Exact scaling for the mean first-passage time of random walks on a generalized Koch network with a trap

In this paper, we study the scaling for the mean first-passage time （MFPT） of the random walks on a generalized Koch network with a trap. Through the network construction, where the initial state is transformed from a triangle to a polygon, we obtain the exact scaling for the MFPT. We show that the MFPT grows linearly with the number of nodes and the dimensions of the polygon in the large limit of the network order. In addition, we determine the exponents of scaling efficiency characterizing the random walks. Our results are the generalizations of those derived for the Koch network, which shed light on the analysis of random walks over various fractal networks.

Transient Properties of a Bistable System with Delay Time Driven by Non-Gaussian and Gaussian Noises:Mean First-Passage Time

The mean first-passage time of a bistable system with time-delayed feedback driven by multiplicative non-Gaussian noise and additive Gaussian white noise is investigated. Firstly, the non-Markov process is reduced to the Markov process through a path-integral approach; Secondly, the approximate Fokker-Planck equation is obtained by applying the unified coloured noise approximation, the small time delay approximation and the Novikov Theorem. The functional analysis and simplification are employed to obtain the approximate expressions of MFPT. The effects of non-Gaussian parameter （measures deviation from Gaussian character） r, the delay time τ, the noise correlation time to, the intensities D and a of noise on the MFPT are discussed. It is found that the escape time could be reduced by increasing the delay time τ, the noise correlation time τ0, or by reducing the intensities D and α. As far as we know, this is the first time to consider the effect of delay time on the mean first-passage time in the stochastic dynamical system.

Mean First-Passage Time of an Asymmetric Kinetic Model Driven by Two Different Kinds of Colored Noises

The mean first-passage time （MFPT） of an asymmetric bistable system between multiplicative non-Gaussian noise and additive Gaussian white noise with nonzero cross-correlation time is investigated. Firstly, the non-Markov process is reduced to the Markov process through a path-integral approach; Secondly, the approximate Fokker Planck equation is obtained by applying the unified colored noise approximation and the.Novikov Theorem. The steady-state probability distribution （SPD） is also obtained. The basal functional analysis and simplification are employed to obtain the approximate expressions of MFPT T^±. The effects of the asymmetry parameter β, the non-Gaussian parameter （measures deviation from Gaussian character） r, the noise correlation times τ and τ2, the coupling coefficient A, the intensities D and a of noise on the MFPT are discussed. It is found that the asymmetry parameter β, the non-Gaussian parameter r and the coupling coefficient A can induce phase transition. Moreover, the main findings are that the effect of self-existent parameters （D, α, and τ） of noise and cross-correlation parameters （A, 7-2） between noises on MFPT T^± is different.

Analytical and NumericalMethods to Study the MFPT and SR of a Stochastic Tumor-Immune Model

The Mean First-Passage Time (MFPT) and Stochastic Resonance (SR) of a stochastic tumor-immune model withnoise perturbation are discussed in this paper. Firstly, considering environmental perturbation, Gaussian whitenoise and Gaussian colored noise are introduced into a tumor growth model under immune surveillance. Asfollows, the long-time evolution of the tumor characterized by the Stationary Probability Density (SPD) and MFPTis obtained in theory on the basis of the Approximated Fokker-Planck Equation (AFPE). Herein the recurrenceof the tumor from the extinction state to the tumor-present state is more concerned in this paper. A moreefficient algorithmof Back-Propagation Neural Network (BPNN) is utilized in order to testify the correction of thetheoretical SPDandMFPT.With the existence of aweak signal, the functional relationship between Signal-to-NoiseRatio (SNR), noise intensities and correlation time is also studied. Numerical results show that both multiplicativeGaussian colored noise and additive Gaussian white noise can promote the extinction of the tumors, and themultiplicative Gaussian colored noise can lead to the resonance-like peak on MFPT curves, while the increasingintensity of the additiveGaussian white noise results in theminimum of MFPT. In addition, the correlation timesare negatively correlated with MFPT. As for the SNR, we find the intensities of both the Gaussian white noise andthe Gaussian colored noise, as well as their correlation intensity can induce SR. Especially, SNR is monotonouslyincreased in the case ofGaussian white noisewith the change of the correlation time.At last, the optimal parametersin BPNN structure are analyzed for MFPT from three aspects: the penalty factors, the number of neural networklayers and the number of nodes in each layer.

Moments and Mean First—Passage Time of Parabolic—Bistable Potential System Driven by Colored Noise

A parabolic-bistable potential system driven by colored noise is studied. The exact analytical expressions of the stationary probability distribution (SPD) and the moments of the system are derived. Furthermore, the mean first-passage time is calculated by the use of two approximate methods, respectively. It is found that (i) the double peaks of SPD are rubbed-down into a flat single peak with the increasing of noise intensity; (ii) a minimum occurs on the curve of the second-order moment of the system vs. noise intensity at the point ; (iii) the results obtained by our approximate approach are in good agreement with the numerical calculations for either small or large correlation time , while the conventional steepest descent approximation leads to poor results.

Modulation of first-passage time for gene expression via asymmetric cell division

How to balance the size of exponentially growing cells has always been a focus of biologists.Recent experiments have uncovered that the cell is divided into two daughter cells only when the level of time-keeper protein reaches a fixed threshold and cell divi-sion in prokaryote is not completely symmetric.The timing of cell division is essentially random because gene expression is stochastic,but cells seen to manage to have precise timing of cell division events.Although the inter cellular variability of gene expression has attracted much attention,the randomness of event timing has been rarely studied.In our analysis,the timing of cell division is formulated as the first-passage time (denoted hy FPT) for time-keeper protein’s level to cross a critical threshold firstly,we derive exact analytical formulae for the mean and noise of FPT based on stochastic gene expression model with asymmetric cell division.The results of numerical simulation show that the regulatory factors (division rate,newborn cell size,exponential growth rate and threshold) have significant influence on the mean and noise of FPT.We also show that both the increase of division rate and newborn cell size could reduce the mean of FPT and increase the noise of FPT,the larger the exponential growth rate is,the smaller the mean and noise of FPT will be;and the larger the threshold value is,the higher the mean of FPT is and the lower the noise is.In addition,compared with symmetric divi sion,asymmetric division can reduce the mean of FPT and improve the noise of FPT.In summary,our results provide insight into the relationship between regulatory factors and FPT and reveal that asymmetric division is an effective mechanism to shorten the mean of FPT.

单链高分子通过薄膜上纳米孔隙的输运过程:链间相互作用的影响

从理论上研究了高分子链通过位于薄膜上的一个纳米孔隙的输运问题.考虑在膜的一侧引入与传输高分子链有特殊相互作用的高分子链段,研究链间相互作用致使高分子链构型的改变对其平均首次通过时间的影响,进而在不同条件下进行了相应的数值模拟,并讨论了其它一些相关问题.

基于期望首达时间的形状距离学习算法

由于逐对形状匹配不能很好地反映形状间相似度,因此需要引入后期处理步骤提升检索精度.为了得到上下文敏感的形状相似度,本文提出了一种基于期望首达时间(Mean first-passage time,MFPT)的形状距离学习方法.在利用标准形状匹配方法得到距离矩阵的基础上,建立离散时间马尔可夫链对形状流形结构进行分析.将形状样本视作状态,利用不同状态之间完成一次状态转移的平均时间步长,即期望首达时间,表示形状间的距离.期望首达时间能够结合测地距离发掘空间流形结构,并可以通过线性方程进行有效求解.分别对不同数据进行实验分析,本文所提出的方法在相同条件下能够达到更高的形状检索精度.

色关联噪声激励可控双稳态势模型的非线性共振研究

研究了色关联乘性色噪声和加性白噪声激励可控双稳态势模型的粒子逃逸过程,推导出近似Fokker-Planck方程和平均首通时间(mean first-passage time,MFPT)表达式,分析可控阱深和可控阱宽下系统参数对平均首通时间的影响。研究结果表明:平均首通时间作为乘性色噪声强度和加性白噪声强度的函数展现出单峰共振和反共振现象,两噪声间的色关联性能激发系统跃迁行为,自(互)关联时间和互关联强度可以诱导系统的共振现象;可控势阱因子对左势阱和右势阱的初始位置逃逸行为的影响不同,能增强可控双稳态势模型稳定性,大尺度可控势阱因子在可控阱宽情形下占据了粒子跃迁行为的主导作用。

Energy diffusion controlled reaction rate in dissipative Hamiltonian systems

In this paper the energy diffusion controlled reaction rate in dissipative Hamiltonian systems is investigated by using the stochastic averaging method for quasi Hamiltonian systems. The boundary value problem of mean first- passage time （MFPT） of averaged system is formulated and the energy diffusion controlled reaction rate is obtained as the inverse of MFPT. The energy diffusion controlled reaction rate in the classical Kramers bistable potential and in a two-dimensional bistable potential with a heat bath are obtained by using the proposed approach respectively. The obtained results are then compared with those from Monte Carlo simulation of original systems and from the classical Kraraers theory. It is shown that the reaction rate obtained by using the proposed approach agrees well with that from Monte Carlo simulation and is more accurate than the classical Kramers rate.

Random walks in generalized delayed recursive trees

Recently a great deal of effort has been made to explicitly determine the mean first-passage time （MFPT） between two nodes averaged over all pairs of nodes on a fractal network. In this paper, we first propose a family of generalized delayed recursive trees characterized by two parameters, where the existing nodes have a time delay to produce new nodes. We then study the MFPT of random walks on this kind of recursive tree and investigate the effect of the time delay on the MFPT. By relating random walks to electrical networks, we obtain an exact formula for the MFPT and verify it by numerical calculations. Based on the obtained results, we further show that the MFPT of delayed recursive trees is much shorter, implying that the efficiency of random walks is much higher compared with the non-delayed counterpart. Our study provides a deeper understanding of random walks on delayed fractal networks.

两嵌段共聚高分子链通过薄膜上纳米孔隙输运的研究

针对两嵌段高分子链的跨膜输运过程,分别给出与不同输运次序相对应的高分子链的自由能,进而通过求解Fokker-Planck方程并在不同条件下对平均首次通过时间进行了数值计算.计算结果表明,当共聚高分子链由良溶剂区向不良溶剂区输运时,不能发生线团-链滴转变的链首先输运总是有利于整个高分子链的输运.而在给定输运次序的情况下,化学势、线团-链滴转变、共聚链的组成以及输运速率等因素对输运时间可产生显著影响.相关研究结果可为调控实际生物高分子链的输运时间提供可能的理论线索.

Quantum model for understanding protein misfolding behavior——phase diagram and manual intervention

The study of protein folding is fundamental and important in the multidisciplinary field because a diversity of diseases,like Alzheimer’s and Parkinson’s are relevant to protein misfolding.The current thermodynamic and geometric approaches only phenomenologically describe but do not provide a mechanistic understanding of the competition between correct folding and misfolding.Here we present a model to understand the misfolding behavior.Considering the influence of dissipative strength for all possible sequences and comparing the folding time toward different compact structures,we obtain a phase diagram of the dissipative quantum phase transition that enables us to model the behavior.We also investigate how a perturbation in the Hamiltonian affects the transition point,which motivates us to explore possible manual interventions.Our results indicate that the manual intervention may be effective for some specific sequence but not for everyone.This approach is expected to lay a foundation for further studies on manual intervention in protein misfolding.