Reversible phosphorylation subserves robust circadian rhythms by creating a switch in inactivating the positive element

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.

Control of tissue homeostasis, tumorigenesis, and degeneration by coupled bidirectional bistable switches

The Hippo-YAP/TAZ signaling pathway plays a critical role in tissue homeostasis, tumorigenesis, and degeneration disorders. The regulation of YAP/TAZ levels is controlled by a complex regulatory network, where several feedback loops have been identified. However, it remains elusive how these feedback loops contain the YAP/TAZ levels and maintain the system in a healthy physiological state or trap the system in pathological conditions. Here, a mathematical model was developed to represent the YAP/TAZ regulatory network. Through theoretical analyses, three distinct states that designate the one physiological and two pathological outcomes were found. The transition from the physiological state to the two pathological states is mechanistically controlled by coupled bidirectional bistable switches, which are robust to parametric variation and stochastic fluctuations at the molecular level. This work provides a mechanistic understanding of the regulation and dysregulation of YAP/TAZ levels in tissue state transitions.

Tissue development and homeostasis require well-controlled cell proliferation. Lack of this control could lead to degenerative or tumorigenic diseases. Signaling pathways have been explored in promoting or inhibiting these diseases. The Hippo signaling pathway is one of these, which has been found to control tissue homeostasis and organ size through cell proliferation and apoptosis, as evidenced by extensive experimental data. However, the question remains of how tissue can transition from a homeostatic state to either a degenerative or tumorigenic state. By theoretically analyzing a mathematical model of its regulatory network, we present a mechanism that underlies Hippo signaling to control tissue transition from a homeostatic state to a disease state. This provides us with a mechanistic understanding of how the parts of the regulatory network are coordinated for the transitions between the homeostasis state and the disease states. In addition, we looked at the role of system noise and found that it could promote the transition to one of the disease states. Our model allows for experimental hypotheses to be generated and could lead to the development of therapeutic strategies by targeting the Hippo signaling pathway.

RECEPTOR-DEPENDENT SENSITIVITY OF NF-κB TO LOW PHYSIOLOGICAL LEVEL

In the NFκB signaling pathway, cells respond to different concentrations of the TNFα signal by means of NFκB transcription factors. Previous studies showed that most cells are activated under high-dose stimulations and NFκB activation results in oscillations in nuclear NFκB abundance. Here, by analyzing sensitivity gain for the response of the nuclear NFκB to the number of cell-surface receptors under low-dose stimulations, we show that changes in the receptor number can give rise to significant changes in the nonsaturation part of the dose–response curve, where the receptor activation rates are very sensitive to stimulations. In addition, the number of the activated receptors tends to increase in a large range of stimulation dose and can significantly influence the expression of the downstream genes. These results imply that the number of cell-surface receptors plays a role of information encoding like frequency or amplitude encoding described in previous studies.

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.

Stochastic pulse regulation in bacterial stress response

Gene regulatory circuits can use dynamic, and even stochastic, strategies to respond to environmental conditions. We examined activation of the general stress response mediated by the alternative sigma factor, σ(B), in individual Bacillus subtilis cells. We observed that energy stress activates σ(B) in discrete stochastic pulses, with increasing levels of stress leading to higher pulse frequencies. By perturbing and rewiring the endogenous system, we found that this behavior results from three key features of the σ(B) circuit: an ultrasensitive phosphorylation switch; stochasticity ("noise"), which activates that switch; and a mixed (positive and negative) transcriptional feedback, which can both amplify a pulse and switch it off. Together, these results show how prokaryotes encode signals using stochastic pulse frequency modulation through a compact regulatory architecture.