Mathematical modelling of p53 basal dynamics and DNA damage response

Basal State Calibration of a Chemical Reaction Network Model for Autophagy

The modulation of autophagy plays a dual role in tumor cells, with the potential to both promote and suppress tumor proliferation. In order to gain a deeper understanding of the nature of autophagy, we have developed a chemical reaction kinetic model of autophagy and apoptosis based on the mass action kinetic models that have been previously described in the literature. It is regrettable that the authors did not provide all of the information necessary to reconstruct their model, which made their simulation results irreproducible. In this study, based on an extensive literature review, we have identified concentrations for each species in the stress-free, homeostatic state. These ranges were randomly sampled to generate sets of initial concentrations, from which the simulations were run. In every case, abnormal behavior was observed, with apoptosis and autophagy being activated, even in the absence of stress. Consequently, the model failed to reproduce even the basal conditions. Detailed examination of the model revealed erroneous reactions, which were corrected. The influential kinetic parameters of the corrected model were identified and optimized using the Optima++ code. The model is now capable of simulating homeostatic states, and provides a suitable basis for further model development to describe cell response to various stresses.

Elucidating the Implications of Diverse Dynamical Responses in p53 Protein

p53 is a well-known tumor suppressor gene that acts as a transcription factor to exhibit a variety of dynamical responses by sensing different types and extent of stress conditions causing DNA damage in Mammalian cells. Mathematical modeling has played a crucial role to correlate cell fate decision-making with some of these dynamic p53 regulations. However, it is extremely challenging to explain the various cell-type and stimulus-specific p53 protein dynamics under different stress conditions by using a single mathematical model. In this article, we propose a simple mathematical model of p53 regulation based on a generic p53 regulatory network that elucidates a range of p53 dynamical responses. By employing bifurcation analysis along with deterministic and stochastic simulations, we explain an array of p53 dynamics by correlating it with the corresponding cell fate regulations in a cell type-specific and stimulus-dependent manner. Moreover, our model makes experimentally testable predictions to fine-tune p53 dynamics under various DNA damage conditions and can be systematically used and improved to analyze complex p53 dynamics in the future.

Encoding and Decoding of p53 Dynamics in Cellular Response to Stresses

In the cellular response to stresses, the tumor suppressor p53 is activated to maintain genomic integrity and fidelity. As a transcription factor, p53 exhibits rich dynamics to allow for discrimination of the type and intensity of stresses and to direct the selective activation of target genes involved in different processes including cell cycle arrest and apoptosis. In this review, we focused on how stresses are encoded into p53 dynamics and how the dynamics are decoded into cellular outcomes. Theoretical modeling may provide a global view of signaling in the p53 network by coupling the encoding and decoding processes. We discussed the significance of modeling in revealing the mechanisms of the transition between p53 dynamic modes. Moreover, we shed light on the crosstalk between the p53 network and other signaling networks. This review may advance the understanding of operating principles of the p53 signaling network comprehensively and provide insights into p53 dynamics-based cancer therapy.

The interplay between DNA damage and autophagy in lung cancer: A mathematical study

The rising mortality in lung cancer, as well as the constraints of the existing drugs, have made it a major research topic. DNA damage marks the early onset of cancer as it often results from vulnerabilities due to UV rays, oxidative stress, ionizing radiation, and various types of genotoxic attacks. p53 plays an unequivocal role in the DNA repair process and has an abiding presence at the crossroads of the pathways linking DNA damage and cancer. p53 also regulates autophagy in a dual manner based on its cellular localization. The plexus of autophagy regulated by p53 includes AMPK and BCL2, which are positive and negative regulators of prime autophagy inducer beclin1, respectively. Although autophagy is a quintessential process, its levels need to be monitored as uncontrolled autophagy may lead to cell death. The association of p53 and autophagic cell death is very vital as the former acts whenever any threat comes to DNA while the latter may play a role in getting rid of the culprit cell. Therefore, in this paper, we have formulated a seven-dimensional mathematical model connecting p53, DNA damage, and autophagy in lung cancer. We performed both local and global sensitivity analysis along with parameter recalibration analysis to understand the system dynamics. We hypothesized that, by the modulation of beclin1 level, the regulation of AMPK and BCL2 could be a possible strategy to mitigate the progression of lung cancer.

MicroRNA-129-2-3p directly targets Wip1 to suppress the proliferation and invasion of intrahepatic cholangiocarcinoma

Accumulated studies showed that numerous microRNAs (miRNAs) were aberrantly expressed in human intrahepatic cholangiocarcinoma (ICC) and contributed to the tumorigenic processes. However, whether miR-129-2-3p is implicated in the ICC initiation and progression is still limited. Here, the results revealed that miR-129-2-3p expression was notably decreased in ICC tissues and cell lines, and that a low miR-129-2-3p expression was obviously associated with distant metastasis and clinical stage. Exogenous miR-129-2-3p expression evidently repressed the proliferative and invasive abilities of ICC cells. Mechanistic studies indicated that Wild-type p53-induced phosphatase 1 (Wip1) was a direct target gene for miR-129-2-3p in ICC cells. Furthermore, silencing Wip1 expression mimicked the suppressive effects of miR-129-2-3p upregulation on ICC cells. Interestingly, reintroduction of Wip1 expression partially abolished the miR-129-2-3p -reduced cell proliferation and invasion in ICC. Moreover, ectopic miR-129-2-3p expression hindered the ICC tumor growth in vivo. To the best of our knowledge, it is the first time to reveal that miR-129-2-3p plays a crucial role in tumor suppression in ICC pathogenesis through directly targeting Wip1. These results will aid in elucidating the roles of miR-129-2-3p in ICC, and suggest that this miRNA may provide a potential target for the treatment of ICC

The effect of stochasticity on repair of DNA double strand breaks throughout non-homologous end joining pathway

DNA double strand breaks (DSBs) are the most lethal lesions of DNA induced by ionizing radiation, industrial chemicals and a wide variety of drugs used in chemotherapy. In the context of DNA damage response system modelling, uncertainty may arise in several ways such as number of induced DSBs, kinetic rates and measurement error in observable quantities. Therefore, using the stochastic approaches is imperative to gain further insight into the dynamic behaviour of DSBs repair process. In this article, a continuous-time Markov chain (CTMC) model of the non-homologous end joining (NHEJ) mechanism is formulated according to the DSB complexity. Additionally, a Metropolis Monte Carlo method is used to perform maximum likelihood estimation of the kinetic rate constants. Here, the effects of fluctuating kinetic rates and DSBs induction rate of the NHEJ mechanism are investigated. The stochastic realizations of the total yield of simple and complex DSBs ligation are simulated to compare their asymptotic dynamics. Furthermore, it has been proved that the total yield of DSBs has a normal distribution for sufficiently large number of DSBs. In order to estimate the expected duration of repairing DSBs, the probability distribution of DSBs lifetime is calculated based on the CTMC NHEJ model. Moreover, the variability of total yield of DSBs during constant low-dose radiation is evaluated in the presented model. The findings indicate that in stochastic NHEJ model, when there is no new DSBs induction through the repair process, all DSBs are eventually repaired. However, when DSBs are induced by constant low-dose radiation, a number of DSBs remains un-repaired.

Mathematical model identifies effective P53 accumulation with target gene binding affinity in DNA damage response for cell fate decision

Functional p53 signaling is essential for appropriate responses to diverse stimuli. P53 dynamics employs the information from the stimulus leading to selective gene expression and cell fate decision. However, the decoding mechanism of p53 dynamics under DNA damage challenge remains poorly understood. Here we mathematically modeled the recently dual-phase p53 dynamics under doxorubicin treatment. We found that p53 could perform sequential pulses followed by a high-amplitude terminal pulse at relatively low doxorubicin treatment, whereas p53 became steadily accumulated when damage level was high. The effective p53 integral above a threshold but not the absolute accumulation of p53 precisely discriminated survival and death. Silencing negative regulators in p53 network might promote the occurrence of terminal pulse. Furthermore, lower binding affinity and degradation rate of p53 target genes could favorably discriminate high and low dose doxorubicin treatment. Grouping by temporal profiles suggested that the p53 dynamics rather than the doxorubicin doses could better discriminate cellular outcomes and confer less variation for effective p53 integral reemphasizing the importance of p53 level regulation. Our model has established a theoretical framework that p53 dynamics can work cooperatively with its binding affinity to target genes leading to cell fate choice, providing new clues of optimized clinical design by manipulating p53 dynamics.

Modeling hepatitis C virus protein and p53 interactions in hepatocytes: Implications for carcinogenesis

Hepatitis C virus (HCV) infection has reached epidemic proportions worldwide. Individuals with chronic HCV infection and without access to treatment are at high risk for developing hepatocellular carcinoma (HCC), a liver cancer that is rapidly fatal after diagnosis. A number of factors have been identified that contribute to HCV-driven carcinogenesis such as scarring of the liver, and chronic inflammation. Recent evidence indicates a direct role for HCV-encoded proteins themselves in oncogenesis of infected hepatocytes. The viral protein HCV core has been shown to interact directly with the host tumor suppressor protein p53, and to modulate p53-activity in a biphasic manner. Here, biochemically-motivated mathematical models of HCV-p53 interactions are developed to elucidate the mechanisms underlying this phenomenon. We show that by itself, direct interaction between HCV core and p53 is insufficient to recapitulate the experimental data. We postulate the existence of an additional factor, activated by HCV core that inhibits p53 function. We present experimental evidence in support of this hypothesis. The model including this additional factor reproduces the experimental results, validating our assumptions. Finally, we investigate what effect HCV core-p53 interactions could have on the capacity of an infected hepatocyte to repair damage to its DNA. Integrating our model with an existing model of the oscillatory response of p53 to DNA damage predicts a biphasic relationship between HCV core and the transformative potential of infected hepatocytes. In addition to providing mechanistic insights, these results suggest a potential biomarker that could help in identifying those HCV patients most at risk of progression to HCC.

Dynamical analysis of cellular ageing by modeling of gene regulatory network based attractor landscape

Ageing is a natural phenomenon that is inherently complex and remains a mystery. Conceptual model of cellular ageing landscape was proposed for computational studies of ageing. However, there is a lack of quantitative model of cellular ageing landscape. This study aims to investigate the mechanism of cellular ageing in a theoretical model using the framework of Waddington's epigenetic landscape. We construct an ageing gene regulatory network (GRN) consisting of the core cell cycle regulatory genes (including p53). A model parameter (activation rate) is used as a measure of the accumulation of DNA damage. Using the bifurcation diagrams to estimate the parameter values that lead to multi-stability, we obtained a conceptual model for capturing three distinct stable steady states (or attractors) corresponding to homeostasis, cell cycle arrest, and senescence or apoptosis. In addition, we applied a Monte Carlo computational method to quantify the potential landscape, which displays: I) one homeostasis attractor for low accumulation of DNA damage; II) two attractors for cell cycle arrest and senescence (or apoptosis) in response to high accumulation of DNA damage. Using the Waddington's epigenetic landscape framework, the process of ageing can be characterized by state transitions from landscape I to II. By in silico perturbations, we identified the potential landscape of a perturbed network (inactivation of p53), and thereby demonstrated the emergence of a cancer attractor. The simulated dynamics of the perturbed network displays a landscape with four basins of attraction: homeostasis, cell cycle arrest, senescence (or apoptosis) and cancer. Our analysis also showed that for the same perturbed network with low DNA damage, the landscape displays only the homeostasis attractor. The mechanistic model offers theoretical insights that can facilitate discovery of potential strategies for network medicine of ageing-related diseases such as cancer.

Understanding non-linear effects from Hill-type dynamics with application to decoding of p53 signaling

Analytical equations are derived depicting four possible scenarios resulting from pulsed signaling of a system subject to Hill-type dynamics. Pulsed Hill-type dynamics involves the binding of multiple signal molecules to a receptor and occurs e.g., when transcription factor p53 orchestrates cancer prevention, during calcium signaling, and during circadian rhythms. The scenarios involve: (i) enhancement of high-affinity binders compared to low-affinity ones, (ii) slowing reactions involving high-affinity binders, (iii) transfer of the clocking of low-affinity binders from the signal molecule to the products, and (iv) a unique clocking process that produces incremental increases in the activity of high-affinity binders with each signal pulse. In principle, these mostly non-linear effects could control cellular outcomes. An applications to p53 signaling is developed, with binding to most gene promoters identified as category (iii) responses. However, currently unexplained enhancement of high-affinity promoters such as CDKN1a (p21) by pulsed signaling could be an example of (i). In general, provision for all possible scenarios is required in the design of mathematical models incorporating pulsed Hill-type signaling as some aspect.

Control of apoptosis by SMAR1

The nuclear matrix associated protein SMAR1 is sensitive to p53 and acts as a stress inducer as well as a regulator in the p53 regulatory network. Depending on the amount of stress SMAR1 stimulates, it can drive the p53 dynamics in the system to various dynamical states which correspond to various cellular states. The behavior of p53 in these dynamical states is found to be multifractal, due to the mostly long range correlations and large scale fluctuations imparted by stress. This fractal behavior is exhibited in the topological properties of the networks constructed from these dynamical states, and is a signature of self-organization to optimize information flow in the dynamics. The assortativity found in these networks is due to perturbation induced by stress, and indicates that the hubs in the time series play a significant role in stress management. SMAR1 can also regulate apoptosis in the presence of HDAC1, depending on the stress induced by it.

Wip1 is associated with tumorigenity and metastasis through MMP-2 in human intrahepatic cholangiocarcinoma

Wip1 has been shown to correlate with the metastasis/invasion of several tumors. This study was designed to investigate the clinical significance and biological function of Wip1 in intrahepatic cholangiocarcinoma (ICC). The expression of Wip1 was investigated in sixty human ICC biopsy samples by immunohistochemistry. Transient and stable knockdown of Wip1 in two human ICC cells (ICC-9810 and SSP25) were established using short hairpin RNA expression vector. Immunohistochemistry revealed that Wip1 was up-regulated in human ICC tissues (47/60, 78.3%). High levels of Wip1 in human ICC correlated with metastasis to the lymph metastasis (P=0.022). Genetic depletion of Wip1 in ICC cells resulted in significantly inhibited proliferation and invasion compared with controls. Most importantly, Wip1 down-regulation impaired tumor migration capacity of ICC cells in vivo. Subsequent investigations revealed that matrix metalloproteinase-2 (MMP-2) is an important target of Wip1. Consistently, in human ICC tissues, Wip1 level was positively correlated with MMP-2 expression. Taken together, our founding indicates that Wip1 may be a crucial regulator in the tumorigenicity and invasion of human ICC, Wip1 exerts its pro-invasion function at least in part through the MMP-2 signaling pathway, suggesting Wip1 as a potential therapeutic target for ICC.

Excitability in the p53 network mediates robust signaling with tunable activation thresholds in single cells

Cellular signaling systems precisely transmit information in the presence of molecular noise while retaining flexibility to accommodate the needs of individual cells. To understand design principles underlying such versatile signaling, we analyzed the response of the tumor suppressor p53 to varying levels of DNA damage in hundreds of individual cells and observed a switch between distinct signaling modes characterized by isolated pulses and sustained oscillations of p53 accumulation. Guided by dynamic systems theory we show that this requires an excitable network structure comprising positive feedback and provide experimental evidence for its molecular identity. The resulting data-driven model reproduced all features of measured signaling responses and is sufficient to explain their heterogeneity in individual cells. We present evidence that heterogeneity in the levels of the feedback regulator Wip1 sets cell-specific thresholds for p53 activation, providing means to modulate its response through interacting signaling pathways. Our results demonstrate how excitable signaling networks can provide high specificity, sensitivity and robustness while retaining unique possibilities to adjust their function to the physiology of individual cells.

Recent progress and open challenges in modeling p53 dynamics in single cells

In mammalian cells, the tumor suppressor p53 is activated upon a variety of cellular stresses and ensures an appropriate response ranging from arrest and repair to the induction of senescence and apoptosis. Quantitative measurements in individual living cells showed stimulus-dependent dynamics of p53 accumulation upon stress induction. Due to the complexity of the underlying biochemical interactions, mathematical models were indispensable for understanding the topology of the network regulating p53 dynamics. Recent work provides furhter insights into the causes of heterogeneous responses in individual cells, the rewiring of the network in response to different inputs and the role of the downstream processes in determining the cellular fate upon stress.

A system of recurrent neural networks for modularising, parameterising and dynamic analysis of cell signalling networks

In this paper, we show how to extend our previously proposed novel continuous time Recurrent Neural Networks (RNN) approach that retains the advantage of continuous dynamics offered by Ordinary Differential Equations (ODE) while enabling parameter estimation through adaptation, to larger signalling networks using a modular approach. Specifically, the signalling network is decomposed into several sub-models based on important temporal events in the network. Each sub-model is represented by the proposed RNN and trained using data generated from the corresponding ODE model. Trained sub-models are assembled into a whole system RNN which is then subjected to systems dynamics and sensitivity analyses. The concept is illustrated by application to G1/S transition in cell cycle using Iwamoto et al. (2008) ODE model. We decomposed the G1/S network into 3 sub-models: (i) E2F transcription factor release; (ii) E2F and CycE positive feedback loop for elevating cyclin levels; and (iii) E2F and CycA negative feedback to degrade E2F. The trained sub-models accurately represented system dynamics and parameters were in good agreement with the ODE model. The whole system RNN however revealed couple of parameters contributing to compounding errors due to feedback and required refinement to sub-model 2. These related to the reversible reaction between CycE/CDK2 and p27, its inhibitor. The revised whole system RNN model very accurately matched dynamics of the ODE system. Local sensitivity analysis of the whole system model further revealed the most dominant influence of the above two parameters in perturbing G1/S transition, giving support to a recent hypothesis that the release of inhibitor p27 from Cyc/CDK complex triggers cell cycle stage transition. To make the model useful in a practical setting, we modified each RNN sub-model with a time relay switch to facilitate larger interval input data (≈20min) (original model used data for 30s or less) and retrained them that produced parameters and protein concentrations similar to the original RNN system. Results thus demonstrated the reliability of the proposed RNN method for modelling relatively large networks by modularisation for practical settings. Advantages of the method are its ability to represent accurate continuous system dynamics and ease of: parameter estimation through training with data from a practical setting, model analysis (40% faster than ODE), fine tuning parameters when more data are available, sub-model extension when new elements and/or interactions come to light and model expansion with addition of sub-models.

A novel mathematical model of ATM/p53/NF- κB pathways points to the importance of the DDR switch-off mechanisms

Background

Ataxia telangiectasia mutated (ATM) is a detector of double-strand breaks (DSBs) and a crucial component of the DNA damage response (DDR) along with p53 and NF- κB transcription factors and Wip1 phosphatase. Despite the recent advances in studying the DDR, the mechanisms of cell fate determination after DNA damage induction is still poorly understood.

Results

To investigate the importance of various DDR elements with particular emphasis on Wip1, we developed a novel mathematical model of ATM/p53/NF- κB pathways. Our results from in silico and in vitro experiments performed on U2-OS cells with Wip1 silenced to 25 % (Wip1-RNAi) revealed a strong dependence of cellular response to DNA damages on this phosphatase. Notably, Wip1-RNAi cells exhibited lower resistance to ionizing radiation (IR) resulting in smaller clonogenicity and higher apoptotic fraction.

Conclusions

In this article, we demonstrated that Wip1 plays a role as a gatekeeper of apoptosis and influences the pro-survival behaviour of cells – the level of Wip1 increases to block the apoptotic decision when DNA repair is successful. Moreover, we were able to verify the dynamics of proteins and transcripts, apoptotic fractions and cells viability obtained from stochastic simulations using in vitro approaches. Taken together, we demonstrated that the model can be successfully used in prediction of cellular behaviour after exposure to IR. Thus, our studies may provide further insights into key elements involved in the underlying mechanisms of the DDR.

Electronic supplementary material

The online version of this article (doi:10.1186/s12918-016-0293-0) contains supplementary material, which is available to authorized users.

Bifurcation in Cell Cycle Dynamics Regulated by p53

We study the regulating mechanism of p53 on the properties of cell cycle dynamics in the light of the proposed model of interacting p53 and cell cycle networks via p53. Irradiation (IR) introduce to p53 compel p53 dynamics to suffer different phases, namely oscillating and oscillation death (stabilized) phases. The IR induced p53 dynamics undergo collapse of oscillation with collapse time Δt which depends on IR strength. The stress p53 via IR drive cell cycle molecular species MPF and cyclin dynamics to different states, namely, oscillation death, oscillations of periods, chaotic and sustain oscillation in their bifurcation diagram. We predict that there could be a critical Δtc induced by p53 via IRc, where, if Δt〈Δtc the cell cycle may come back to normal state, otherwise it will go to cell cycle arrest (apoptosis).