Periodic enzyme synthesis: reconsideration of the theory of oscillatory repression

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.

The Goodwin model revisited: Hopf bifurcation, limit-cycle, and periodic entrainment

The three-variable Goodwin oscillator is a minimal model demonstrating the emergence of oscillations in simple biochemical feedback systems. As a prototypical oscillator, this model was extensively studied from a theoretical point of view and applied to various biological systems, including circadian clocks. Here, we reexamine this model, derive analytically the amplitude equation near the Hopf bifurcation and investigate the effect of a periodic modulation of the oscillator. In particular, we compare the entrainment performance when the free oscillator displays either self-sustained or damped oscillations. We discuss the results in the context of circadian oscillators.

The Goodwin model: behind the Hill function

The Goodwin model is a 3-variable model demonstrating the emergence of oscillations in a delayed negative feedback-based system at the molecular level. This prototypical model and its variants have been commonly used to model circadian and other genetic oscillators in biology. The only source of non-linearity in this model is a Hill function, characterizing the repression process. It was mathematically shown that to obtain limit-cycle oscillations, the Hill coefficient must be larger than 8, a value often considered unrealistic. It is indeed difficult to explain such a high coefficient with simple cooperative dynamics. We present here molecular models of the standard Goodwin model, based on single or multisite phosphorylation/dephosphorylation processes of a transcription factor, which have been previously shown to generate switch-like responses. We show that when the phosphorylation/dephosphorylation processes are fast enough, the limit-cycle obtained with a multisite phosphorylation-based mechanism is in very good quantitative agreement with the oscillations observed in the Goodwin model. Conditions in which the detailed mechanism is well approximated by the Goodwin model are given. A variant of the Goodwin model which displays sharp thresholds and relaxation oscillations is also explained by a double phosphorylation/dephosphorylation-based mechanism through a bistable behavior. These results not only provide rational support for the Goodwin model but also highlight the crucial role of the speed of post-translational processes, whose response curve are usually established at a steady state, in biochemical oscillators.

Self-regulation in a minimal model of chemical self-replication

In biological systems, regulation plays an important role in keeping metabolite concentrations within physiological ranges. To study the dynamical implications of self-regulation, we consider a functional form used in genetic networks and couple it to a mechanism associated with chemical self-replication. For the two-variable minimal model, we find that activation can yield chemical toggles similar to those reported for gene repression in E. coli as well as more complex dynamics.

Modelling dynamic processes in yeast

Yeast molecular and cell biology has accumulated large amounts of qualitative and quantitative data of diverse cellular processes. The results are often summarized as verbal or graphical descriptions. Moreover, a series of mathematical models has been developed that should help to interpret such data, to integrate them into a coherent picture and to allow for an understanding of the underlying processes. Dynamic modelling of regulatory processes in yeast focuses on central carbon metabolism, on a number of selected signalling pathways and on cell cycle regulation. These models can explain questions of general relevance, such as whether the dynamics of a network can be understood from the combination of in vitro kinetics of its individual reactions. They help to elucidate complicated dynamic features, such as glycolytic oscillations, effects of feedback regulation or the optimal regulation of gene expression. The availability of comprehensive qualitative information, such as protein interactions or pathway composition, and sets of quantitative data make yeast a perfect model organism. Therefore, yeast-related data are often used to develop and examine computational approaches and modelling methods.

A systems- and signal-oriented approach to intracellular dynamics

A mathematical understanding of regulation, and, in particular, the role of feedback, has been central to the advance of the physical sciences and technology. In this article, the framework provided by systems biology is used to argue that the same can be true for molecular biology. In particular, and using basic modular methods of mathematical modelling which are standard in control theory, a set of dynamic models is developed for some illustrative cell signalling processes. These models, supported by recent experimental evidence, are used to argue that a control theoretical approach to the mechanisms of feedback in intracellular signalling is central to furthering our understanding of molecular communication. As a specific example, a MAPK (mitogen-activated protein kinase) signalling pathway is used to show how potential feedback mechanisms in the signalling process can be investigated in a simulated environment. Such ‘what if’ modelling/simulation studies have been an integral part of physical science research for many years. Using tools of control systems analysis, as embodied in the disciplines of systems biology, similar predictive modelling/simulation studies are now bearing fruit in cell signalling research.

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.

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).