Oscillations and temporal signalling in cells

The development of new techniques to quantitatively measure gene expression in cells has shed light on a number of systems that display oscillations in protein concentration. Here we review the different mechanisms which can produce oscillations in gene expression or protein concentration using a framework of simple mathematical models. We focus on three eukaryotic genetic regulatory networks which show 'ultradian' oscillations, with a time period of the order of hours, and involve, respectively, proteins important for development (Hes1), apoptosis (p53) and immune response (NF-kappaB). We argue that underlying all three is a common design consisting of a negative feedback loop with time delay which is responsible for the oscillatory behaviour.     

Roles of protein ubiquitination and degradation kinetics in biological oscillations

Protein ubiquitination and degradation play important roles in many biological functions and are associated with many human diseases. It is well known that for biochemical oscillations to occur, proper degradation rates of the participating proteins are needed. In most mathematical models of biochemical reactions, linear degradation kinetics has been used. However, the degradation kinetics in real systems may be nonlinear, and how nonlinear degradation kinetics affects biological oscillations are not well understood. In this study, we first develop a biochemical reaction model of protein ubiquitination and degradation and calculate the degradation rate against the concentration of the free substrate. We show that the protein degradation kinetics mainly follows the Michaelis-Menten formulation with a time delay caused by ubiquitination and deubiquitination. We then study analytically how the Michaelis-Menten degradation kinetics affects the instabilities that lead to oscillations using three generic oscillation models: 1) a positive feedback mediated oscillator; 2) a positive-plus-negative feedback mediated oscillator; and 3) a negative feedback mediated oscillator. In all three cases, nonlinear degradation kinetics promotes oscillations, especially for the negative feedback mediated oscillator, resulting in much larger oscillation amplitudes and slower frequencies than those observed with linear kinetics. However, the time delay due to protein ubiquitination and deubiquitination generally suppresses oscillations, reducing the amplitude and increasing the frequency of the oscillations. These theoretical analyses provide mechanistic insights into the effects of specific proteins in the ubiquitination-proteasome system on biological oscillations.     

Stochastic Models of Gene Expression with Delayed Degradation

In many biochemical reactions occurring in living cells, number of various molecules might be low which results in significant stochastic fluctuations. In addition, most reactions are not instantaneous, there exist natural time delays in the evolution of cell states. It is a challenge to develop a systematic and rigorous treatment of stochastic dynamics with time delays and to investigate combined effects of stochasticity and delays in concrete models.We propose a new methodology to deal with time delays in biological systems and apply it to simple models of gene expression with delayed degradation. We show that time delay of protein degradation does not cause oscillations as it was recently argued. It follows from our rigorous analysis that one should look for different mechanisms responsible for oscillations observed in biological experiments.We develop a systematic analytical treatment of stochastic models of time delays. Specifically we take into account that some reactions, for example degradation, are consuming, that is: once molecules start to degrade they cannot be part in other degradation processes.We introduce an auxiliary stochastic process and calculate analytically the variance and the autocorrelation function of the number of protein molecules in stationary states in basic models of delayed protein degradation.     

Intrinsic noise and Hill dynamics in the p53 system

The p53 feedback loop can induce cellular senescence, cell cycle arrest and apoptosis in response to various stresses, including DNA damage, hypoxia and nutrient deprivation. Using a stochastic model of the negative feedback circuit involving p53 and its inhibitor Mdm2, we present the different oscillatory dynamics at the single-cell and population-cell levels as described in the experiments, and the resonant nature of the oscillations is captured. The stationary distributions of protein populations are characterized by non-Gaussian nature which is due to the interplay between time delay and nonlinearity of reactions.     

MAPK cascade possesses decoupled controllability of signal amplification and duration

The three important characteristics of the output signal of mitogen activated protein kinase (MAPK) cascade are time delay between stimulus and response, amplitude gain, and duration of the output signal. In this study, we performed a sensitivity analysis on the computational model of epidermal growth factor receptor (EGFR) activated MAPK cascade developed by Schoeberl and co-workers (1) to identify the sensitive steps of the pathway affecting these characteristics. We show that the signaling network is sensitive in a decoupled manner, which provides the ability to control its output amplitude and duration one at a time. Signal duration is found sensitive only to the phosphatase reactions at the MEK level. In contrast, signal amplitude is found most sensitive to the phosphatase reactions at the ERK level. Time delay is found to be a robust characteristic of the system.     

Oscillatory phosphorylation of yeast Fus3 MAP kinase controls periodic gene expression and morphogenesis

Signal-transduction networks can display complex dynamic behavior such as oscillations in the activity of key components [1-6], but it is often unclear whether such dynamic complexity is actually important for the network's regulatory functions [7, 8]. Here, we found that the mitogen-activated protein kinase (MAPK) Fus3, a key regulator of the yeast mating-pheromone response, undergoes sustained oscillations in its phosphorylation and activation state during continuous pheromone exposure. These MAPK activity oscillations led to corresponding oscillations in mating-gene expression. Oscillations in MAPK activity and gene expression required the negative regulator of G protein signaling Sst2 and partially required the MAPK phosphatase Msg5. Peaks in Fus3 activation correlated with periodic rounds of cell morphogenesis, with each peak preceding the formation of an additional mating projection. Preventing projection formation did not eliminate MAPK oscillation, but preventing MAPK oscillation blocked the formation of additional projections. A mathematical model was developed that reproduced several features of the observed oscillatory dynamics. These observations demonstrate a role for MAPK activity oscillation in driving a periodic downstream response and explain how the pheromone signaling pathway, previously thought to desensitize after 1-3 hr, controls morphology changes that continue for a much longer time.     

Reduction theories elucidate the origins of complex biological rhythms generated by interacting delay-induced oscillations

Time delay is known to induce sustained oscillations in many biological systems such as electroencephalogram (EEG) activities and gene regulations. Furthermore, interactions among delay-induced oscillations can generate complex collective rhythms, which play important functional roles. However, due to their intrinsic infinite dimensionality, theoretical analysis of interacting delay-induced oscillations has been limited. Here, we show that the two primary methods for finite-dimensional limit cycles, namely, the center manifold reduction in the vicinity of the Hopf bifurcation and the phase reduction for weak interactions, can successfully be applied to interacting infinite-dimensional delay-induced oscillations. We systematically derive the complex Ginzburg-Landau equation and the phase equation without delay for general interaction networks. Based on the reduced low-dimensional equations, we demonstrate that diffusive (linearly attractive) coupling between a pair of delay-induced oscillations can exhibit nontrivial amplitude death and multimodal phase locking. Our analysis provides unique insights into experimentally observed EEG activities such as sudden transitions among different phase-locked states and occurrence of epileptic seizures.     

Promoters Architecture-Based Mechanism for Noise-Induced Oscillations in a Single-Gene Circuit

It is well known that single-gene circuits with negative feedback loop can lead to oscillatory gene expression when they operate with time delay. In order to generate these oscillations many processes can contribute to properly timing such delay. Here we show that the time delay coming from the transitions between internal states of the cis-regulatory system (CRS) can drive sustained oscillations in an auto-repressive single-gene circuit operating in a small volume like a cell. We found that the cooperative binding of repressor molecules is not mandatory for a oscillatory behavior if there are enough binding sites in the CRS. These oscillations depend on an adequate balance between the CRS kinetic, and the synthesis/degradation rates of repressor molecules. This finding suggest that the multi-site CRS architecture can play a key role for oscillatory behavior of gene expression. Finally, our results can also help to synthetic biologists on the design of the promoters architecture for new genetic oscillatory circuits.     

Machines vs. Ensembles: Effective MAPK Signaling through Heterogeneous Sets of Protein Complexes

Despite the importance of intracellular signaling networks, there is currently no consensus regarding the fundamental nature of the protein complexes such networks employ. One prominent view involves stable signaling machines with well-defined quaternary structures. The combinatorial complexity of signaling networks has led to an opposing perspective, namely that signaling proceeds via heterogeneous pleiomorphic ensembles of transient complexes. Since many hypotheses regarding network function rely on how we conceptualize signaling complexes, resolving this issue is a central problem in systems biology. Unfortunately, direct experimental characterization of these complexes has proven technologically difficult, while combinatorial complexity has prevented traditional modeling methods from approaching this question. Here we employ rule-based modeling, a technique that overcomes these limitations, to construct a model of the yeast pheromone signaling network. We found that this model exhibits significant ensemble character while generating reliable responses that match experimental observations. To contrast the ensemble behavior, we constructed a model that employs hierarchical assembly pathways to produce scaffold-based signaling machines. We found that this machine model could not replicate the experimentally observed combinatorial inhibition that arises when the scaffold is overexpressed. This finding provides evidence against the hierarchical assembly of machines in the pheromone signaling network and suggests that machines and ensembles may serve distinct purposes in vivo. In some cases, e.g. core enzymatic activities like protein synthesis and degradation, machines assembled via hierarchical energy landscapes may provide functional stability for the cell. In other cases, such as signaling, ensembles may represent a form of weak linkage, facilitating variation and plasticity in network evolution. The capacity of ensembles to signal effectively will ultimately shape how we conceptualize the function, evolution and engineering of signaling networks.