Phase separation organizes the site of autophagosome formation.

Many biomolecules undergo liquid-liquid phase separation to form liquid-like condensates that mediate diverse cellular functions1,2. Autophagy is able to degrade such condensates using autophagosomes-double-membrane structures that are synthesized de novo at the pre-autophagosomal structure (PAS) in yeast3-5. Whereas Atg proteins that associate with the PAS have been characterized, the physicochemical and functional properties of the PAS remain unclear owing to its small size and fragility. Here we show that the PAS is in fact a liquid-like condensate of Atg proteins. The autophagy-initiating Atg1 complex undergoes phase separation to form liquid droplets in vitro, and point mutations or phosphorylation that inhibit phase separation impair PAS formation in vivo. In vitro experiments show that Atg1-complex droplets can be tethered to membranes via specific protein-protein interactions, explaining the vacuolar membrane localization of the PAS in vivo. We propose that phase separation has a critical, active role in autophagy, whereby it organizes the autophagy machinery at the PAS.

Conversion of graded phosphorylation into switch-like nuclear translocation via autoregulatory mechanisms in ERK signalling.

The phosphorylation cascade in the extracellular signal-regulated kinase (ERK) pathway is a versatile reaction network motif that can potentially act as a switch, oscillator or memory. Nevertheless, there is accumulating evidence that the phosphorylation response is mostly linear to extracellular signals in mammalian cells. Here we find that subsequent nuclear translocation gives rise to a switch-like increase in nuclear ERK concentration in response to signal input. The switch-like response disappears in the presence of ERK inhibitor, suggesting the existence of autoregulatory mechanisms for ERK nuclear translocation involved in conversion from a graded to a switch-like response. In vitro reconstruction of ERK nuclear translocation indicates that ERK-mediated phosphorylation of nucleoporins regulates ERK translocation. A mathematical model and knockdown experiments suggest a contribution of nucleoporins to regulation of the ERK nuclear translocation response. Taken together, this study provides evidence that nuclear translocation with autoregulatory mechanisms acts as a switch in ERK signalling.

Optimality conditions for cell-fate heterogeneity that maximize the effects of growth factors in PC12 cells.

Recently, the heterogeneity that arises from stochastic fate decisions has been reported for several types of cancer-derived cell lines and several types of clonal cells grown under constant environmental conditions. However, the relation between this stochasticity and the responsiveness to extracellular stimuli remains largely unknown. Here we focused on the fate decisions of the PC12 cell line, which was derived from rat pheochromocytoma, and is a model system to study differentiation into sympathetic neurons. Whereas epidermal growth factor (EGF) stimulates the proliferation of populations of PC12 cells, nerve growth factor (NGF) promotes the differentiation of neurites to neuron-like cells. We found that phenotypic heterogeneity increased with time at several surrounding serum concentrations, suggesting stochastic cell-fate decisions in single cells. We made a simple mathematical model assuming Markovian transitions of the cell fates, and estimated the transition rates based on Bayes' theorem. The model suggests that depending on the serum concentration, EGF (NGF) even directs differentiation (proliferation) at the single-cell level. The maximum effects of the growth factors were ensured when the transition rates were appropriately controlled by the serum concentration to produce a nonextremal, moderate amount of cell-fate heterogeneity. Our model was validated by the experimental finding that the means and variances of the local cell densities obey a power-law relationship. These results suggest that even when efficient responses to growth factors are observed at the population level, the growth factors stochastically direct the cell-fate decisions in different directions at the single-cell level.

A mathematical model for the detection mechanism of DNA double-strand breaks depending on autophosphorylation of ATM.

BACKGROUND: After IR stress, DNA double-strand breaks (DSBs) occur and repair proteins (RPs) bind to them, generating DSB-RP complexes (DSBCs), which results in repaired DSBs (RDSBs). In recent experimental studies, it is suggested that the ATM proteins detect these DNA lesions depending on the autophosphorylation of ATM which exists as a dimer before phosphorylation. Interestingly, the ATM proteins can work as a sensor for a small number of DSBs (approximately 18 DSBs in a cell after exposure to IR). Thus the ATM proteins amplify the small input signals based on the phosphorylation of the ATM dimer proteins. The true DSB-detection mechanism depending on ATM autophosphorylation has yet to be clarified. METHODOLOGY/PRINCIPAL FINDINGS: We propose a mathematical model for the detection mechanism of DSBs by ATM. Our model includes both a DSB-repair mechanism and an ATM-phosphorylation mechanism. We model the former mechanism as a stochastic process, and obtain theoretical mean values of DSBs and DSBCs. In the latter mechanism, it is known that ATM autophosphorylates itself, and we find that the autophosphorylation induces bifurcation of the phosphorylated ATM (ATM*). The bifurcation diagram depends on the total concentration of ATM, which makes three types of steady state diagrams of ATM*: monostable, reversible bistable, and irreversible bistable. Bistability exists depending on the Hill coefficient in the equation of ATM autophosphorylation, and it emerges as the total concentration of ATM increases. Combining these two mechanisms, we find that ATM* exhibits switch-like behaviour in the presence of bistability, and the detection time after DNA damage decreases when the total concentration of ATM increases. CONCLUSIONS/SIGNIFICANCE: This work provides a mathematical model that explains the DSB-detection mechanism depending on ATM autophosphorylation. These results indicate that positive auto-regulation works both as a sensor and amplifier of small input signals.

The Fokker-Planck approach for the cooperative molecular motor model with finite number of motors.

We provide the methodology for the analysis of the cooperative molecular motor model with finite number of motors, which are linearly and rigidly coupled, based on the Fokker-Planck approach. The probability density functions for the position of motors are solved numerically from the stationary Fokker-Planck equations. By using these probability density functions, we provide the analytical expressions, such as the velocity, the rate of the ATP consumption, the energetic efficiency, and the dissipation energy rates. Furthermore, we investigate three specific examples, such as single motor model, 2-motor model, and infinitely coupled motor model. Numerical algorithm to solve the Fokker-Planck equations is also provided.