Stochastic behavior, switching

Stochastic resonance is a kinetic effect universally inherent to bi- or multistable dynamic systems exposed to either white or color noise. Its main manifestation is the appearance of a maximum on the noise intensity dependencies of the signal-to-noise ratio in a system subject to a weak driving force. Essentially, this behavior is due to the presence of an exponential Kramers time x cx exp(AU/3>) of the system switching between energy minima here AU is the effective height of the energy barrier separating the potential wells and 3> is the noise intensity. 

This study proves that the stochastic single-molecule event of complete dissociation of the tetrameric lac repressor from DNA is solely responsible for the life changing decision of the cell, switching from one phenotype to another. This finding highlights the importance of single-molecule behaviors in biology [41]. 

Furthermore, it has recently been found that the discrete nature of a molecule population leads to qualitatively different behavior than in the continuum case in a simple autocatalytic reaction network [29]. In a simple autocatalytic reaction system with a small number of molecules, a novel steady state is found when the number of molecules is small, which is not described by a continuum rate equation of chemical concentrations. This novel state is first found by stochastic particle simulations. The mechanism is now understood in terms of fluctuation and discreteness in molecular numbers. Indeed, some state with extinction of specific molecule species shows a qualitatively different behavior from that with very low concentration of the molecule. This difference leads to a transition to a novel state, referred to as discreteness-induced transition. This phase transition appears by decreasing the system size or flow to the system, and it is analyzed from the stochastic process, where a single-molecule switch changes the distributions of molecules drastically. 

The spontaneous switching between the stabilizing and destabilizing modes (i.e., the transition among solutions) was observed as nonperiodic and stochastic. As one of its possible origins, we suppose the combination of intrinsic microscopic fluctuations and spatiotemporal chaos characterized as nonperiodic but nonrandom behavior. Indeed, the system s behavior is chaotic as its time evolution is unstable and unreproducible. Its capability of successively searching for multiple solutions, however, is robustly maintained and qualitatively reproducible. This resembles the robustness of strange attractors of chaotic systems. 

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