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The p53 transcription factor is a regulator of key cellular processes including DNA repair, cell cycle arrest, and apoptosis. In this theoretical study, we investigate how the complex circuitry of the p53 network allows for stochastic yet unambiguous cell fate decision-making. The proposed Markov chain model consists of the regulatory core and two subordinated bistable modules responsible for cell cycle arrest and apoptosis. The regulatory core is controlled by two negative feedback loops (regulated by Mdm2 and Wip1) responsible for oscillations, and two antagonistic positive feedback loops (regulated by phosphatases Wip1 and PTEN) responsible for bistability. By means of bifurcation analysis of the deterministic approximation we capture the recurrent solutions (i.e., steady states and limit cycles) that delineate temporal responses of the stochastic system. Direct switching from the limit-cycle oscillations to the “apoptotic” steady state is enabled by the existence of a subcritical Neimark—Sacker bifurcation in which the limit cycle loses its stability by merging with an unstable invariant torus. Our analysis provides an explanation why cancer cell lines known to have vastly diverse expression levels of Wip1 and PTEN exhibit a broad spectrum of responses to DNA damage: from a fast transition to a high level of p53 killer (a p53 phosphoform which promotes commitment to apoptosis) in cells characterized by high PTEN and low Wip1 levels to long-lasting p53 level oscillations in cells having PTEN promoter methylated (as in, e.g., MCF-7 cell line)., Author Summary Cancers are diseases of signaling networks. Transcription factor p53 is a pivotal node of a network that integrates a variety of stress signals and governs critical processes of DNA repair, cell cycle arrest, and apoptosis. Somewhat paradoxically, despite the fact that carcinogenesis is prevalently caused by p53 network malfunction, most of our knowledge about p53 signaling is based on cancer or immortalized cell lines. In this paper, we construct a mathematical model of intact p53 network to understand dynamics of non-cancerous cells and then dynamics of cancerous cells by introducing perturbations to the regulatory system. Cell fate decisions are enabled by the presence of interlinked feedback loops which give rise to a rich repertoire of behaviors. We explain and analyze by means of numerical simulations how the dynamical structure of the regulatory system allows for generating unambiguous single-cell fate decisions, also in the case when the cell population splits into an apoptotic and a surviving subpopulation. Perturbation analysis provides an explanation why cancer cell lines known to have vastly diverse expression levels of p53 regulators can exhibit a broad spectrum of responses to DNA damage.

We proposed a spatially extended model of early events of B cell receptors (BCR) activation, which is based on mutual kinase-receptor interactions that are characteristic for the immune receptors and the Src family kinases. These interactions lead to the positive feedback which, together with two nonlinearities resulting from the double phosphorylation of receptors and Michaelis-Menten dephosphorylation kinetics, are responsible for the system bistability. We demonstrated that B cell can be activated by a formation of a tiny cluster of receptors or displacement of the nucleus. The receptors and Src kinases are activated, first locally, in the locus of the receptor cluster or the region where the cytoplasm is the thinnest. Then the traveling wave of activation propagates until activity spreads over the whole cell membrane. In the models in which we assume that the kinases are free to diffuse in the cytoplasm, we found that the fraction of aggregated receptors, capable to initiate B cell activation decreases with the decreasing thickness of cytoplasm and decreasing kinase diffusion. When kinases are restricted to the cell membrane - which is the case for most of the Src family kinases - even a cluster consisting of a tiny fraction of total receptors becomes activatory. Interestingly, the system remains insensitive to the modest changes of total receptor level. The model provides a plausible mechanism of B cells activation due to the formation of small receptors clusters collocalized by binding of polyvalent antigens or arising during the immune synapse formation., Author Summary B cells are activated in response to binding of appropriate ligands, which induces the aggregation of B cell receptors. The formation of even small clusters containing less than 1% of all the receptors is sufficient for activation. This observation led us to a model in which the receptor cluster serves only as a switch that turns on the activation process involving also the remaining receptors. The idea of the model exploits the fact the Src kinase - BCR system is bistable, and thus its local activation may start the propagation of a traveling wave, which spreads activation over the entire membrane. We found that the minimal size of the activatory cluster decreases with the thickness of the cytoplasm and kinase diffusion coefficient. It is particularly small when kinases are restricted to the membrane. These findings are consistent with the properties of B cells, which prior to activation have extremely thin cytoplasmic layer and in which Src family kinases (interacting with the receptors) are tethered to the membrane.

The p53 regulatory pathway controls cell responses, which include cell cycle arrest, DNA repair, apoptosis and cellular senescence. We propose a stochastic model of p53 regulation, which is based on two feedback loops: the negative, coupling p53 with its immediate downregulator Mdm2, and the positive, which involves PTEN, PIP3 and Akt. Existence of the negative feedback assures homeostasis of healthy cells and oscillatory responses of DNA-damaged cells, which are persistent when DNA repair is inefficient and the positive feedback loop is broken. The positive feedback destroys the negative coupling between Mdm2 and p53 by sequestering most of Mdm2 in cytoplasm, so it may no longer prime the nuclear p53 for degradation. It works as a clock, giving the cell some time for DNA repair. However, when DNA repair is inefficient, the active p53 rises to a high level and triggers transcription of proapoptotic genes. As a result, small DNA damage may be repaired and the cell may return to its initial "healthy" state, while the extended damage results in apoptosis. The stochasticity of p53 regulation, introduced at the levels of gene expression, DNA damage and repair, leads to high heterogeneity of cell responses and causes cell population split after irradiation into subpopulations of apoptotic and surviving cells, with fraction of apoptotic cells growing with the irradiation dose.

Background Apoptosis is a tightly regulated process: cellular survive-or-die decisions cannot be accidental and must be unambiguous. Since the suicide program may be initiated in response to numerous stress stimuli, signals transmitted through a number of checkpoints have to be eventually integrated. Results In order to analyze possible mechanisms of the integration of multiple pro-apoptotic signals, we constructed a simple model of the Bcl-2 family regulatory module. The module collects upstream signals and processes them into life-or-death decisions by employing interactions between proteins from three subgroups of the Bcl-2 family: pro-apoptotic multidomain effectors, pro-survival multidomain restrainers, and pro-apoptotic single domain BH3-only proteins. Although the model is based on ordinary differential equations (ODEs), it demonstrates that the Bcl-2 family module behaves akin to a Boolean logic gate of the type dependent on levels of BH3-only proteins (represented by Bad) and restrainers (represented by Bcl-xL). A low level of pro-apoptotic Bad or a high level of pro-survival Bcl-xL implies gate AND, which allows for the initiation of apoptosis only when two stress stimuli are simultaneously present: the rise of the p53 killer level and dephosphorylation of kinase Akt. In turn, a high level of Bad or a low level of Bcl-xL implies gate OR, for which any of these stimuli suffices for apoptosis. Conclusions Our study sheds light on possible signal integration mechanisms in cells, and spans a bridge between modeling approaches based on ODEs and on Boolean logic. In the proposed scheme, logic gates switching results from the change of relative abundances of interacting proteins in response to signals and involves system bistability. Consequently, the regulatory system may process two analogous inputs into a digital survive-or-die decision.