Wu Fengyan, Chen Xiaoli, Zheng Yayun, Duan Jinqiao, Kurths Jürgen, Li Xiaofan
Center for Mathematical Sciences & School of Mathematics and Statistics, Huazhong University of Science and Technology, Wuhan 430074, China.
Wuhan National Laboratory for Optoelectronics, Wuhan 430074, China.
Chaos. 2018 Jul;28(7):075510. doi: 10.1063/1.5025235.
We investigate a quantitative bistable two-dimensional model (MeKS network) of gene expression dynamics describing the competence development in the Bacillus subtilis under the influence of Lévy as well as Brownian motions. To analyze the transitions between the vegetative and the competence regions therein, two dimensionless deterministic quantities, the mean first exit time (MFET) and the first escape probability, are determined from a microscopic perspective, as well as their averaged versions from a macroscopic perspective. The relative contribution factor λ, the ratio of non-Gaussian and Gaussian noise strengths, is adopted to identify an optimum choice in these transitions. Additionally, we use a recent geometric concept, the stochastic basin of attraction (SBA), to exhibit a pictorial comprehension about the influence of the Lévy motion on the basin stability of the competence state. Our main results indicate that (i) the transitions between the vegetative and the competence regions can be induced by the noise intensities, the relative contribution factor λ and the Lévy motion index α; (ii) a higher noise intensity and a larger α with smaller jump magnitude make the MFET shorter, and the MFET as a function of λ exhibits one maximum value, which is a signature of the noise-enhanced stability phenomenon for the vegetative state; (iii) a larger α makes the transition from the vegetative to the adjacent competence region to occur at the highest probability. The Lévy motion index α≈0.5 (a larger jump magnitude with a lower frequency) is an ideal choice to implement the transition to the non-adjacent competence region; (iv) there is an expansion in SBA when α decreases.
我们研究了一个描述枯草芽孢杆菌感受态发育的基因表达动力学定量双稳二维模型（MeKS网络），该模型受 Lévy 运动以及布朗运动的影响。为了分析其中营养态区域和感受态区域之间的转变，从微观角度确定了两个无量纲确定性量，即平均首次退出时间（MFET）和首次逃逸概率，以及从宏观角度确定了它们的平均形式。采用非高斯噪声强度与高斯噪声强度之比的相对贡献因子λ来确定这些转变中的最佳选择。此外，我们使用了一个最近的几何概念，即随机吸引盆（SBA），来直观理解 Lévy 运动对感受态状态的吸引盆稳定性的影响。我们的主要结果表明：（i）营养态区域和感受态区域之间的转变可由噪声强度、相对贡献因子λ和 Lévy 运动指数α诱导；（ii）更高的噪声强度和更大的α以及更小的跳跃幅度会使 MFET 更短，并且作为λ函数的 MFET 呈现一个最大值，这是营养态噪声增强稳定性现象的一个特征；（iii）更大的α使得从营养态到相邻感受态区域的转变以最高概率发生。Lévy 运动指数α≈0.5（较大的跳跃幅度和较低的频率）是实现向非相邻感受态区域转变的理想选择；（iv）当α减小时，SBA 会有扩张。
