A theoretical exploration of birhythmicity in the p53-Mdm2 network

Experimental observations performed in the p53-Mdm2 network, one of the key protein modules involved in the control of proliferation of abnormal cells in mammals, revealed the existence of two frequencies of oscillations of p53 and Mdm2 in irradiated cells depending on the irradiation dose. These observations raised the question of the existence of birhythmicity, i.e. the coexistence of two oscillatory regimes for the same external conditions, in the p53-Mdm2 network which would be at the origin of these two distinct frequencies. A theoretical answer has been recently suggested by Ouattara, Abou-Jaoudé and Kaufman who proposed a 3-dimensional differential model showing birhythmicity to reproduce the two frequencies experimentally observed. The aim of this work is to analyze the mechanisms at the origin of the birhythmic behavior through a theoretical analysis of this differential model. To do so, we reduced this model, in a first step, into a 3-dimensional piecewise linear differential model where the Hill functions have been approximated by step functions, and, in a second step, into a 2-dimensional piecewise linear differential model by setting one autonomous variable as a constant in each domain of the phase space. We find that two features related to the phase space structure of the system are at the origin of the birhythmic behavior: the existence of two embedded cycles in the transition graph of the reduced models; the presence of a bypass in the orbit of the large amplitude oscillatory regime of low frequency. Based on this analysis, an experimental strategy is proposed to test the existence of birhythmicity in the p53-Mdm2 network. From a methodological point of view, this approach greatly facilitates the computational analysis of complex oscillatory behavior and could represent a valuable tool to explore mathematical models of biological rhythms showing sufficiently steep nonlinearities.     

The Dynamics of Stress p53-Mdm2 Network Regulated by p300 and HDAC1

We construct a stress p53-Mdm2-p300-HDAC1 regulatory network that is activated and stabilised by two regulatory proteins, p300 and HDAC1. Different activation levels of observed due to these regulators during stress condition have been investigated using a deterministic as well as a stochastic approach to understand how the cell responds during stress conditions. We found that these regulators help in adjusting p53 to different conditions as identified by various oscillatory states, namely fixed point oscillations, damped oscillations and sustain oscillations. On assessing the impact of p300 on p53-Mdm2 network we identified three states: first stabilised or normal condition where the impact of p300 is negligible, second an interim region where p53 is activated due to interaction between p53 and p300, and finally the third regime where excess of p300 leads to cell stress condition. Similarly evaluation of HDAC1 on our model led to identification of the above three distinct states. Also we observe that noise in stochastic cellular system helps to reach each oscillatory state quicker than those in deterministic case. The constructed model validated different experimental findings qualitatively.     

Emergence of long‐term balanced polymorphism under cyclic selection of spatially variable magnitude

A fundamental question in evolutionary biology is what promotes genetic variation at nonneutral loci, a major precursor to adaptation in changing environments. In particular, balanced polymorphism under realistic evolutionary models of temporally varying environments in finite natural populations remains to be demonstrated. Here, we propose a novel mechanism of balancing selection under temporally varying fitnesses. Using forward‐in‐time computer simulations and mathematical analysis, we show that cyclic selection that spatially varies in magnitude, such as along an environmental gradient, can lead to elevated levels of nonneutral genetic polymorphism in finite populations. Balanced polymorphism is more likely with an increase in gene flow, magnitude and period of fitness oscillations, and spatial heterogeneity. This polymorphism‐promoting effect is robust to small systematic fitness differences between competing alleles or to random environmental perturbation. Furthermore, we demonstrate analytically that protected polymorphism arises as spatially heterogeneous cyclic fitness oscillations generate a type of storage effect that leads to negative frequency dependent selection. Our findings imply that spatially variable cyclic environments can promote elevated levels of nonneutral genetic variation in natural populations.     

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.     

Hopf bifurcation in the presomitic mesoderm during the mouse segmentation

Vertebrae and ribs arise from embryonic tissues called somites. Somites arise sequentially from the unsegmented embryo tail, called presomitic mesoderm (PSM). The pace of somite formation is controlled by gene products such as hairy and enhancer of split 7 (Hes7) whose expression oscillates in the PSM. In addition to the cyclic genes, there is a gradient of fibroblast growth factor 8 (Fgf8) mRNA from posterior to anterior PSM. Recent experiments have shown that in the absence of Fgf signaling, Hes7 oscillations in the anterior and posterior PSM are lost. On the other hand, Notch mutants reduce the amplitude of posterior Hes7 oscillations and abolish anterior Hes7 oscillations. To understand these phenotypes, we delineated and simulated a logical and a delay differential equation (DDE) model with similar network topology in wild-type and mutant situations. Both models reproduced most wild-type and mutant phenotypes suggesting that the chosen topology is robust to explain these phenotypes. Numerical continuation of the model showed that even in the wild-type situation, the system changed from sustained to damped, i.e. a Hopf bifurcation occurred, when the Fgf concentration decreased in the PSM. This numerical continuation analysis further indicated that the most sensitive parameters for the oscillations are the parameters of Hes7 followed by those of Lunatic fringe (Lfng) and Notch1. In the wild-type, the damping of Hes7 oscillations was not so strong so that cells reached the new somites before they lose Hes7 oscillations. By contrast, in the fibroblast growth factor receptor 1 (Fgfr1) conditional knock-out (cKO) mutant simulation, Notch signaling was not able to maintain sustained Hes7 oscillations. Our analysis suggests that Fgf signaling makes cells enter an oscillatory state of Hes7 expression. After moving to the anterior PSM, where Fgf signaling is missing, Notch signaling compensates the damping of Hes7 oscillations in the anterior PSM.     

A theoretical model for p53 dynamics: Identifying optimal therapeutic strategy for its activation and stabilization

The activation and stabilization of tumor suppressor p53 are very important in preventing cells from becoming cancerous. Hence, many experimental works have been carried out to investigate p53’s dynamics through its interactions with other proteins and its therapeutic applications for the treatment of cancers. In this work, by analyzing a theoretical model, we attempt to search for an optimal therapeutic strategy that guarantees the activation and stabilization of p53. For this purpose, we introduce a new mathematical model including oncogene activation and ARF, which are recognized as crucial for tumor suppression but have not yet been considered in most theoretical works. Through mathematical modeling and numerical simulations, we confirm several important properties of p53 dynamics: the role of the oncogene-mediated activation of ARF as an important factor for the activation and stabilization of p53, the necessity of time delays in negative feedback loops to guarantee sustained p53 oscillations, and the digital behavior of p53 pulses. Furthermore, we propose that the binding of ARF to Mdm2 and enhancing the degradation of Mdm2 is an efficient strategy for therapeutic targeting, which may assure the activation and stabilization of p53.     

p53-Mdm2 loop controlled by a balance of its feedback strength and effective dampening using ATM and delayed feedback

When the genomic integrity of a cell is challenged, its fate is determined in part by signals conveyed by the p53 tumour suppressor protein. It was observed recently that such signals are not simple gradations of p53 concentration, but rather a counter-intuitive limit-cycle behaviour. Based on a careful mathematical interpretation of the experimental body of knowledge, we propose a model for the p53 signalling network and characterise the p53 stability and oscillatory dynamics. In our model, ATM, a protein that senses DNA damage, activates p53 by phosphorylation. In its active state, p53 has a decreased degradation rate and an enhanced transactivation of Mdm2, a gene whose protein product Mdm2 tags p53 for degradation. Thus the p53-Mdm2 system forms a negative feedback loop. However, the feedback in this loop is delayed, as the pool of Mdm2 molecules being induced by p53 at a given time will mark for degradation the pool of p53 molecules at some later time, after the Mdm2 molecules have been transcribed, exported out of the nucleus, translated and transported back into the nucleus. The analysis of our model demonstrates how this time lag combines with the ATM-controlled feedback strength and effective dampening of the negative feedback loop to produce limit-cycle oscillations. The picture that emerges is that ATM, once activated by DNA damage, makes the p53-Mdm2 oscillator undergo a supercritical Hopf bifurcation. This approach yields an improved understanding of the global dynamics and bifurcation structure of our time-delayed, negative feedback model and allows for predictions of the behaviour of the p53 system under different perturbations.     

Analysis of a minimal model for p53 oscillations

Oscillatory behaviours in genetic networks are important examples for studying the principles underlying the dynamics of cellular regulation. Recently the team of Alon has reported a surprisingly rich oscillatory response of the p53 tumor suppressor to irradiation stress et al. [Lahav, G., Rosenfeld, N., Sigal, A., Geva-Zatorsky, N., Levine, A.J., Elowitz, M.B., Alon, U., 2004. Dynamics of the p53-Mdm2 feedback loop in individual cells. Nat. Genet. 36 (2), 147-150; Geva-Zatorsky, N., Rosenfeld, N., Itzkovitz, S., Milo, R., Sigal, A., Dekel, E., Yarnitzky, T., Liron, Y., Polak, P., Lahav, G., Alon, U., 2006. Oscillations and variability in the p53 system. Mol. Syst. Biol. 2, 2006.0033]. Several models for this system have been proposed by different groups, based essentially on negative feedback loops. In this paper we investigate in detail oscillations and stability in a deterministic time delayed differential model of the core circuit for p53 expression. This model is representative of a class of modelling approaches of this system, based on a "minimal" set of well-established biomolecular regulations. Depending on the protein degradation rates we show the existence of bifurcations between a stable steady state and oscillations both in presence and absence of stress.     

Novel recurrent neural network for modelling biological networks: Oscillatory p53 interaction dynamics

Understanding the control of cellular networks consisting of gene and protein interactions and their emergent properties is a central activity of Systems Biology research. For this, continuous, discrete, hybrid, and stochastic methods have been proposed. Currently, the most common approach to modelling accurate temporal dynamics of networks is ordinary differential equations (ODE). However, critical limitations of ODE models are difficulty in kinetic parameter estimation and numerical solution of a large number of equations, making them more suited to smaller systems. In this article, we introduce a novel recurrent artificial neural network (RNN) that addresses above limitations and produces a continuous model that easily estimates parameters from data, can handle a large number of molecular interactions and quantifies temporal dynamics and emergent systems properties. This RNN is based on a system of ODEs representing molecular interactions in a signalling network. Each neuron represents concentration change of one molecule represented by an ODE. Weights of the RNN correspond to kinetic parameters in the system and can be adjusted incrementally during network training. The method is applied to the p53-Mdm2 oscillation system – a crucial component of the DNA damage response pathways activated by a damage signal. Simulation results indicate that the proposed RNN can successfully represent the behaviour of the p53-Mdm2 oscillation system and solve the parameter estimation problem with high accuracy. Furthermore, we presented a modified form of the RNN that estimates parameters and captures systems dynamics from sparse data collected over relatively large time steps. We also investigate the robustness of the p53-Mdm2 system using the trained RNN under various levels of parameter perturbation to gain a greater understanding of the control of the p53-Mdm2 system. Its outcomes on robustness are consistent with the current biological knowledge of this system. As more quantitative data become available on individual proteins, the RNN would be able to refine parameter estimation and mapping of temporal dynamics of individual signalling molecules as well as signalling networks as a system. Moreover, RNN can be used to modularise large signalling networks.     

A spatio-temporal model of Notch signalling in the zebrafish segmentation clock: conditions for synchronised oscillatory dynamics

In the vertebrate embryo, tissue blocks called somites are laid down in head-to-tail succession, a process known as somitogenesis. Research into somitogenesis has been both experimental and mathematical. For zebrafish, there is experimental evidence for oscillatory gene expression in cells in the presomitic mesoderm (PSM) as well as evidence that Notch signalling synchronises the oscillations in neighbouring PSM cells. A biological mechanism has previously been proposed to explain these phenomena. Here we have converted this mechanism into a mathematical model of partial differential equations in which the nuclear and cytoplasmic diffusion of protein and mRNA molecules is explicitly considered. By performing simulations, we have found ranges of values for the model parameters (such as diffusion and degradation rates) that yield oscillatory dynamics within PSM cells and that enable Notch signalling to synchronise the oscillations in two touching cells. Our model contains a Hill coefficient that measures the co-operativity between two proteins (Her1, Her7) and three genes (her1, her7, deltaC) which they inhibit. This coefficient appears to be bounded below by the requirement for oscillations in individual cells and bounded above by the requirement for synchronisation. Consistent with experimental data and a previous spatially non-explicit mathematical model, we have found that signalling can increase the average level of Her1 protein. Biological pattern formation would be impossible without a certain robustness to variety in cell shape and size; our results possess such robustness. Our spatially-explicit modelling approach, together with new imaging technologies that can measure intracellular protein diffusion rates, is likely to yield significant new insight into somitogenesis and other biological processes.     

Delay Hill dynamics in regulatory biological systems

We explore one of the best-studied protein circuits in human cells, the negative feedback loop between the tumor suppressor p53 and the oncogene Mdm2 following nuclear irradiation. Using stochastic delay differential equations and the Gillespie algorithm, we illustrate the distinct oscillatory dynamics at the single-cell and population-cell levels which were found in the recent experiments. The oscillatory dynamics of p53-Mdm2 interaction appears as coherent resonance with delay and noise in individual cells. Dephasing mechanisms provide the origin of damped oscillation at the population level out of the sustained one at the single-cell level. The non-Gaussian nature of distributions of protein populations results from the interplay between time delay and nonlinearity of reaction processes. Our findings may lead to new insights related to the effects of noise and cancer therapy.     

A dynamic role of HAUSP in the p53-Mdm2 pathway

Our previous study showed that ubiquitination of p53 is reversible and that the ubiquitin hydrolase HAUSP can stabilize p53 by deubiquitination. Here, we found that partial reduction of endogenous HAUSP levels by RNAi indeed destabilizes endogenous p53; surprisingly, however, nearly complete ablation of HAUSP stabilizes and activates p53. We further show that this phenomenon occurs because HAUSP stabilizes Mdm2 in a p53-independent manner, providing an interesting feedback loop in p53 regulation. Notably, HAUSP is required for Mdm2 stability in normal cells; in HAUSP-ablated cells, self-ubiquitinated-Mdm2 becomes extremely unstable, leading to indirect p53 activation. Furthermore, this feedback regulation is specific to Mdm2; in HeLa cells, where p53 is preferentially degraded by viral E6-dependent ubiquitination, depletion of HAUSP fails to activate p53. This study provides an example of an ubiquitin ligase (Mdm2) that is directly regulated by a deubiquitinase (HAUSP) and also reveals a dynamic role of HAUSP in the p53-Mdm2 pathway.