractal) network operating simultaneously over several timescales.

Mitochondrial chaotic dynamics: Redox-energetic behavior at the edge of stability

Mitochondria serve multiple key cellular functions, including energy generation, redox balance, and regulation of apoptotic cell death, thus making a major impact on healthy and diseased states. Increasingly recognized is that biological network stability/instability can play critical roles in determining health and disease. We report for the first-time mitochondrial chaotic dynamics, characterizing the conditions leading from stability to chaos in this organelle. Using an experimentally validated computational model of mitochondrial function, we show that complex oscillatory dynamics in key metabolic variables, arising at the “edge” between fully functional and pathological behavior, sets the stage for chaos. Under these conditions, a mild, regular sinusoidal redox forcing perturbation triggers chaotic dynamics with main signature traits such as sensitivity to initial conditions, positive Lyapunov exponents, and strange attractors. At the “edge” mitochondrial chaos is exquisitely sensitive to the antioxidant capacity of matrix Mn superoxide dismutase as well as to the amplitude and frequency of the redox perturbation. These results have potential implications both for mitochondrial signaling determining health maintenance, and pathological transformation, including abnormal cardiac rhythms.

Functional Implications of Cardiac Mitochondria Clustering

The spatio-temporal organization of mitochondria in cardiac myocytes facilitates myocyte-wide, cluster-bound, mitochondrial inner membrane potential oscillatory depolarizations, commonly triggered by metabolic or oxidative stressors. Local intermitochondrial coupling can be mediated by reactive oxygen species (ROS) that activate inner membrane pores to initiate a ROS-induced-ROS-release process that produces synchronized limit cycle oscillations of mitochondrial clusters within the whole mitochondrial network. The network's dynamic organization, structure and function can be assessed by quantifying dynamic local coupling constants and dynamic functional clustering coefficients, both providing information about the network's response to external stimuli. In addition to its special organization, the mitochondrial network of cardiac myocytes exhibits substrate-sensitive coupling constants and clustering coefficients. The myocyte's ability to form functional clusters of synchronously oscillating mitochondria is sensitive to conditions such as substrate availability (e.g., glucose, pyruvate, β-hydroxybutyrate), antioxidant status, respiratory chain activity, or history of oxidative challenge (e.g., ischemia-reperfusion). This underscores the relevance of quantitative methods to characterize the network's functional status as a way to assess the myocyte's resilience to pathological stressors.

Nucleotide-Dependent Bioautocatalytic Timer Reaction

We describe a biochemical timer composed of three biocatalytic reactions involving three types of adenylate nucleotides: adenosine triphosphate (ATP), adenosine diphosphate (ADP), and adenosine monophosphate (AMP). The timer is triggered by a small amount of ATP or ADP. An abrupt increase in the ATP concentration (following numerous amplification cycles) leads to a sudden increase of luminescence from the reaction mixture. The time point when the luminescence appears is found to be a function of the initial concentration of the triggering nucleotide (5.0 × 10(-8)-1.0 × 10(-6) M), even in the presence of a complex biological matrix. The mechanism of the observed dependence of the time of luminescence increase on the concentration has been confirmed with simple kinetic models. Due to the biocompatibility of the proposed trienzymatic reaction scheme (sensitivity to common nucleotides and occurrence in a neutral pH aqueous environment), the scheme can be used in bioengineered systems that require modulation of the response time (light emission) by concentration.

Network dynamics: quantitative analysis of complex behavior in metabolism, organelles, and cells, from experiments to models and back

Advancing from two core traits of biological systems: multilevel network organization and nonlinearity, we review a host of novel and readily available techniques to explore and analyze their complex dynamic behavior within the framework of experimental-computational synergy. In the context of concrete biological examples, analytical methods such as wavelet, power spectra, and metabolomics-fluxomics analyses, are presented, discussed, and their strengths and limitations highlighted. Further shown is how time series from stationary and nonstationary biological variables and signals, such as membrane potential, high-throughput metabolomics, O₂ and CO₂ levels, bird locomotion, at the molecular, (sub)cellular, tissue, and whole organ and animal levels, can reveal important information on the properties of the underlying biological networks. Systems biology-inspired computational methods start to pave the way for addressing the integrated functional dynamics of metabolic, organelle and organ networks. As our capacity to unravel the control and regulatory properties of these networks and their dynamics under normal or pathological conditions broadens, so is our ability to address endogenous rhythms and clocks to improve health-span in human aging, and to manage complex metabolic disorders, neurodegeneration, and cancer. WIREs Syst Biol Med 2017, 9:e1352. doi: 10.1002/wsbm.1352 For further resources related to this article, please visit the WIREs website.

Biological Timekeeping: The Business of a Blind Watchmaker

Fundamental understanding of life depends on both structural and functional details at the molecular level. Continually improving means of measurement of spatial and dynamic properties of biochemical constituents and cellular components complement studies of whole organisms. Integration of the interaction of components to provide coherent behaviour depends on highly elaborate orchestration in space and time. Whereas spatial information on a nanometre resolution is available, and fast dynamic analyses provide biochemical reaction rates measured in nanoseconds, functional coordination of the system requires integrated time dependence. While we are well aware of the special complexity of living organisms, appreciation of temporal scales and their organisation in time is still fragmentary. This article summarises current developments in research on biological time on scales from nanoseconds to years, the networks that connect different time domains and the oscillations, rhythms and biological clocks that coordinate and synchronise the complexity of the living state.

“It is the pattern maintained by this homeostasis, which is the touchstone of our personal identity. Our tissues change as we live: the food we eat and the air we breathe become flesh of our flesh, and bone of our bone, and the momentary elements of our flesh and bone pass out of our body every day with our excreta. We are but whirlpools in a river of ever-flowing water. We are not the stuff that abides, but patterns that perpetuate themselves”60.

Wiener, 1954

“What are called structures are slow processes of long duration, functions are quick processes of short duration”61.

Von Bertalanffy, 1952

Function of metabolic and organelle networks in crowded and organized media

(Macro)molecular crowding and the ability of the ubiquitous cytoskeleton to dynamically polymerize–depolymerize are prevalent cytoplasmic conditions in prokaryotic and eukaryotic cells. Protein interactions, enzymatic or signaling reactions - single, sequential or in complexes - whole metabolic pathways and organelles can be affected by crowding, the type and polymeric status of cytoskeletal proteins (e.g., tubulin, actin), and their imparted organization. The self-organizing capability of the cytoskeleton can orchestrate metabolic fluxes through entire pathways while its fractal organization can frame the scaling of activities in several levels of organization. The intracellular environment dynamics (e.g., biochemical reactions) is dominated by the orderly cytoskeleton and the intrinsic randomness of molecular crowding. Existing evidence underscores the inherent capacity of intracellular organization to generate emergent global behavior. Yet unknown is the relative impact on cell function provided by organelle or functional compartmentation based on transient proteins association driven by weak interactions (quinary structures) under specific environmental challenges or functional conditions (e.g., hypoxia, division, differentiation). We propose a qualitative, integrated structural–functional model of cytoplasmic organization based on a modified version of the Sierspinsky–Menger–Mandelbrot sponge, a 3D representation of a percolation cluster, and examine its capacity to accommodate established experimental facts.

Complex oscillatory redox dynamics with signaling potential at the edge between normal and pathological mitochondrial function

The time-keeping properties bestowed by oscillatory behavior on functional rhythms represent an evolutionarily conserved trait in living systems. Mitochondrial networks function as timekeepers maximizing energetic output while tuning reactive oxygen species (ROS) within physiological levels compatible with signaling. In this work, we explore the potential for timekeeping functions dependent on mitochondrial dynamics with the validated two-compartment mitochondrial energetic-redox (ME-R) computational model, that takes into account (a) four main redox couples [NADH, NADPH, GSH, Trx(SH)₂], (b) scavenging systems (glutathione, thioredoxin, SOD, catalase) distributed in matrix and extra-matrix compartments, and (c) transport of ROS species between them. Herein, we describe that the ME-R model can exhibit highly complex oscillatory dynamics in energetic/redox variables and ROS species, consisting of at least five frequencies with modulated amplitudes and period according to power spectral analysis. By stability analysis we describe that the extent of steady state—as against complex oscillatory behavior—was dependent upon the abundance of Mn and Cu, Zn SODs, and their interplay with ROS production in the respiratory chain. Large parametric regions corresponding to oscillatory dynamics of increasingly complex waveforms were obtained at low Cu, Zn SOD concentration as a function of Mn SOD. This oscillatory domain was greatly reduced at higher levels of Cu, Zn SOD. Interestingly, the realm of complex oscillations was located at the edge between normal and pathological mitochondrial energetic behavior, and was characterized by oxidative stress. We conclude that complex oscillatory dynamics could represent a frequency- and amplitude-modulated H₂O₂ signaling mechanism that arises under intense oxidative stress. By modulating SOD, cells could have evolved an adaptive compromise between relative constancy and the flexibility required under stressful redox/energetic conditions.

Dynamics of oscillatory phenotypes in Saccharomyces cerevisiae reveal a network of genome-wide transcriptional oscillators

Genetic and environmental factors are well-studied influences on phenotype; however, time is a variable that is rarely considered when studying changes in cellular phenotype. Time-resolved microarray data revealed genome-wide transcriptional oscillation in a yeast continuous culture system with ∼ 2 and ∼ 4 h periods. We mapped the global patterns of transcriptional oscillations into a 3D map to represent different cellular phenotypes of redox cycles. This map shows the dynamic nature of gene expression in that transcripts are ordered and coupled to each other through time and concentration space. Although cells differed in oscillation periods, transcripts involved in certain processes were conserved in a deterministic way. When oscillation period lengthened, the peak to trough ratio of transcripts increased and the fraction of cells in the unbudded (G0/G1) phase of the cell division cycle increased. Decreasing the glucose level in the culture medium was one way to increase the redox cycle, possibly from changes in metabolic flux. The period may be responding to lower glucose levels by increasing the fraction of cells in G1 and reducing S-phase gating so that cells can spend more time in catabolic processes. Our results support that gene transcripts are coordinated with metabolic functions and the cell division cycle.

What yeast and cardiomyocytes share: ultradian oscillatory redox mechanisms of cellular coherence and survival

The coherent and robust, yet sensitively adaptable, nature of organisms is an astonishing phenomenon that involves massive parallel processing and concerted network performance at the molecular level. Unravelling the dynamic complexities of the living state underlines the essential operation of ultradian oscillations, rhythms and clocks for the establishment and maintenance of functional order simultaneously on fast and slower timescales. Non-invasive monitoring of respiration, mitochondrial inner membrane potentials, and redox states (especially those of NAD(P)H, flavin, and the monochlorobimane complex of glutathione), even after more than 50 years research, continue to provide both new insights and biomedical applications. Experiments with yeast and in cardiac cells reveal astonishing parallels and similarities in their dynamic biochemical organization.

Redox regulation in respiring Saccharomyces cerevisiae

BACKGROUND

In biological systems, redox reactions are central to most cellular processes and the redox potential of the intracellular compartment dictates whether a particular reaction can or cannot occur. Indeed the widespread use of redox reactions in biological systems makes their detailed description outside the scope of one review.

SCOPE OF THE REVIEW

Here we will focus on how system-wide redox changes can alter the reaction and transcriptional landscape of Saccharomyces cerevisiae. To understand this we explore the major determinants of cellular redox potential, how these are sensed by the cell and the dynamic responses elicited.

MAJOR CONCLUSIONS

Redox regulation is a large and complex system that has the potential to rapidly and globally alter both the reaction and transcription landscapes. Although we have a basic understanding of many of the sub-systems and a partial understanding of the transcriptional control, we are far from understanding how these systems integrate to produce coherent responses. We argue that this non-linear system self-organises, and that the output in many cases is temperature-compensated oscillations that may temporally partition incompatible reactions in vivo.

GENERAL SIGNIFICANCE

Redox biochemistry impinges on most of cellular processes and has been shown to underpin ageing and many human diseases. Integrating the complexity of redox signalling and regulation is perhaps one of the most challenging areas of biology. This article is part of a Special Issue entitled Systems Biology of Microorganisms.

PSK2 coordinates glucose metabolism and utilization to maintain ultradian clock-coupled respiratory oscillation in Saccharomyces cerevisiae yeast

Ultradian clock-coupled respiratory oscillation (UCRO) in an aerobic continuous culture of Saccharomyces cerevisiae S288C is principally regulated by control of certain redox reactions of energy metabolism. It is also modulated by the metabolism of storage carbohydrates during adaptation to environmental change. However, the mechanism of cell sensing and response to environmental nutrients in UCRO is unknown. The purpose of the present study was to determine the role of PSK2 kinase in UCRO in yeast. S. cerevisiae in culture showed oscillation in PSK2 mRNA levels with a definite phase relationship to the respiratory oscillation. Furthermore, inactivation of Psk2 by gene disruption severely affected UCRO and its decline to undetectable levels within 2days. In addition, the extracellular and intracellular glucose concentrations of PSK2 deletion mutants in culture were higher and lower, respectively, than those of the wild type. PSK2 mutant cells showed no alteration in redox state. Furthermore, the levels of storage carbohydrates such as glycogen and trehalose fluctuated in PSK2 mutants with attenuated amplitudes comparable to those in the wild type. The results indicated that PSK2 kinase is important for the uptake of glucose and regulation of storage-carbohydrate synthesis and hence the maintenance of an unperturbed continuously oscillating state.

The scale-free dynamics of eukaryotic cells

Temporal organization of biological processes requires massively parallel processing on a synchronized time-base. We analyzed time-series data obtained from the bioenergetic oscillatory outputs of Saccharomyces cerevisiae and isolated cardiomyocytes utilizing Relative Dispersional (RDA) and Power Spectral (PSA) analyses. These analyses revealed broad frequency distributions and evidence for long-term memory in the observed dynamics. Moreover RDA and PSA showed that the bioenergetic dynamics in both systems show fractal scaling over at least 3 orders of magnitude, and that this scaling obeys an inverse power law. Therefore we conclude that in S. cerevisiae and cardiomyocytes the dynamics are scale-free in vivo. Applying RDA and PSA to data generated from an in silico model of mitochondrial function indicated that in yeast and cardiomyocytes the underlying mechanisms regulating the scale-free behavior are similar. We validated this finding in vivo using single cells, and attenuating the activity of the mitochondrial inner membrane anion channel with 4-chlorodiazepam to show that the oscillation of NAD(P)H and reactive oxygen species (ROS) can be abated in these two evolutionarily distant species. Taken together these data strongly support our hypothesis that the generation of ROS, coupled to redox cycling, driven by cytoplasmic and mitochondrial processes, are at the core of the observed rhythmicity and scale-free dynamics. We argue that the operation of scale-free bioenergetic dynamics plays a fundamental role to integrate cellular function, while providing a framework for robust, yet flexible, responses to the environment.

Redox rhythmicity: clocks at the core of temporal coherence

Ultradian rhythms are those that cycle many times in a day and are therefore measured in hours, minutes, seconds or even fractions of a second. In yeasts and protists, a temperature-compensated clock with a period of about an hour (30-90 minutes) provides the time base upon which all central processes are synchronized. A 40-minute clock in yeast times metabolic, respiratory and transcriptional processes, and controls cell division cycle progression. This system has at its core a redox cycle involving NAD(P)H and dithiol-disulfide interconversions. It provides an archetype for biological time keeping on longer time scales (e.g. the daily cycles driven by circadian clocks) and underpins these rhythms, which cannot be understood in isolation. Ultradian rhythms are the foundation upon which the coherent functioning of the organism depends.