An entrainment model for timed enzyme syntheses in bacteria

Nuclear-cytoplasmic compartmentalization of cyclin B1-Cdk1 promotes robust timing of mitotic events

The cyclin-dependent kinase (Cdk1) oscillator is widely characterized in homogenized cytosolic extracts, leaving unclear the impact of nucleocytoplasmic compartmentalization. Here, by developing a Förster resonance energy transfer (FRET) biosensor, we track Cdk1 spatiotemporal dynamics in reconstituted cells with or without side by side and find compartmentalization significantly modulates clock properties previously found in bulk studies. Although nucleus-absent cells display highly tunable frequency, the nucleus-present cells maintain constant frequency against cyclin B1 variations. Despite high expression variability, cyclin degraded within the same duration, enabling a robust mitotic phase. Moreover, Cdk1 and cyclin B1 cycle rigorously out-of-phase, ensuring wide phase-plane orbits, essential for oscillation robustness. Although Cdk1 in homogeneous extracts is well known for delayed switch-like activation, we find active cyclin B1-Cdk1 accumulates in nuclei, without delay, until the nuclear envelope breakdown (NEB) when another abrupt activation triggers anaphase. Cdk1 biphasic activation and spatial compartmentalization may together coordinate the accurate ordering of different downstream events.

Dissecting cell fate dynamics in pediatric glioblastoma through the lens of complex systems and cellular cybernetics

Cancers are complex dynamic ecosystems. Reductionist approaches to science are inadequate in characterizing their self-organized patterns and collective emergent behaviors. Since current approaches to single-cell analysis in cancer systems rely primarily on single time-point multiomics, many of the temporal features and causal adaptive behaviors in cancer dynamics are vastly ignored. As such, tools and concepts from the interdisciplinary paradigm of complex systems theory are introduced herein to decode the cellular cybernetics of cancer differentiation dynamics and behavioral patterns. An intuition for the attractors and complex networks underlying cancer processes such as cell fate decision-making, multiscale pattern formation systems, and epigenetic state-transitions is developed. The applications of complex systems physics in paving targeted therapies and causal pattern discovery in precision oncology are discussed. Pediatric high-grade gliomas are discussed as a model-system to demonstrate that cancers are complex adaptive systems, in which the emergence and selection of heterogeneous cellular states and phenotypic plasticity are driven by complex multiscale network dynamics. In specific, pediatric glioblastoma (GBM) is used as a proof-of-concept model to illustrate the applications of the complex systems framework in understanding GBM cell fate decisions and decoding their adaptive cellular dynamics. The scope of these tools in forecasting cancer cell fate dynamics in the emerging field of computational oncology and patient-centered systems medicine is highlighted.

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

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

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

The Goodwin Oscillator and its Legacy

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

Resonating neurons stabilize heterogeneous grid-cell networks

A central theme that governs the functional design of biological networks is their ability to sustain stable function despite widespread parametric variability. Here, we investigated the impact of distinct forms of biological heterogeneities on the stability of a two-dimensional continuous attractor network (CAN) implicated in grid-patterned activity generation. We show that increasing degrees of biological heterogeneities progressively disrupted the emergence of grid-patterned activity and resulted in progressively large perturbations in low-frequency neural activity. We postulated that targeted suppression of low-frequency perturbations could ameliorate heterogeneity-induced disruptions of grid-patterned activity. To test this, we introduced intrinsic resonance, a physiological mechanism to suppress low-frequency activity, either by adding an additional high-pass filter (phenomenological) or by incorporating a slow negative feedback loop (mechanistic) into our model neurons. Strikingly, CAN models with resonating neurons were resilient to the incorporation of heterogeneities and exhibited stable grid-patterned firing. We found CAN models with mechanistic resonators to be more effective in targeted suppression of low-frequency activity, with the slow kinetics of the negative feedback loop essential in stabilizing these networks. As low-frequency perturbations (1/f noise) are pervasive across biological systems, our analyses suggest a universal role for mechanisms that suppress low-frequency activity in stabilizing heterogeneous biological networks.

An integrate-and-fire model for pulsatility in the neuroendocrine system

A model for pulsatility in neuroendocrine regulation is proposed which combines Goodwin-type feedback control with impulsive input from neurons located in the hypothalamus. The impulsive neural input is modeled using an integrate-and-fire mechanism; namely, inputs are generated only when the membrane potential crosses a threshold, after which it is reset to baseline. The resultant model takes the form of a functional-differential equation with continuous and impulsive components. Despite the impulsive nature of the inputs, realistic hormone profiles are generated, including ultradian and circadian rhythms, pulsatile secretory patterns, and even chaotic dynamics.

Modeling the dynamic behavior of biochemical regulatory networks

Strategies for modeling the complex dynamical behavior of gene/protein regulatory networks have evolved over the last 50 years as both the knowledge of these molecular control systems and the power of computing resources have increased. Here, we review a number of common modeling approaches, including Boolean (logical) models, systems of piecewise-linear or fully non-linear ordinary differential equations, and stochastic models (including hybrid deterministic/stochastic approaches). We discuss the pro's and con's of each approach, to help novice modelers choose a modeling strategy suitable to their problem, based on the type and bounty of available experimental information. We illustrate different modeling strategies in terms of some abstract network motifs, and in the specific context of cell cycle regulation.

Entrainment of a cellular circadian oscillator by light in the presence of molecular noise

In this paper, we consider a stochastic molecular circadian oscillator described by a sequence of biological reactions and its deterministic kinetics governed by a system of ordinary differential equations in the limit of large numbers of molecules. The oscillations in the model are generated by negative feedback regulation of a gene. The focus of this paper is the entrainment of the oscillator by a periodic light signal that affects the maximal transcription rate of the gene. We introduce two scalings of the model parameters that provide independent control over the natural frequency of the oscillator and the relative noise level. We study entrainment in two ways: by visualizing the stochastic limit cycle in various projections of the discrete phase space of the system and by evaluating the maximum of the normalized cross correlation of the light signal with the number of protein molecules in the cell. The visualization method ignores the phase of the oscillator, and we find in this way that entrainment has a subtle organizing effect on the limit cycle as a whole. The cross correlation results reveal an interval of natural frequencies of the oscillator surrounding the frequency of the light signal within which maximal entrainment occurs with rather sharp drops in entrainment at the edges of this interval. The width of the interval of maximal entrainment increases with the amplitude of the light signal. These statements are applicable both to the stochastic oscillator and to its deterministic limit, but the results are most clear-cut in the deterministic case and degrade from there as the relative noise level increases.

Grip on complexity in chemical reaction networks

A new discipline of "systems chemistry" is emerging, which aims to capture the complexity observed in natural systems within a synthetic chemical framework. Living systems rely on complex networks of chemical reactions to control the concentration of molecules in space and time. Despite the enormous complexity in biological networks, it is possible to identify network motifs that lead to functional outputs such as bistability or oscillations. To truly understand how living systems function, we need a complete understanding of how chemical reaction networks (CRNs) create function. We propose the development of a bottom-up approach to design and construct CRNs where we can follow the influence of single chemical entities on the properties of the network as a whole. Ultimately, this approach should allow us to not only understand such complex networks but also to guide and control their behavior.

Serotyping, antibiotic susceptibility, and virulence genes screening of Escherichia coli isolates obtained from diarrheic buffalo calves in Egyptian farms

AIM

In Egypt as in many other countries, river water buffalo (Bubalus bubalis) is considered an important source of high-quality milk and meat supply. The objective of this study was to investigate serotypes, virulence genes, and antibiotic resistance determinants profiles of Escherichia coli isolated from buffalo at some places in Egypt; noticibly, this issue was not discussed in the country yet.

MATERIALS AND METHODS

A number of 58 rectal samples were collected from diarrheic buffalo calves in different regions in Egypt, and bacteriological investigated for E. coli existence. The E. coli isolates were biochemically, serologicaly identified, tested for antibiotic susceptibility, and polymerase chain reaction (PCR) analyzed for the presence of antibiotic resistance determinants and virulence genes.

RESULTS

Overall 14 isolates typed as E. coli (24.1%); 6 were belonged to serogroup O78 (10.3%), followed by O125 (4 isolates, 6.9%), then O158 (3 isolates, 5.2%) and one isolate O8 (1.7%), among them, there were 5 E. coli isolates showed a picture of hemolysis (35.7%). The isolates exhibited a high resistance to β lactams over 60%, followed by sulfa (50%) and aminoglucoside (42.8%) group, in the same time the isolates were sensitive to quinolone, trimethoprim-sulfamethoxazole, tetracycline (100%), and cephalosporine groups (71.4%). A multiplex PCR was applied to the 14 E. coli isolates revealed that all were carrying at least one gene, as 10 carried blaTEM (71.4%), 8 Sul1 (57.1%), and 6 aadB (42.8%), and 9 isolates could be considered multidrug resistant (MDR) by an incidence of 64.3%. A PCR survey was stratified for the most important E. coli virulence genes, and showed the presence of Shiga toxins in 9 isolates carried either one or the two Stx genes (64.3%), 5 isolates carried hylA gene (35.7%), and eae in 2 isolates only (14.3%), all isolates carried at least one virulence gene except two (85.7%).

CONCLUSION

The obtained data displayed that in Egypt, buffalo as well as other ruminants could be a potential source of MDR pathogenic E. coli variants which have a public health importance.

Dynamic information routing in complex networks

Flexible information routing fundamentally underlies the function of many biological and artificial networks. Yet, how such systems may specifically communicate and dynamically route information is not well understood. Here we identify a generic mechanism to route information on top of collective dynamical reference states in complex networks. Switching between collective dynamics induces flexible reorganization of information sharing and routing patterns, as quantified by delayed mutual information and transfer entropy measures between activities of a network's units. We demonstrate the power of this mechanism specifically for oscillatory dynamics and analyse how individual unit properties, the network topology and external inputs co-act to systematically organize information routing. For multi-scale, modular architectures, we resolve routing patterns at all levels. Interestingly, local interventions within one sub-network may remotely determine nonlocal network-wide communication. These results help understanding and designing information routing patterns across systems where collective dynamics co-occurs with a communication function.

Flexible information routing underlies the function of many biological and artificial networks. Here, the authors present a theoretical framework that shows how information can be flexibly routed across networks using collective reference dynamics and how local changes may induce remote rerouting.

Robustness and period sensitivity analysis of minimal models for biochemical oscillators

Biological systems exhibit numerous oscillatory behaviors from calcium oscillations to circadian rhythms that recur daily. These autonomous oscillators contain complex feedbacks with nonlinear dynamics that enable spontaneous oscillations. The detailed nonlinear dynamics of such systems remains largely unknown. In this paper, we investigate robustness and dynamical differences of five minimal systems that may underlie fundamental molecular processes in biological oscillatory systems. Bifurcation analyses of these five models demonstrate an increase of oscillatory domains with a positive feedback mechanism that incorporates a reversible reaction, and dramatic changes in dynamics with small modifications in the wiring. Furthermore, our parameter sensitivity analysis and stochastic simulations reveal different rankings of hierarchy of period robustness that are determined by the number of sensitive parameters or network topology. In addition, systems with autocatalytic positive feedback loop are shown to be more robust than those with positive feedback via inhibitory degradation regardless of noise type. We demonstrate that robustness has to be comprehensively assessed with both parameter sensitivity analysis and stochastic simulations.

It is not the parts, but how they interact that determines the behaviour of circadian rhythms across scales and organisms

Biological rhythms, generated by feedback loops containing interacting genes, proteins and/or cells, time physiological processes in many organisms. While many of the components of the systems that generate biological rhythms have been identified, much less is known about the details of their interactions. Using examples from the circadian (daily) clock in three organisms, Neurospora, Drosophila and mouse, we show, with mathematical models of varying complexity, how interactions among (i) promoter sites, (ii) proteins forming complexes, and (iii) cells can have a drastic effect on timekeeping. Inspired by the identification of many transcription factors, for example as involved in the Neurospora circadian clock, that can both activate and repress, we show how these multiple actions can cause complex oscillatory patterns in a transcription–translation feedback loop (TTFL). Inspired by the timekeeping complex formed by the NMO–PER–TIM–SGG complex that regulates the negative TTFL in the Drosophila circadian clock, we show how the mechanism of complex formation can determine the prevalence of oscillations in a TTFL. Finally, we note that most mathematical models of intracellular clocks model a single cell, but compare with experimental data from collections of cells. We find that refitting the most detailed model of the mammalian circadian clock, so that the coupling between cells matches experimental data, yields different dynamics and makes an interesting prediction that also matches experimental data: individual cells are bistable, and network coupling removes this bistability and causes the network to be more robust to external perturbations. Taken together, we propose that the interactions between components in biological timekeeping systems are carefully tuned towards proper function. We also show how timekeeping can be controlled by novel mechanisms at different levels of organization.

Noise induced oscillations and coherence resonance in a generic model of the nonisothermal chemical oscillator

Oscillating chemical reactions are common in biological systems and they also occur in artificial non-biological systems. Generally, these reactions are subject to random fluctuations in environmental conditions which translate into fluctuations in the values of physical variables, for example, temperature. We formulate a mathematical model for a nonisothermal minimal chemical oscillator containing a single negative feedback loop and study numerically the effects of stochastic fluctuations in temperature in the absence of any deterministic limit cycle or periodic forcing. We show that noise in temperature can induce sustained limit cycle oscillations with a relatively narrow frequency distribution and some characteristic frequency. These properties differ significantly depending on the noise correlation. Here, we have explored white and colored (correlated) noise. A plot of the characteristic frequency of the noise induced oscillations as a function of the correlation exponent shows a maximum, therefore indicating the existence of autonomous stochastic resonance, i.e. coherence resonance.

Multi-scale modeling in biology: how to bridge the gaps between scales?

Human physiological functions are regulated across many orders of magnitude in space and time. Integrating the information and dynamics from one scale to another is critical for the understanding of human physiology and the treatment of diseases. Multi-scale modeling, as a computational approach, has been widely adopted by researchers in computational and systems biology. A key unsolved issue is how to represent appropriately the dynamical behaviors of a high-dimensional model of a lower scale by a low-dimensional model of a higher scale, so that it can be used to investigate complex dynamical behaviors at even higher scales of integration. In the article, we first review the widely-used different modeling methodologies and their applications at different scales. We then discuss the gaps between different modeling methodologies and between scales, and discuss potential methods for bridging the gaps between scales.

Signal processing in cellular clocks

Many biochemical events within a cell need to be timed properly to occur at specific times of day, after other events have happened within the cell or in response to environmental signals. The cellular biochemical feedback loops that time these events have already received much recent attention in the experimental and modeling communities. Here, we show how ideas from signal processing can be applied to understand the function of these clocks. Consider two signals from the network s(t) and r(t), either two variables of a model or two experimentally measured time courses. We show how s(t) can be decomposed into two parts, the first being a function of r(t), and the second the derivative of a function of r(t). Geometric principles are then derived that can be used to understand when oscillations appear in biochemical feedback loops, the period of these oscillations, and their time course. Specific examples of this theory are provided that show how certain networks are prone or not prone to oscillate, how individual biochemical processes affect the period, and how oscillations in one chemical species can be deduced from oscillations in other parts of the network.

Multiscale complexity in the mammalian circadian clock

The field of systems biology studies how the interactions among individual components (e.g. genes and proteins) yield interesting and complex behavior. The circadian (daily) timekeeping system in mammals is an ideal system to study complexity because of its many biological scales (from genes to animal behavior). A wealth of data at each of these scales has recently been discovered. Within each scale, modeling can advance our understanding of challenging problems that arise in studying mammalian timekeeping. However, future work must focus on bridging the multiple spatial and temporal scales in the modeling of SCN network. Here we review recent advances, and then delve into a few areas that are promising research directions. We also discuss the flavor of modeling needed (simple or detailed) as well as new techniques that are needed to meet the challenges in modeling data across scales.

The effects of cascade length, kinetics and feedback loops on biological signal transduction dynamics in a simplified cascade model

How intracellular signals are propagated with appropriate strength, duration and fidelity over time is poorly understood. To address these issues, intracellular signal transduction was studied both analytically and numerically using a simplified cascade model. The main observations can be summarized as follows: when the response kinetics is of the Michaelis-Menten type, the signal strength will always reach the same magnitude as the cascade length increases, regardless of the type of stimulus applied (i.e. either continuous or unitary pulse). However, when the response kinetics is of the Hill type (Hill coefficient >1), there exists a stimulation threshold. If the stimulus is below the threshold, the signal decays toward zero; in contrast, if the stimulus is above the threshold, the signal amplitude reaches a nonzero steady state. The time taken for the signal to proceed through the cascade increases as the half-maximum point, or Hill coefficient, increases, whereas the duration of the output signal at the end of the cascade decreases as the half-maximum point increases. In the presence of positive feedback, the stimulation threshold increases; under these conditions, the feedback strength necessary for bistability changes (with power-law characteristics) inversely related to the length of the cascade. In the presence of negative feedback, oscillations are induced when the Hill coefficient is greater than 1 and the cascade has more than two steps. Likewise, the feedback strength required to generate oscillations changes (again with power-law characteristics) inversely with the length of the cascade.

Design principles of biochemical oscillators

Cellular rhythms are generated by complex interactions among genes, proteins and metabolites. They are used to control every aspect of cell physiology, from signalling, motility and development to growth, division and death. We consider specific examples of oscillatory processes and discuss four general requirements for biochemical oscillations: negative feedback, time delay, sufficient 'nonlinearity' of the reaction kinetics and proper balancing of the timescales of opposing chemical reactions. Positive feedback is one mechanism to delay the negative-feedback signal. Biological oscillators can be classified according to the topology of the positive- and negative-feedback loops in the underlying regulatory mechanism.

Synchronization of genetic oscillators

Synchronization of genetic or cellular oscillators is a central topic in understanding the rhythmicity of living organisms at both molecular and cellular levels. Here, we show how a collective rhythm across a population of genetic oscillators through synchronization-induced intercellular communication is achieved, and how an ensemble of independent genetic oscillators is synchronized by a common noisy signaling molecule. Our main purpose is to elucidate various synchronization mechanisms from the viewpoint of dynamics, by investigating the effects of various biologically plausible couplings, several kinds of noise, and external stimuli. To have a comprehensive understanding on the synchronization of genetic oscillators, we consider three classes of genetic oscillators: smooth oscillators (exhibiting sine-like oscillations), relaxation oscillators (displaying jump dynamics), and stochastic oscillators (noise-induced oscillation). For every class, we further study two cases: with intercellular communication (including phase-attractive and repulsive coupling) and without communication between cells. We find that an ensemble of smooth oscillators has different synchronization phenomena from those in the case of relaxation oscillators, where noise plays a different but key role in synchronization. To show differences in synchronization between them, we make comparisons in many aspects. We also show that a population of genetic stochastic oscillators have their own synchronization mechanisms. In addition, we present interesting phenomena, e.g., for relaxation-type stochastic oscillators coupled to a quorum-sensing mechanism, different noise intensities can induce different periodic motions (i.e., inhomogeneous limit cycles).

Dynamic properties of a delayed protein cross talk model

In this paper we investigate how the inclusion of time delay alters the dynamical properties of the Jacob-Monod model, describing the control of the beta-galactosidase synthesis by the lac repressor protein in E. coli. The consequences of a time delay on the dynamics of this system are analysed using Hopf's theorem and Lyapunov-Andronov's theory applied to the original mathematical model and to an approximated version. Our analytical calculations predict that time delay acts as a key bifurcation parameter. This is confirmed by numerical simulations. A critical value of time delay, which depends on the values of the model parameters, is analytically established. Around this critical value, the properties of the system change drastically, allowing under certain conditions the emergence of stable limit cycles, that is self-sustained oscillations. In addition, the features of the end product repression play an essential role in the characterisation of these limit cycles: if cooperativity is considered in the end product repression, time delay higher than the mentioned critical value induce differentiated responses during the oscillations, provoking cycles of all-or-nothing response in the concentration of the species.

Model of calcium oscillations due to negative feedback in olfactory cilia

We present a mathematical model for calcium oscillations in the cilia of olfactory sensory neurons. The underlying mechanism is based on direct negative regulation of cyclic nucleotide-gated channels by calcium/calmodulin and does not require any autocatalysis such as calcium-induced calcium release. The model is in quantitative agreement with available experimental data, both with respect to oscillations and to fast adaptation. We give predictions for the ranges of parameters in which oscillations should be observable. Relevance of the model to calcium oscillations in other systems is discussed.

Sniffers, buzzers, toggles and blinkers: dynamics of regulatory and signaling pathways in the cell

The physiological responses of cells to external and internal stimuli are governed by genes and proteins interacting in complex networks whose dynamical properties are impossible to understand by intuitive reasoning alone. Recent advances by theoretical biologists have demonstrated that molecular regulatory networks can be accurately modeled in mathematical terms. These models shed light on the design principles of biological control systems and make predictions that have been verified experimentally.

A cycle of enzymatic reactions that behaves like a neuronal circuit

A cycle of four enzymatic reactions with repression or allosteric inhibition of the enzymes has been proposed by analogy to a neural oscillator. The system is analysed in a situation remote from full symmetry. Asymmetry has been introduced by treating one of the reagents as a reservoir substance with constant concentration and having essentially different rate constants for forward and backward reactions. It is demonstrated that, for certain values of parameters, the system can work as a strength of stimulus to frequency transducer. For other values of the parameters, it acquires the features of an excitable system.

Network dynamics and cell physiology

Complex assemblies of interacting proteins carry out most of the interesting jobs in a cell, such as metabolism, DNA synthesis, movement and information processing. These physiological properties play out as a subtle molecular dance, choreographed by underlying regulatory networks. To understand this dance, a new breed of theoretical molecular biologists reproduces these networks in computers and in the mathematical language of dynamical systems.

Molecular circadian oscillators: an alternative hypothesis

Results from experiments in different organisms have shown that elements of input pathways can themselves be under circadian control and that outputs might feed back into the oscillator. In addition, it has become clear that there might be redundancies in the generation of circadian rhythmicity, even within single cells. In view of these results, it is worth reevaluating our current working hypotheses about the pacemaker's molecular mechanisms and the involvement of single autoregulatory genes. On one hand, redundancies in the generation of circadian rhythmicity might make the approach of defining a discrete circadian oscillator with the help of single gene mutations extremely difficult. On the other hand, many examples show that components of signal transduction pathways can indeed be encoded by single genes. The authors have constructed a model placing an autoregulatory gene and its products on an input pathway feeding into a separate oscillator. The behavior of this model can explain the majority of results of molecular circadian biology published to date. In addition, it shows that different qualities of the circadian system might be associated with different cellular functions that can exist independently and, only if put together, will lead to the known circadian phenotype.

Does replication-induced transcription regulate synthesis of the myriad low copy number proteins of Escherichia coli?

Over 80% of the genes in the E. coli chromosome express fewer than a hundred copies each of their protein products per cell. It is argued here that transcription of these genes is neither constitutive nor regulated by protein factors, but rather, induced by the act of replication. The utility of such replication-induced (RI) transcription to the temporal regulation of synthesis of determinate quantities of low copy number (LCN) proteins is described. It is suggested that RI transcription may be necessitated, as well as facilitated, by the folding of the bacterial chromosome into a compact nucleoid. Mechanistic aspects of the induction of transcription by replication are discussed with respect to the modulation of transcriptional initiation by negative supercoiling effects, promoter methylation status and derepression. It is shown that RI transcription offers plausible explanations for the constancy of the C period of the E. coli cell cycle and the remarkable conservation of gene order in the chromosomes of enteric bacteria. Some experimental tests of the hypothesis are proposed.

Complex behaviour of the repressible operon

The repressor-mediated repression process in bacteria is modelled using a gene-enzyme-endproduct control unit. A combined analytical-numerical study shows that the system, though stable normally, becomes unstable for super-repressing strains even at low values of the cooperativity of repression, provided demand for the endproduct saturates at large endproduct concentrations. In addition the system also shows bistability, i.e., the co-existence of a stable steady-state and a stable limit cycle. The tryptophan operon is used as a model system and the results are discussed in the light of differential regulation of gene expression in lower organisms, especially in mutant strains.

Mathematical modelling of dynamics and control in metabolic networks: VI. Dynamic bifurcations in single biochemical control loops

A dynamic stability analysis of an extended form of the Goodwin equations is presented. The Goodwin equations are extended to include Michaelis-Menten kinetics for the removal of the end-product. Inclusion of saturation kinetic behavior substantially increases the likelihood of dynamic instability in this model control loop. Oscillations are found for reaction chains of low order, as low as second order, and low degrees of co-operativity, as low as v = 2, simultaneously, thus indicating that dynamic instability in this system exists for physiologically realistic parameter values. The branches of bifurcated solutions are computed numerically and unstable Hopf bifurcations are found. Further, solution branches from stable Hopf bifurcation points are found to "fold back", i.e. have periodic limit points, producing situations where multiple stable limit cycles exist.

Why are so many biological systems periodic?

The ubiquity of oscillations in biological systems is well established. Oscillations are observed in all types of organisms from the simplest to the most complex. Periods can range from fractions of a second to months or years. From time to time, it has been suggested that many biological oscillations are the result of the breakdown of effective self-regulation. The opposite view is defended here. It is argued that most periodic behavior is not pathological but rather constitutes the normal operation for these systems. They are present because they confer positive functional advantages for the organism. The advantages fall into five general categories: temporal organization, spatial organization, prediction of repetitive events, efficiency and precision of control.

Periodic enzyme synthesis and oscillatory repression: why is the period of oscillation close to the cell cycle time?

During exponential growth of a cell culture, some enzymes are synthesized periodically. In a synchronous culture, in which all cells undergo DNA synthesis and division more-or-less synchronously, the burst of enzyme synthesis also occurs synchronously in each cell once per division cycle. However, there are a number of interesting cases in which periodic enzyme synthesis continues in the absence of synchronous DNA replication or cell division. In all cases of periodic enzyme synthesis in asynchronous cultures, the time between bursts of enzyme synthesis, though no longer identical to the cell cycle time, is still close to the interdivision time of the growing, replicating cells. The theory of oscillatory repression looks for an explanation of this phenomenon in the periodic repression of gene transcription caused by periodic fluctuations in the concentration of the endproduct of the metabolic pathway of which the enzyme is a part. A major difficulty with this theory is that there is no obvious relationship between the periodicity of the negative feedback loop, which is determined by the kinetics of synthesis and degradation of the individual components of the feedback loop, and the periodicity of the cell cycle, which is determined by overall net synthetic rates of cellular macromolecules. Why should the period of oscillation of a repressible gene transcription system be close to the interdivision time of a population of growing cells? In this paper, I show that the relationship may be coincidental: the two fundamental periods are close to each other because they are both close to the mass-doubling time of the cell culture. That the mean interdivision time must be close to the mass-doubling time is a consequence of "balanced" growth: there is a stable size distribution of cells in a growing culture. That the period of oscillation of the negative feedback loop is also close to the mass-doubling time is shown to be a consequence of the large, nearly constant demand for endproduct and the assumed stability of the enzyme. The period of oscillation is largely attributable to the slow dilution of the stable enzyme by cell growth. For reasonable values of the parameters describing the gene-control system, I show that the enzyme must be diluted by a factor of two (approximately), that is, by the growth accomplished by one mass-doubling (nearly).

Regulation of accumulation and turnover of an inducible glutamate dehydrogenase in synchronous cultures of Chlorella

Earlier studies indicated that the gene of an ammonium-inducible glutamate dehydrogenase (GDH) was inducible throughout the cell cycle and was expressible shortly after replication early in the S-phase in synchronous Chlorella cells growing at a rate of 13% per h in the absence of inducer. In the present study, synchronous cells cultured at the same growth rate in the continuous presence of inducer accumulated this enzyme in a linear manner, with a positive rate change observed late instead of early in the S-phase. At a growth rate of 26% per h, the positive rate change appeared to be displaced to 1.5 h before the S-phase in the next cell cycle. With 2'-deoxyadenosine, an in vivo inhibitor of deoxyribonucleic acid (DNA) synthesis, the magnitude of the positive rate change was shown to be proportional to the relative increase in DNA in the previous cell cycle. Collectively, these data support the idea that expression of newly replicated genes of this enzyme can be delayed into the subsequent cell cycle in cells in the continuous presence of inducer. Studies with cycloheximide indicated that the inducible GDH and another GDH isozyme were stable in fully induced cells in the absence of protein synthesis. However, after ammonium was removed from the culture medium, the activity of the inducible GDH decreased rapidly in vivo, with a half-time of 5 to 10 min at 38.5 degrees C, whereas the rate of accumulation of the other GDH isozyme did not change. Addition of cycloheximide, at the time of inducer removal, prevented this loss in activity of the inducible GDH. The inability to rescue the activity of the inducible GDH, by readdition of ammonium during the deinduction period, indicates that this enzyme probably underwent irreversible inactivation and/or proteolytic degradation.

Evidence for a new kind of regulatory gene controlling expression of genes for morphogenesis during the cell cycle in Ustilago violacea

The first part of the paper provides strong supportive evidence for the previous findings (Cummins & Day, 1973; Day & Cummins, 1973) that the two alleles of the mating-type locus of the basidiomyceteUstilago violaceahave different periods of inducibility during a cell cycle, and that the cell cycle characteristics of each allele are maintained in freshly isolated diploids. This difference in temporal properties of the alleles appears to be the basis of the dominance of allele a2as it is inducible during a phase of the cell cycle when allele a1is non-inducible. During G1both alleles appear to be inducible and apparently ‘neutralize’ each other so that the cell cannot mate.

The second part of the paper provides evidence for a unique genetic control mechanism. The evidence suggests that the period of cell cycle inducibility of a locus governing a morphogenetic pathway may be regulated by a separate control gene thecclocus, with two known allelesccstr(a stringent or restricted period of inducibility) andccrel(a relaxed or non-restricted period of inducibility). This hypothesis stems from analysis of a diploid that wasa1·ccstr/a2·ccreland showed dominance of allelea2during the S and G2phases when freshly isolated, but which became incapable of mating after a period of subculturing. Analysis of haploids derived from this diploid strain showed that both mating-type alleles were functional but that it was now homozygous forccstr, i.e. of genotypea1·ccstr/a2·ccstr· Thus the temporal and functional aspects of the mating type alleles are determined by different loci. It is postulated that cell cycle control loci may be widespread and serve to regulate the action of genes concerned with morphogenesis in relation to other cell cycle events.