A stochastic hybrid framework for obtaining statistics of many random walkers in a switching environment

We analyze a population of randomly walking particles in a stochastically switching environment by formulating the model as a stochastic hybrid system. The latter describes the evolution of the probability distribution of the particles, which is a random variable depending on realizations of the random environment. We derive a hierarchy of moment equations for the probability distribution, which allows us to extract statistics of the multiparticle system. As a specific example, we consider a population of particles walking on a one-dimensional lattice with a dynamic gate at some unknown location, which stochastically switches between an open and closed state according to a two-state Markov process. This type of model has two levels of stochasticity: one due to the jump process describing the evolution of each particle on the lattice, and the other due to the switching of the gate. By solving the moment equations for the stochastic hybrid system, we extract statistical information about the location and dynamics of the gate in terms of how the mean and variance of site occupancies varies with distance of a given site from the gate. This has potential applications in the analysis of time series data obtained from biophysical experiments on the diffusion of particles in the plasma membrane of cells

Mathematical Modelling of Cell Migration and Polarization

Cell migration plays a fundamental role in both development and disease. It is a complex process during which cells interact with one another and with their local environment. Mathematical modelling offers tools to investigate such processes and can give insights into the underlying biological details, and can also guide new experiments.The first two papers of this thesis are concerned with modelling durotaxis, which is the phenomena where cells migrate preferentially up a stiffness gradient. Two distinct mechanisms which potentially drive durotaxis are investigated. One is based on the hypothesis that adhesion sites of migrating cells become reinforced and have a longer lifespan on stiffer substrates. The second mechanism is based on cells being able to generate traction forces, the magnitude of which depend on the stiffness of the substrate. We find that both mechanisms can indeed give rise to biased migration up a stiffness gradient. Our results encourages new experiments which could determine the importance of the two mechanisms in durotaxis.The third paper is devoted to a population-level model of cancer cells in the brain of mice. The model incorporates diffusion tensor imaging data, which is used to guide the migration of the cells. Model simulations are compared to experimental data, and highlights the model’s difficulty in producing irregular growth patterns observed in the experiments. As a consequence, the findings encourage further model development.The fourth paper is concerned with modelling cell polarization, in the absence of environmental cues, referred to as spontaneous symmetry breaking. Polarization is an important part of cell migration, but also plays a role during division and differentiation. The model takes the form of a reaction diffusion system in 3D and describes the spatio-temporal evolution of three forms of Cdc42 in the cell. The model is able to produce biologically relevant patterns, and numerical simulations show how model parameters influence key features such as pattern formation and time to polarization

Tractable Quantification of Metastability for Robust Bipedal Locomotion

This work develops tools to quantify and optimize performance metrics for bipedal walking, toward enabling improved practical and autonomous operation of two-legged robots in real-world environments. While speed and energy efficiency of legged locomotion are both useful and straightforward to quantify, measuring robustness is arguably more challenging and at least as critical for obtaining practical autonomy in variable or otherwise uncertain environmental conditions, including rough terrain. The intuitive and meaningful robustness quantification adopted in this thesis begins by stochastic modeling of disturbances such as terrain variations, and conservatively defining what a failure is, for example falling down, slippage, scuffing, stance foot rotation, or a combination of such events. After discretizing the disturbance and state sets by meshing, step-to-step dynamics are studied to treat the system as a Markov chain. Then, failure rates can be easily quantified by calculating the expected number of steps before failure. Once robustness is measured, other performance metrics can also be easily incorporated into the cost function for optimization.For high performance and autonomous operation under variations, we adopt a capacious framework, exploiting a hierarchical control structure. The low-level controllers, which use only proprioceptive (internal state) information, are optimized by a derivative-free method without any constraints. For practicability of this process, developing an algorithm for fast and accurate computation of our robustness metric was a crucial and necessary step. While the outcome of optimization depends on capabilities of the controller scheme employed, the convenient and time-invariant parameterization presented in this thesis ensures accommodating large terrain variations. In addition, given environment estimation and state information, the high-level control is a behavioral policy to choose the right low-level controller at each step. In this thesis, optimal switching policies are determined by applying dynamic programming tools on Markov decision processes obtained through discretization. For desirable performance in practice from policies that are formed using meshing-based approximation to the true dynamics, robustness of high-level control to environment estimation and discretization errors are ensured by modeling stochastic noise in the terrain information and belief state while solving for behavioral policies

Control and Data Analysis of Complex Networks

abstract: This dissertation treats a number of related problems in control and data analysis of complex networks. First, in existing linear controllability frameworks, the ability to steer a network from any initiate state toward any desired state is measured by the minimum number of driver nodes. However, the associated optimal control energy can become unbearably large, preventing actual control from being realized. Here I develop a physical controllability framework and propose strategies to turn physically uncontrollable networks into physically controllable ones. I also discover that although full control can be guaranteed by the prevailing structural controllability theory, it is necessary to balance the number of driver nodes and control energy to achieve actual control, and my work provides a framework to address this issue. Second, in spite of recent progresses in linear controllability, controlling nonlinear dynamical networks remains an outstanding problem. Here I develop an experimentally feasible control framework for nonlinear dynamical networks that exhibit multistability. The control objective is to apply parameter perturbation to drive the system from one attractor to another. I introduce the concept of attractor network and formulate a quantifiable framework: a network is more controllable if the attractor network is more strongly connected. I test the control framework using examples from various models and demonstrate the beneficial role of noise in facilitating control. Third, I analyze large data sets from a diverse online social networking (OSN) systems and find that the growth dynamics of meme popularity exhibit characteristically different behaviors: linear, “S”-shape and exponential growths. Inspired by cell population growth model in microbial ecology, I construct a base growth model for meme popularity in OSNs. Then I incorporate human interest dynamics into the base model and propose a hybrid model which contains a small number of free parameters. The model successfully predicts the various distinct meme growth dynamics. At last, I propose a nonlinear dynamics model to characterize the controlling of WNT signaling pathway in the differentiation of neural progenitor cells. The model is able to predict experiment results and shed light on the understanding of WNT regulation mechanisms.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

Stochastic switching in biology: from genotype to phenotype

There has been a resurgence of interest in non-equilibrium stochastic processes in recent years, driven in part by the observation that the number of molecules (genes, mRNA, proteins) involved in gene expression are often of order 1–1000. This means that deterministic mass-action kinetics tends to break down, and one needs to take into account the discrete, stochastic nature of biochemical reactions. One of the major consequences of molecular noise is the occurrence of stochastic biological switching at both the genotypic and phenotypic levels. For example, individual gene regulatory networks can switch between graded and binary responses, exhibit translational/transcriptional bursting, and support metastability (noise-induced switching between states that are stable in the deterministic limit). If random switching persists at the phenotypic level then this can confer certain advantages to cell populations growing in a changing environment, as exemplified by bacterial persistence in response to antibiotics. Gene expression at the single-cell level can also be regulated by changes in cell density at the population level, a process known as quorum sensing. In contrast to noise-driven phenotypic switching, the switching mechanism in quorum sensing is stimulus-driven and thus noise tends to have a detrimental effect. A common approach to modeling stochastic gene expression is to assume a large but finite system and to approximate the discrete processes by continuous processes using a system-size expansion. However, there is a growing need to have some familiarity with the theory of stochastic processes that goes beyond the standard topics of chemical master equations, the system-size expansion, Langevin equations and the Fokker–Planck equation. Examples include stochastic hybrid systems (piecewise deterministic Markov processes), large deviations and the Wentzel–Kramers–Brillouin (WKB) method, adiabatic reductions, and queuing/renewal theory. The major aim of this review is to provide a self-contained survey of these mathematical methods, mainly within the context of biological switching processes at both the genotypic and phenotypic levels. However, applications to other examples of biological switching are also discussed, including stochastic ion channels, diffusion in randomly switching environments, bacterial chemotaxis, and stochastic neural networks

Robustness of stochastic chemical reaction networks to extrinsic noise: the role of deficiency

The biochemical systems inside a living cell are able to reliably perform complex tasks while subjected to various sources of noise. In this study we consider stochastic models of biochemical networks evolving in the presence of dynamic random environments. These environments are themselves modeled as chemical reaction networks so that the full system can be viewed as a multiscale chemical reaction network. The multiscale structure arises from the fact that the environment and the internal system may operate on different timescales. While previous results in chemical reaction network theory have established that certain dynamic behavior can be ruled out when a topological parameter, known as the network deficiency, is zero, these results fail to capture the behavior that can be observed in multiscale networks. We demonstrate that the deficiency of the network has implications for how robust it is to environmental noise. We then show how our results can be used to prove that correlations in a population of chemical reaction networks in a random environment vanish given certain topological constraints

A First-Passage Kinetic Monte Carlo Algorithm for Complex Diffusion-Reaction Systems

We develop an asynchronous event-driven First-Passage Kinetic Monte Carlo (FPKMC) algorithm for continuous time and space systems involving multiple diffusing and reacting species of spherical particles in two and three dimensions. The FPKMC algorithm presented here is based on the method introduced in [Phys. Rev. Lett., 97:230602, 2006] and is implemented in a robust and flexible framework. Unlike standard KMC algorithms such as the n-fold algorithm, FPKMC is most efficient at low densities where it replaces the many small hops needed for reactants to find each other with large first-passage hops sampled from exact time-dependent Green's functions, without sacrificing accuracy. We describe in detail the key components of the algorithm, including the event-loop and the sampling of first-passage probability distributions, and demonstrate the accuracy of the new method. We apply the FPKMC algorithm to the challenging problem of simulation of long-term irradiation of metals, relevant to the performance and aging of nuclear materials in current and future nuclear power plants. The problem of radiation damage spans many decades of time-scales, from picosecond spikes caused by primary cascades, to years of slow damage annealing and microstructure evolution. Our implementation of the FPKMC algorithm has been able to simulate the irradiation of a metal sample for durations that are orders of magnitude longer than any previous simulations using the standard Object KMC or more recent asynchronous algorithms.Comment: See also arXiv:0905.357

Pedestrian Dynamics: Modeling and Analyzing Cognitive Processes and Traffic Flows to Evaluate Facility Service Level

Walking is the oldest and foremost mode of transportation through history and the prevalence of walking has increased. Effective pedestrian model is crucial to evaluate pedestrian facility service level and to enhance pedestrian safety, performance, and satisfaction. The objectives of this study were to: (1) validate the efficacy of utilizing queueing network model, which predicts cognitive information processing time and task performance; (2) develop a generalized queueing network based cognitive information processing model that can be utilized and applied to construct pedestrian cognitive structure and estimate the reaction time with the first moment of service time distribution; (3) investigate pedestrian behavior through naturalistic and experimental observations to analyze the effects of environment settings and psychological factors in pedestrians; and (4) develop pedestrian level of service (LOS) metrics that are quick and practical to identify improvement points in pedestrian facility design. Two empirical and two analytical studies were conducted to address the research objectives. The first study investigated the efficacy of utilizing queueing network in modeling and predicting the cognitive information processing time. Motion capture system was utilized to collect detailed pedestrian movement. The predicted reaction time using queueing network was compared with the results from the empirical study to validate the performance of the model. No significant difference between model and empirical results was found with respect to mean reaction time. The second study endeavored to develop a generalized queueing network system so the task can be modeled with the approximated queueing network and its first moment of any service time distribution. There was no significant difference between empirical study results and the proposed model with respect to mean reaction time. Third study investigated methods to quantify pedestrian traffic behavior, and analyze physical and cognitive behavior from the real-world observation and field experiment. Footage from indoor and outdoor corridor was used to quantify pedestrian behavior. Effects of environmental setting and/or psychological factor on travel performance were tested. Finally, adhoc and tailor-made LOS metrics were presented for simple realistic service level assessments. The proposed methodologies were composed of space revision LOS, delay-based LOS, preferred walking speed-based LOS, and ‘blocking probability’