First-passage-time distribution in a moving parabolic potential with spatial roughness

Computing probability density of the first passage time for state transition in stochastic dynamical systems driven by Brownian motions: A singular integral method

Nonlinear dynamical systems, such as climate systems, often switch from one metastable state to another when subject to noise. The first occurrence of such state transition, which is usually characterized by the first passage time, has gained enormous interest in many engineering and scientific fields. We develop an efficient numerical method to compute the probability density of the first passage time for state transitions in stochastic dynamical systems driven by Brownian motions. The proposed method involves solving a singular integral equation, which determines probability density of the first passage time. Some numerical examples, with application to a simplified thermohaline circulation system, are provided to illustrate and verify the proposed method.

Microscopic origin of diffusive dynamics in the context of transition path time distributions for protein folding and unfolding

Experimentally measured transition path time distributions are usually analyzed theoretically in terms of a diffusion equation over a free energy barrier. It is though well understood that the free energy profile separating the folded and unfolded states of a protein is characterized as a transition through many stable micro-states which exist between the folded and unfolded states. Why is it then justified to model the transition path dynamics in terms of a diffusion equation, namely the Smoluchowski equation (SE)? In principle, van Kampen has shown that a nearest neighbor Markov chain of thermal jumps between neighboring microstates will lead in a continuum limit to the SE, such that the friction coefficient is proportional to the mean residence time in each micro-state. However, the practical question of how many microstates are needed to justify modeling the transition path dynamics in terms of an SE has not been addressed. This is a central topic of this paper where we compare numerical results for transition paths based on the diffusion equation on the one hand and the nearest neighbor Markov jump model on the other. Comparison of the transition path time distributions shows that one needs at least a few dozen microstates to obtain reasonable agreement between the two approaches. Using the Markov nearest neighbor model one also obtains good agreement with the experimentally measured transition path time distributions for a DNA hairpin and PrP protein. As found previously when using the diffusion equation, the Markov chain model used here also reproduces the experimentally measured long time tail and confirms that the transition path barrier height is ∼3k_BT. This study indicates that in the future, when attempting to model experimentally measured transition path time distributions, one should perhaps prefer a nearest neighbor Markov model which is well defined also for rough energy landscapes. Such studies can also shed light on the minimal number of microstates needed to unravel the experimental data.

Variational inference of the drift function for stochastic differential equations driven by Lévy processes

In this work, we consider the nonparametric estimation problem of the drift function of stochastic differential equations driven by the α-stable Lévy process. We first optimize the Kullback-Leibler divergence between the path probabilities of two stochastic differential equations with different drift functions. We then construct the variational formula based on the stationary Fokker-Planck equation using the Lagrangian multiplier. Moreover, we apply the empirical distribution to replace the stationary density, combining it with the data information, and we present the estimator of the drift function from the perspective of the process. In the numerical experiment, we investigate the effect of the different amounts of data and different α values. The experimental results demonstrate that the estimation result of the drift function is related to both and that the exact drift function agrees well with the estimated result. The estimation result will be better when the amount of data increases, and the estimation result is also better when the α value increases.

Stochastic sensitivity analysis and early warning signals of critical transitions in a tri-stable prey-predator system with noise

Near a tipping point, small changes in a certain parameter cause an irreversible shift in the behavior of a system, called critical transitions. Critical transitions can be observed in a variety of complex dynamical systems, ranging from ecology to financial markets, climate change, molecular bio-systems, health, and disease. As critical transitions can occur suddenly and are hard to manage, it is important to predict their occurrence. Although it is very tough to predict such critical transitions, various recent works suggest that generic early warning signals can detect the situation when systems approach a critical point. The most important indicator that predicts the risk of an upcoming critical transition is critical slowing down (CSD). CSD indicates a slow recovery rate from external perturbations of the stable state close to a bifurcation point. In this contribution, we study a two dimensional prey-predator model. Without any noise, the prey-predator model shows bistability and tri-stability due to the Allee effect in predators. We explore the critical transitions when external noise is added to the prey-predator system. We investigate early warning indicators, e.g., recovery rate, lag-1 autocorrelation, variance, and skewness to predict the critical transition. We explore the confidence domain method using the stochastic sensitivity function (SSF) technique near a stable equilibrium point to find a threshold value of noise intensity for a transition. The SSF technique in a two stage transition through confidence ellipse is described. We also show that the possibility of a transition to the predator-free state is independent of initial conditions. Our result may serve as a paradigm to understand and predict the critical transition in a two dimensional system.

Most probable transitions from metastable to oscillatory regimes in a carbon cycle system

Global climate changes are related to the ocean's store of carbon. We study a carbonate system of the upper ocean, which has metastable and oscillatory regimes, under small random fluctuations. We calculate the most probable transition path via a geometric minimum action method in the context of the large deviation theory. By examining the most probable transition paths from metastable to oscillatory regimes for various external carbon input rates, we find two different transition patterns, which gives us an early warning sign for the dramatic change in the carbonate state of the ocean.

Heterogeneous diffusion processes and nonergodicity with Gaussian colored noise in layered diffusivity landscapes

Heterogeneous diffusion processes (HDPs) with space-dependent diffusion coefficients D(x) are found in a number of real-world systems, such as for diffusion of macromolecules or submicron tracers in biological cells. Here, we examine HDPs in quenched-disorder systems with Gaussian colored noise (GCN) characterized by a diffusion coefficient with a power-law dependence on the particle position and with a spatially random scaling exponent. Typically, D(x) is considered to be centerd at the origin and the entire x axis is characterized by a single scaling exponent α. In this work we consider a spatially random scenario: in periodic intervals ("layers") in space D(x) is centerd to the midpoint of each interval. In each interval the scaling exponent α is randomly chosen from a Gaussian distribution. The effects of the variation of the scaling exponents, the periodicity of the domains ("layer thickness") of the diffusion coefficient in this stratified system, and the correlation time of the GCN are analyzed numerically in detail. We discuss the regimes of superdiffusion, subdiffusion, and normal diffusion realisable in this system. We observe and quantify the domains where nonergodic and non-Gaussian behaviors emerge in this system. Our results provide new insights into the understanding of weak ergodicity breaking for HDPs driven by colored noise, with potential applications in quenched layered systems, typical model systems for diffusion in biological cells and tissues, as well as for diffusion in geophysical systems.

Atomic Force Microscopy-Based Force Spectroscopy and Multiparametric Imaging of Biomolecular and Cellular Systems

During the last three decades, a series of key technological improvements turned atomic force microscopy (AFM) into a nanoscopic laboratory to directly observe and chemically characterize molecular and cell biological systems under physiological conditions. Here, we review key technological improvements that have established AFM as an analytical tool to observe and quantify native biological systems from the micro- to the nanoscale. Native biological systems include living tissues, cells, and cellular components such as single or complexed proteins, nucleic acids, lipids, or sugars. We showcase the procedures to customize nanoscopic chemical laboratories by functionalizing AFM tips and outline the advantages and limitations in applying different AFM modes to chemically image, sense, and manipulate biosystems at (sub)nanometer spatial and millisecond temporal resolution. We further discuss theoretical approaches to extract the kinetic and thermodynamic parameters of specific biomolecular interactions detected by AFM for single bonds and extend the discussion to multiple bonds. Finally, we highlight the potential of combining AFM with optical microscopy and spectroscopy to address the full complexity of biological systems and to tackle fundamental challenges in life sciences.

The influences of correlated spatially random perturbations on first passage time in a linear-cubic potential

The influences of correlated spatially random perturbations (SRPs) on the first passage problem are studied in a linear-cubic potential with a time-changing external force driven by a Gaussian white noise. First, the escape rate in the absence of SRPs is obtained by Kramers' theory. For the random potential case, we simplify the escape rate by multiplying the escape rate of smooth potentials with a specific coefficient, which is to evaluate the influences of randomness. Based on this assumption, the escape rates are derived in two scenarios, i.e., small/large correlation lengths. Consequently, the first passage time distributions (FPTDs) are generated for both smooth and random potential cases. We find that the position of the maximal FPTD has a very good agreement with that of numerical results, which verifies the validity of the proposed approximations. Besides, with increasing the correlation length, the FPTD shifts to the left gradually and tends to the smooth potential case. Second, we investigate the most probable passage time (MPPT) and mean first passage time (MFPT), which decrease with increasing the correlation length. We also find that the variation ranges of both MPPT and MFPT increase nonlinearly with increasing the intensity. Besides, we briefly give constraint conditions to guarantee the validity of our approximations. This work enables us to approximately evaluate the influences of the correlation length of SRPs in detail, which was always ignored previously.

Predicting noise-induced critical transitions in bistable systems

Critical transitions from one dynamical state to another contrasting state are observed in many complex systems. To understand the effects of stochastic events on critical transitions and to predict their occurrence as a control parameter varies are of utmost importance in various applications. In this paper, we carry out a prediction of noise-induced critical transitions using a bistable model as a prototype class of real systems. We find that the largest Lyapunov exponent and the Shannon entropy can act as general early warning indicators to predict noise-induced critical transitions, even for an earlier transition due to strong fluctuations. Furthermore, the concept of the parameter dependent basin of the unsafe regime is introduced via incorporating a suitable probabilistic notion. We find that this is an efficient tool to approximately quantify the range of the control parameter where noise-induced critical transitions may occur. Our method may serve as a paradigm to understand and predict noise-induced critical transitions in multistable systems or complex networks and even may be extended to a broad range of disciplines to address the issues of resilience.