Deterministic and stochastic approaches to a minimal model for the transition from autophagy to apoptosis.

Autophagy and apoptosis are crucial cellular mechanisms. The cytoprotective function of autophagy is mediated by the negative regulation of apoptosis, which in turn inhibits autophagy. Although research into the molecular connection between autophagy and apoptosis is booming, the intricate regulatory mechanisms of this process are still not completely understood. Therefore, the objective of this study was to develop a minimal model to explore the transition from autophagy to apoptosis. This biological system was analyzed by comprehensively integrating both the deterministic and the stochastic dynamics of the cells. The system exhibited bistability, and the statistical properties of cells undergoing autophagy and apoptosis were analyzed at two different stress levels with varying noise strengths. Moreover, we investigated how noise affected the double negative feedback loops between autophagy and apoptosis and further triggered transitions at two different stress levels and initial conditions. Finally, the effect of noise on transition was comprehensively studied under continuous stress variations and the two different initial conditions, showing that stronger noise results in more randomness during the switching process. Our work may provide novel insights for further experiments and modeling.

Dynamic modelling and tristability analysis of misfolded α-synuclein degraded via autophagy in Parkinson's disease.

The widely-accepted hallmark pathology of Parkinson's disease (PD) is the presence of Lewy bodies with characteristic abnormal aggregated α-synuclein (αSyn). Growing physiological evidence suggests that there is a pivotal role for the autophagy-lysosome pathway (ALP) in the clearance of misfolded αSyn (αSyn∗). This work establishes a mathematical model for αSyn∗ degradation through the ALP. Qualitative simulations are used to uncover the tristable behavior of αSyn∗, i.e., the lower, medium, and upper steady states, which correspond to the healthy, critical, and disease stages of PD, respectively. Time series and codimension-1 bifurcation analysis suggest that the system shows tristability dynamics. Furthermore, variations in the key parameters influence the tristable dynamic behavior, and the distribution of tristable regions is exhibited more comprehensively in codimension-2 bifurcation diagrams. In addition, robustness analysis demonstrates that tristability is a robust property of the system. These results may be valuable in therapeutic strategies for the prevention and treatment of PD.

Dynamics of a model for the degradation mechanism of aggregated α-synuclein in Parkinson's disease.

Accumulation of the misfolded synaptic protein α-synuclein (αSyn*) is a hallmark of neurodegenerative disease in Parkinson's disease (PD). Recent studies suggest that the autophagy lysosome pathway (ALP) including both the Beclin1-associated and mTOR-signaling pathways is involved in the αSyn* clearance mechanism. In this study, a mathematical model is proposed for the degradation of αSyn* by ALP with the crosstalk element of mTOR. Using codimension-1 bifurcation analysis, the tri-stability of αSyn* is surveyed under three different stress signals and, in addition, consideration is given to the regulatory mechanisms for the Beclin1- and mTOR-dependent rates on αSyn* degradation using the codimension-1 and-2 bifurcation diagrams. It was found that, especially under internal and external oxidative stresses (S 1), the bistable switch of the aggregation of αSyn* can be transformed from an irreversible to a reversible condition through the ALP degradation pathways. Furthermore, the robustness of the tri-stable state for the stress S 1 to the parameters related to mTOR-mediated ALP was probed. It was confirmed that mTOR-mediated ALP is important for maintaining the essential dynamic features of the tri-stable state. This study may provide a promising avenue for conducting further experiments and simulations of the degradation mechanism of dynamic modeling in PD.

Dynamical Mechanisms for Gene Regulation Mediated by Two Noncoding RNAs in Long-Term Memory Formation.

Noncoding RNAs such as miRNAs and piRNAs have long-lasting effects on the regulation of gene expression involved in long-term synaptic changes. To characterize gene regulation mediated by small noncoding RNAs associated with long-term memory in Aplysia, we consider two noncoding RNAs stimulated by 5-HT into a gene regulatory network motif model, including miR-124 that binds to and inhibits the mRNA of CREB1 and piR-F that facilitates serotonin-dependent DNA methylation to lead to repression of CREB2. Codimension-1 and -2 bifurcation analyses of 5-HT regulating both miR-124 and piR-F and a negative feedback strength for oscillation reveal rich dynamical properties of bistability and oscillations robust to variations in all other parameters. More importantly, we verify three stimulus protocols of 5-HT in experiments by our model and find that application of five pulses of 5-HT leads to a transient decrease of miR-124 but increase of piR-F concentrations, which matters sustained high level of CREB1 concentration associated with long-term memory. Furthermore, we perform bifurcation analyses for the concentrations of miR-124 and piR-F as two parameters to explore dynamical mechanisms underlying the epigenetic regulation in long-term memory formation. This study provides insights into revealing regulatory roles of epigenetic changes in gene expression involving noncoding RNAs associated with synaptic plasticity.

DNA Methylation Heterogeneity Induced by Collaborations Between Enhancers.

During mammalian embryo development, reprogramming of DNA methylation plays important roles in the erasure of parental epigenetic memory and the establishment of naive pluripotent cells. Multiple enzymes that regulate the processes of methylation and demethylation work together to shape the pattern of genome-scale DNA methylation and guide the process of cell differentiation. Recent availability of methylome information from single-cell whole genome bisulfite sequencing (scBS-seq) provides an opportunity to study DNA methylation dynamics in the whole genome in individual cells, which reveal the heterogeneous methylation distributions of enhancers in embryo stem cells. In this study, we developed a computational model of enhancer methylation inheritance to study the dynamics of genome-scale DNA methylation reprogramming during exit from pluripotency. The model enables us to track genome-scale DNA methylation reprogramming at single-cell level during the embryo development process and reproduce the DNA methylation heterogeneity reported by scBS-seq. Model simulations show that DNA methylation heterogeneity is an intrinsic property driven by cell division along the development process, and the collaboration between neighboring enhancers is required for heterogeneous methylation. Our study suggests that the mechanism of genome-scale oscillation might not be necessary for the DNA methylation heterogeneity during exit from pluripotency.

Dynamic Analysis of the Time-Delayed Genetic Regulatory Network Between Two Auto-Regulated and Mutually Inhibitory Genes.

Time delays play important roles in genetic regulatory networks. In this paper, a gene regulatory network model with time delays and mutual inhibition is considered, where time delays are regarded as bifurcation parameters. In the first part of this paper, we analyze the associated characteristic equations and obtain the conditions for the stability of the system and the existence of Hopf bifurcations in five special cases. Explicit formulas are given to determine the direction and stability of the Hopf bifurcation by using the normal form method and the center manifold theorem. Numerical simulations are then performed to illustrate the results we obtained. In the second part of the paper, using time-delayed stochastic numerical simulations, we study the impact of biological fluctuations on the system and observe that, in modest noise regimes, unexpectedly, noise acts to stabilize the otherwise destabilized oscillatory system.

From reversible to irreversible bistable switches via bifurcations in a gene regulatory network.

The interplay of small, noncoding microRNAs (miRNAs), mRNAs and proteins plays crucial roles in almost all cellular processes. MiR-124, widely known as a memory-related miRNA, can regulate LTM by binding to the mRNA of CREB1 stimulated with 5-HT. In this paper, we establish a regulatory network model of CREB1 and miR-124 stimulated by 5-HT, in which miR-124 inhibits CREB1, which in turn enhances miR-124. Our model validates three protocols based on 5-HT in experiments on the induction of LTM in Aplysia. A steady-state analysis and numerical bifurcations of the abstracted system beyond memory formation, when the fast reaction has been in the equilibrium, can facilitate more abundant dynamical behaviors such as bistability and oscillation. The original system also exhibits bistability under appropriate feedback strengths, which is relevant to the mechanism of LTM formation. Furthermore, we specifically show a change in the transition from a reversible switch to an irreversible switch via bifurcations of the negative regulation of miR-124 on CREB1, which eventually maintains a high phosphorylated CREB1 level after initially elevated by 5-HT. These findings indicate that miR-124 provides an inhibitory constraint on long-term synaptic plasticity through the regulation of CREB1.

Stochastic Telomere Shortening and the Route to Limitless Replicative Potential.

In human tissues, the replicative potential of stem cells is limited by the shortening of telomere, limitless replicative potential is a hallmark of cancer. Telomere length changes stochastically during cell division mainly due to the competition between the end replication problem and telomerase, short telomere can lead to replicative senescence and cell apoptosis. Here, we investigate how stochastic changes of telomere length in individual cells may affect the population dynamics of clonal growth. We established a computational model that couples telomerase-regulated stochastic telomere length changes with the replicative potential of clones. Model simulations reveal qualitative dependence of clone proliferation potential with activities of telomerase; mutations in cells to alter the activities of telomerase and its inhibitors can induce abnormal tissue growth and lead to limitless replicative potential.

Stability and Hopf bifurcation analysis in a delayed three-node circuit involving interlinked positive and negative feedback loops.

A system involving interlinked positive and negative feedback loops is a flexible motif that can tune itself in gene regulatory networks. It is well known that time delay is inevitable in gene regulatory networks due to transcription and translation not being physically co-located. In this paper, we systematically consider the effect of time delay on the dynamical behavior of the three-node circuit with three time delays. Based on linear stability analysis and bifurcation theory, sufficient conditions for stability of equilibria and oscillatory behaviors via Hopf bifurcation are derived when choosing positive and negative feedback strengths as well as time delays τ1, τ2, τ3 as the bifurcation parameters, respectively. Moreover, stability and direction of the Hopf bifurcation of time delay are studied by using the normal form method and the center manifold theorem. Finally, several examples are performed to illustrate some analytical results we obtained.

Maintenance of postsynaptic neuronal excitability by a positive feedback loop of postsynaptic BDNF expression.

Experiments have demonstrated that in mice, the PVT strongly projects to the CeL and participates in the formation of fear memories by synaptic potentiation in the amygdala. Herein, we propose a mathematical model based on a positive feedback loop of BDNF expression and signaling to investigate PVT manipulation of synaptic potentiation. The model is validated by comparisons with experimental observations. We find that a high postsynaptic firing frequency after stimulation is induced by presynaptic Ca2+ when the rates of BDNF secretion from PVT and LA neurons to the CeL are above a threshold value. Moreover, the positive feedback of postsynaptic BDNF production is important for the maintenance of the high excitability of the SOM+ CeL neuron after stimulation. The model brings insight into the underlying mechanisms of PVT modulation of synaptic potentiation at LA-CeL synapses and provides a framework of understanding other similar processes associated with synaptic plasticity.

Underlying Mechanisms of Cooperativity, Input Specificity, and Associativity of Long-Term Potentiation Through a Positive Feedback of Local Protein Synthesis.

Long-term potentiation (LTP) is a specific form of activity-dependent synaptic plasticity that is a leading mechanism of learning and memory in mammals. The properties of cooperativity, input specificity, and associativity are essential for LTP; however, the underlying mechanisms are unclear. Here, based on experimentally observed phenomena, we introduce a computational model of synaptic plasticity in a pyramidal cell to explore the mechanisms responsible for the cooperativity, input specificity, and associativity of LTP. The model is based on molecular processes involved in synaptic plasticity and integrates gene expression involved in the regulation of neuronal activity. In the model, we introduce a local positive feedback loop of protein synthesis at each synapse, which is essential for bimodal response and synapse specificity. Bifurcation analysis of the local positive feedback loop of brain-derived neurotrophic factor (BDNF) signaling illustrates the existence of bistability, which is the basis of LTP induction. The local bifurcation diagram provides guidance for the realization of LTP, and the projection of whole system trajectories onto the two-parameter bifurcation diagram confirms the predictions obtained from bifurcation analysis. Moreover, model analysis shows that pre- and postsynaptic components are required to achieve the three properties of LTP. This study provides insights into the mechanisms underlying the cooperativity, input specificity, and associativity of LTP, and the further construction of neural networks for learning and memory.

Fabrication of large micro-structured high-numerical-aperture optofluidic compound eyes with tunable angle of view.

A large optofluidic compound eye is developed by using a straightforward, rapid, and low-cost technique. The compound eye's angle of view can be adjusted by injecting deionized water/calcium chloride solution of different volume into the optofluidic chip. Optofluidic compound eyes containing about 78,000 microlenses of 50 µm diameter are fabricated for analysis. The angle of view can be tuned up to 104°. With the compound eye's deformation, the microlenses' focal length increases, due to the variation in profile. Owing to the non-uniform strain over the compound eye, the central lenses experience more variation. Furthermore, optical imaging of the compound eye is demonstrated and sharp images can be obtained from the omnidirectional microlenses.

Stochasticity and bifurcations in a reduced model with interlinked positive and negative feedback loops of CREB1 and CREB2 stimulated by 5-HT.

The cyclic AMP (cAMP)-response element-binding protein (CREB) family of transcription factors is crucial in regulating gene expression required for long-term memory (LTM) formation. Upon exposure of sensory neurons to the neurotransmitter serotonin (5-HT), CREB1 is activated via activation of the protein kinase A (PKA) intracellular signaling pathways, and CREB2 as a transcriptional repressor is relieved possibly via phosphorylation of CREB2 by mitogen-activated protein kinase (MAPK). Song et al. [18] proposed a minimal model with only interlinked positive and negative feedback loops of transcriptional regulation by the activator CREB1 and the repressor CREB2. Without considering feedbacks between the CREB proteins, Pettigrew et al. [8] developed a computational model characterizing complex dynamics of biochemical pathways downstream of 5-HT receptors. In this work, to describe more simply the biochemical pathways and gene regulation underlying 5-HT-induced LTM, we add the important extracellular sensitizing stimulus 5-HT as well as the product Ap-uch into the Song's minimal model. We also strive to examine dynamical properties of the gene regulatory network under the changing concentration of the stimulus, [5-HT], cooperating with the varying positive feedback strength in inducing a high state of CREB1 for the establishment of long-term memory. Different dynamics including monostability, bistability and multistability due to coexistence of stable steady states and oscillations is investigated by means of codimension-2 bifurcation analysis. At the different positive feedback strengths, comparative analysis of deterministic and stochastic dynamics reveals that codimension-1 bifurcation with respect to [5-HT] as the parameter can predict diverse stochastic behaviors resulted from the finite number of molecules, and the number of CREB1 molecules more and more preferentially resides near the high steady state with increasing [5-HT], which contributes to long-term memory formation.

Bifurcation analysis and potential landscapes of the p53-Mdm2 module regulated by the co-activator programmed cell death 5.

The dynamics of p53 play important roles in the regulation of cell fate decisions in response to various stresses, and programmed cell death 5 (PDCD5) functions as a co-activator of p53 that modulates p53 dynamics. In the present paper, we investigated how p53 dynamics are modulated by PDCD5 during the deoxyribose nucleic acid damage response using methods of bifurcation analysis and potential landscape. Our results revealed that p53 activities display rich dynamics under different PDCD5 levels, including monostability, bistability with two stable steady states, oscillations, and the coexistence of a stable steady state (or two states) and an oscillatory state. The physical properties of the p53 oscillations were further demonstrated by the potential landscape in which the potential force attracts the system state to the limit cycle attractor, and the curl flux force drives coherent oscillation along the cyclic trajectory. We also investigated the efficiency with which PDCD5 induced p53 oscillations. We show that Hopf bifurcation can be induced by increasing the PDCD5 efficiency and that the system dynamics exhibited clear transition features in both barrier height and energy dissipation when the efficiency was close to the bifurcation point.

Effects of elongation delay in transcription dynamics.

In the transcription process, elongation delay is induced by the movement of RNA polymerases (RNAP) along the DNA sequence, and can result in changes in the transcription dynamics. This paper studies the transcription dynamics that involved the elongation delay and effects of cell division and DNA replication. The stochastic process of gene expression is modeled with delay chemical master equation with periodic coefficients, and is studied numerically through the stochastic simulation algorithm with delay. We show that the average transcription level approaches to a periodic dynamics over cell cycles at homeostasis, and the elongation delay can reduce the transcription level and increase the transcription noise. Moreover, the transcription elongation can induce bimodal distribution of mRNA levels that can be measured by the techniques of flow cytometry.

Subtle interplay of stochasticity and deterministic dynamics pervades an evolutionary plausible genetic circuit for Bacillus subtilis competence.

Here we study the interplay of stochastic and deterministic dynamics in an evolutionary plausible candidate core genetic circuit for Bacillus subtilis competence. We find that high noise would not necessarily be detrimental to the circuit's ability to deliver the phenotype, due to an unexpected built-in robustness that we further investigate. Also, we find that seemingly subtle deterministic dynamical features of the regulation, unstable and stable limit cycles, while in the presence of biochemical noise, would result in a distinctive new observable in the phenotype. We conduct mathematical analyses of the system's stability at the fixed points and derive some general model-independent consequences. We also show how imperfect time-scale separation in the system would result in observables detrimental to the phenotype, that nature could have harnessed for selection.