Download or read book Model Order Reduction for Stochastic Models of Biomolecular Systems with Time scale Separation written by Narmada Kumari Herath and published by . This book was released on 2018 with total page 183 pages. Available in PDF, EPUB and Kindle. Book excerpt: Biomolecular systems often involve reactions that take place on different time-scales, giving rise to 'slow' and 'fast' system variables. This property is widely used in the analysis of systems to obtain dynamical models with reduced dimensions. In deterministic systems, methods to obtain such reduced-order models are well defined by the singular perturbation or averaging techniques. However, model reduction of stochastic systems remains an ongoing area of research. In particular, existing model reduction methods for stochastic models of biomolecular systems lack rigorous error quantifications between the full and reduced dynamics. Furthermore, they only provide approximations for the slow variable dynamics, making the application of such methods to biomolecular systems difficult since the variables of interest are typically mixed (i.e., they encompass both fast and slow variables). In this thesis, we consider biomolecular systems modeled using the chemical Langevin equation (CLE) and the Linear Noise Approximation (LNA). Specifically, we consider biomolecular systems with linear propensity functions modeled by the CLE and systems with arbitrary propensity functions modeled by the LNA. For these systems, we obtain reduced-order models that approximate both the slow and fast variables under time-scale separation conditions. In particular, with suitable assumptions, we prove that the moments of the reduced-order models converge to those of the full systems as the time-scale separation becomes large. Our results further provide a rigorous justification for the accuracy of the stochastic total quasi-steady state approximation (tQSSA). We then consider two applications of these reduced-order models. In the first application, we analyze the trade-offs between modularity and signal noise in biomolecular networks. In the second application, we consider the application of the reduced-order LNA developed in this work to obtain reduced-order stochastic models for gene-regulatory networks.

Download or read book Stochastic Modeling of Biomolecular Systems Using the Data driven Langevin Equation written by Norbert Schaudinnus and published by . This book was released on 2015 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Zusammenfassung: Nowadays, molecular dynamics (MD) simulations are widely used to understand the dynamics of biomolecular systems.Although MD simulations provide a description in atomistic detail, they are limited with respect to both system size and simulation time for computational reasons.However, dynamics which take place on rather long time scales can often be captured along few collective coordinates.The data-driven Langevin Equation (dLE) is a systematic approach to capture dynamical information in this reduced space, establishing local field estimates from an underlying time series.These fields are in turn used to propagate dynamics.The dLE therefore relies on a clear timescale separation of a low-dimensional coordinate that enables to treat the remaining degrees of freedom as a stochastic bath.This thesis focuses on the characteristics and capabilities of the dLE approach. The property of the dLE to operate in a self-consistent manner can be applied to test the validity of the model assumptions.In particular, an important extension of the dLE is developed which accounts for inertia effects.This more generalized form of the dLE is shown to suitably recover the conformational dynamics of a small peptide system, showing a nontrivial dynamical behavior.Aside from establishing a technical guideline on how to construct a dLE model and verify its functionality, several promising applications are demonstrated.For instance, a dLE-based enhanced sampling scheme is introduced which facilitates to forecast the long-time behavior of a reduced system from short spatially overlapping trajectory samples

Download or read book Stochastic Models in Biology written by Narendra S. Goel and published by Elsevier. This book was released on 2013-10-22 with total page 282 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic Models in Biology describes the usefulness of the theory of stochastic process in studying biological phenomena. The book describes analysis of biological systems and experiments though probabilistic models rather than deterministic methods. The text reviews the mathematical analyses for modeling different biological systems such as the random processes continuous in time and discrete in state space. The book also discusses population growth and extinction through Malthus' law and the work of MacArthur and Wilson. The text then explains the dynamics of a population of interacting species. The book also addresses population genetics under systematic evolutionary pressures known as deterministic equations and genetic changes in a finite population known as stochastic equations. The text then turns to stochastic modeling of biological systems at the molecular level, particularly the kinetics of biochemical reactions. The book also presents various useful equations such as the differential equation for generating functions for birth and death processes. The text can prove valuable for biochemists, cellular biologists, and researchers in the medical and chemical field who are tasked to perform data analysis.

Download or read book Stochastic Dynamics in Computational Biology written by Stefanie Winkelmann and published by Springer Nature. This book was released on 2021-01-04 with total page 284 pages. Available in PDF, EPUB and Kindle. Book excerpt: The aim of this book is to provide a well-structured and coherent overview of existing mathematical modeling approaches for biochemical reaction systems, investigating relations between both the conventional models and several types of deterministic-stochastic hybrid model recombinations. Another main objective is to illustrate and compare diverse numerical simulation schemes and their computational effort. Unlike related works, this book presents a broad scope in its applications, from offering a detailed introduction to hybrid approaches for the case of multiple population scales to discussing the setting of time-scale separation resulting from widely varying firing rates of reaction channels. Additionally, it also addresses modeling approaches for non well-mixed reaction-diffusion dynamics, including deterministic and stochastic PDEs and spatiotemporal master equations. Finally, by translating and incorporating complex theory to a level accessible to non-mathematicians, this book effectively bridges the gap between mathematical research in computational biology and its practical use in biological, biochemical, and biomedical systems.

Download or read book Stochastic Biomathematical Models written by Mostafa Bachar and published by Springer. This book was released on 2012-10-19 with total page 216 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic biomathematical models are becoming increasingly important as new light is shed on the role of noise in living systems. In certain biological systems, stochastic effects may even enhance a signal, thus providing a biological motivation for the noise observed in living systems. Recent advances in stochastic analysis and increasing computing power facilitate the analysis of more biophysically realistic models, and this book provides researchers in computational neuroscience and stochastic systems with an overview of recent developments. Key concepts are developed in chapters written by experts in their respective fields. Topics include: one-dimensional homogeneous diffusions and their boundary behavior, large deviation theory and its application in stochastic neurobiological models, a review of mathematical methods for stochastic neuronal integrate-and-fire models, stochastic partial differential equation models in neurobiology, and stochastic modeling of spreading cortical depression.

Download or read book Stochastic Chemical Reaction Systems in Biology written by Hong Qian and published by Springer Nature. This book was released on 2021-10-18 with total page 364 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an introduction to the analysis of stochastic dynamic models in biology and medicine. The main aim is to offer a coherent set of probabilistic techniques and mathematical tools which can be used for the simulation and analysis of various biological phenomena. These tools are illustrated on a number of examples. For each example, the biological background is described, and mathematical models are developed following a unified set of principles. These models are then analyzed and, finally, the biological implications of the mathematical results are interpreted. The biological topics covered include gene expression, biochemistry, cellular regulation, and cancer biology. The book will be accessible to graduate students who have a strong background in differential equations, the theory of nonlinear dynamical systems, Markovian stochastic processes, and both discrete and continuous state spaces, and who are familiar with the basic concepts of probability theory.

Download or read book Biomolecular Feedback Systems written by Domitilla Del Vecchio and published by Princeton University Press. This book was released on 2014-10-26 with total page 287 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book provides an accessible introduction to the principles and tools for modeling, analyzing, and synthesizing biomolecular systems. It begins with modeling tools such as reaction-rate equations, reduced-order models, stochastic models, and specific models of important core processes. It then describes in detail the control and dynamical systems tools used to analyze these models. These include tools for analyzing stability of equilibria, limit cycles, robustness, and parameter uncertainty. Modeling and analysis techniques are then applied to design examples from both natural systems and synthetic biomolecular circuits. In addition, this comprehensive book addresses the problem of modular composition of synthetic circuits, the tools for analyzing the extent of modularity, and the design techniques for ensuring modular behavior. It also looks at design trade-offs, focusing on perturbations due to noise and competition for shared cellular resources. Featuring numerous exercises and illustrations throughout, Biomolecular Feedback Systems is the ideal textbook for advanced undergraduates and graduate students. For researchers, it can also serve as a self-contained reference on the feedback control techniques that can be applied to biomolecular systems. Provides a user-friendly introduction to essential concepts, tools, and applications Covers the most commonly used modeling methods Addresses the modular design problem for biomolecular systems Uses design examples from both natural systems and synthetic circuits Solutions manual (available only to professors at press.princeton.edu) An online illustration package is available to professors at press.princeton.edu

Download or read book Stochastic Processes Multiscale Modeling and Numerical Methods for Computational Cellular Biology written by David Holcman and published by Springer. This book was released on 2017-10-04 with total page 377 pages. Available in PDF, EPUB and Kindle. Book excerpt: This book focuses on the modeling and mathematical analysis of stochastic dynamical systems along with their simulations. The collected chapters will review fundamental and current topics and approaches to dynamical systems in cellular biology. This text aims to develop improved mathematical and computational methods with which to study biological processes. At the scale of a single cell, stochasticity becomes important due to low copy numbers of biological molecules, such as mRNA and proteins that take part in biochemical reactions driving cellular processes. When trying to describe such biological processes, the traditional deterministic models are often inadequate, precisely because of these low copy numbers. This book presents stochastic models, which are necessary to account for small particle numbers and extrinsic noise sources. The complexity of these models depend upon whether the biochemical reactions are diffusion-limited or reaction-limited. In the former case, one needs to adopt the framework of stochastic reaction-diffusion models, while in the latter, one can describe the processes by adopting the framework of Markov jump processes and stochastic differential equations. Stochastic Processes, Multiscale Modeling, and Numerical Methods for Computational Cellular Biology will appeal to graduate students and researchers in the fields of applied mathematics, biophysics, and cellular biology.

Download or read book Model Reduction Using Multiple Time Scales in Stochastic Gene Regulatory Networks written by and published by . This book was released on 2006 with total page 13 pages. Available in PDF, EPUB and Kindle. Book excerpt: Gene network dynamics often involves processes that take place on widely differing time scales -- from the order of nanoseconds to the order of several days. Multiple time scales in mathematical models often lead to serious computational difficulties, such as numerical stiffness in the case of differential equations or excessively redundant Monte Carlo simulations in the case of stochastic processes. We present a method that takes advantage of multiple time scales and dramatically reduces the computational time for a broad class of problems arising in stochastic gene regulatory networks. We illustrate the efficiency of our method in two gene network examples, which describe two substantially different biological processes -- cellular heat shock response and expression of the pap gene in Escherichia coli bacteria.

Download or read book Deterministic Versus Stochastic Modelling in Biochemistry and Systems Biology written by Paola Lecca and published by Woodhead Publishing. This book was released on 2013-04-09 with total page 0 pages. Available in PDF, EPUB and Kindle. Book excerpt: Stochastic kinetic methods are currently considered to be the most realistic and elegant means of representing and simulating the dynamics of biochemical and biological networks. Deterministic versus stochastic modelling in biochemistry and systems biology introduces and critically reviews the deterministic and stochastic foundations of biochemical kinetics, covering applied stochastic process theory for application in the field of modelling and simulation of biological processes at the molecular scale. Following an overview of deterministic chemical kinetics and the stochastic approach to biochemical kinetics, the book goes onto discuss the specifics of stochastic simulation algorithms, modelling in systems biology and the structure of biochemical models. Later chapters cover reaction-diffusion systems, and provide an analysis of the Kinfer and BlenX software systems. The final chapter looks at simulation of ecodynamics and food web dynamics.

Download or read book Model Order Reduction for Stochastic Systems written by Yolanda Rocío Rodríguez Cruz and published by . This book was released on 2018 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt:

Download or read book Stochastic Models for Nonlinear Transport in Multiphase and Multiscale Heterogeneous Media written by Farzaneh Rajabi and published by . This book was released on 2021 with total page pages. Available in PDF, EPUB and Kindle. Book excerpt: Elucidating multiscale, multiphase and multiphysics phenomena of flow and transport processes in porous media is the cornerstone of numerous environmental and engineering applications. Several factors including spatial and temporal heterogeneity on a continuity of scales, the strong coupling of processes at such different scales at least at a localized region within the domain, combined with the nonlinearity of processes calls for a new modeling paradigm called multiscale models, which are able to properly address all such issues while presenting an accurate descriptive model for processes occurring at field scale applications. Furthermore, the typical temporal resolution used in modern simulations significantly exceeds characteristic time scales at which the system is driven and a solution is sought. This is especially so when systems are simulated over time scales that are much longer than the typical temporal scales of forcing factors. In addition to spatial and temporal heterogeneity, mixing and spreading of contaminants in the subsurface is remarkably influenced by oscillatory forcing factors. While the pore-scale models are able to handle the experimentally-observed phenomena, they are not always the best choice due to the high computational burden. Although handling across-scale coupling in environments with several simultaneous physical mechanisms such as advection, diffusion, reaction, and fluctuating boundary forcing factors complicates the theoretical and numerical modeling capabilities at high resolutions, multiscale models come to rescue. To this end, we investigate the impact of space-time upscaling on reactive transport in porous media driven by time-dependent boundary conditions whose characteristic time scale is much smaller than that at which transport is studied or observed at the macroscopic level. We first introduce the concept of spatiotemporal upscaling in the context of homogenization by multiple-scale expansions, and demonstrate the impact of time-dependent forcings and boundary conditions on macroscopic reactive transport. Proposing such a framework, we scrutinize the behavior of porous media for ``quasisteady stage time'' (thousands of years), where there is an interplay between signal frequency and the three physical underlying mechanisms; advection, molecular diffusion and heterogeneous reaction. To this end, we demonstrate that the transient forcing factors augment the solute mixing as they are combined with diffusion at the pore-scale. We then derive the macroscopic equation as well as the corresponding applicability criteria based on the order of magnitude of the dimensionless Peclet and Damkohler numbers. Also, we demonstrate that the dynamics at the continuum scale is strongly influenced by the interplay between signal frequency at the boundary and transport processes at the pore level. We validate such a framework for reactive transport in a planar fracture in which the single-component solute particle is undergoing nonlinear first-order heterogeneous reaction at the solid-liquid interface, while the medium is episodically influenced by time-dependent boundary conditions at the inlet. We also present the alternative effective transport model at a much lower cost, albeit at the regions where the corresponding applicability criteria are satisfied. We perform direct numerical simulations to study several test cases with different controlling parameters i.e. Peclet and Damkohler numbers and the space/time scale separation parameters; the ratio of characteristic transversal and longitudinal lengths $\varepsilon$ and $\omega$; the ratio of period of time-fluctuating boundary conditions to the observation time scale. A rigorous justification of the effective transport model for the given applicability conditions is demonstrated, essentially by comparing the local vertically averaged microscopic simulations with their corresponding macroscopic counterparts. Moreover, as a special case, we employ a singular perturbation technique to look at the effective model for vertical mixing through a narrow and long two-dimensional pore. We obtain explicit expressions for dispersion tensor as well as the other effective coefficients in the coarse-scale homogenized equation. Our analysis manifests robustness of the sufficient and necessary applicability constraints which validate the upscaled model as a solid replacement of the pore-scale one within the accuracy prescribed by homogenization theory. While a deterministic model is sufficiently robust for a plethora of subsurface applications, a more realistic setting is often required when dealing with other scopes of engineering applications, e.g. reservoir engineering and enhanced oil recovery. Rigorous modeling of these systems calls for sophisticated strategies for uncertainty quantification and stochastic treatment of the system under study. Such an uncertainty is inherent to, and critical for any physical modeling, essentially due to the incomplete knowledge of state of the world, noisy observations, and limitations in systematically recasting physical processes in a suitable mathematical framework. To this end, accurate predictions of outputs (e.g. saturation fields) from reservoir simulations guarantee precise oil recovery forecasts. These quantitative predictions rely on the quality of the input measurements/data, such as the reservoir permeability and porosity fields as well as forcings, such as initial and boundary conditions. However, the available information about a particular geologic formation, e.g. from well logs and seismic data of an outcrop, is usually sparse and inaccurate compared to the size of the natural system and the complexity arising from multiscale heterogeneity of the underlying system. Eventually, the uncertainty in the flow prediction can have a huge impact on the oil recovery. Consequently, we also develop a probabilistic approach to map the parametric uncertainty to the output state uncertainty in first-order hyperbolic conservation laws. We analyze this problem for nonlinear immiscible two-phase transport (Buckley-Leverett displacement) in heterogeneous porous media in the presence of a stochastic velocity field, where the uncertainty in the velocity field can arise from the incomplete description of either porosity field, injection flux, or both. Such uncertainty leads to the spatiotemporal uncertainty in the outputs of the problem. Given information about the spatial/temporal statistics of the correlated heterogeneity, we leverage method of distributions (MD) to derive deterministic equations that govern the evolution of single-point CDF of saturation in the form of linear hyperbolic conservation laws. We first derive the semi-analytical solution of the raw CDF of saturation at a given point, for the cases in which two shocks are present due to the gravitational forces. Then, we describe development of the partial differential equation that governs the evolution of the raw CDF of saturation, subject to uniquely specified boundary conditions in the phase space, wherein no closure approximations are required. Hereby, we give routes to circumventing the computational cost of Monte Carlo scheme while obtaining the full statistical description of saturation. This derivation is followed by conducting a set of numerical experiments for horizontal reservoirs and more complex scenarios in which gravity segregation takes place. We then compare the CDFs as well as the first two moments of saturation computed with the method of distributions, against those obtained using the statistical moment equations (SME) approach and kernel density estimation post-processing of exhaustive high-resolution Monte Carlo simulations (MCS). This comparison demonstrates that the CDF equations remain accurate over a wide range of statistical properties, i.e. standard deviation and correlation length of the underlying random fields, while the corresponding low-order statistical moment equations significantly deviate from Monte Carlo results, unless for very small values of standard deviation and correlation length.

Download or read book Stochastic Modeling of Advection diusion reaction Processes in Biological Systems written by TaiJung Choi and published by . This book was released on 2013 with total page 149 pages. Available in PDF, EPUB and Kindle. Book excerpt: This dissertation deals with complex and multi-scale biological processes. In general, these phenomena can be described by ordinary or partial differential equations and treated with deterministic methods such as Runge-Kutta and alternating direction implicit algorithms. However, these approaches cannot predict the random effects caused by the low number of molecules involved and can result in severe stability and accuracy problem due to wide range of time or length scales depending upon the system being studied. In the first part of the dissertation, therefore, we developed the stochastic hybrid algorithm for complex reaction networks. Deterministic models of biochemical processes at the subcellular level might become inadequate when a cascade of chemical reactions is induced by a few molecules. Inherent randomness of such phenomena calls for the use of stochastic simulations. However, being computationally intensive, such simulations become infeasible for large and complex reaction networks. To improve their computational efficiency in handling these networks, we present a hybrid approach, in which slow reactions and fluxes are handled through exact stochastic simulation and their fast counterparts are treated partially deterministically through chemical Langevin equation. The classification of reactions as fast or slow is accompanied by the assumption that in the time-scale of fast reactions, slow reactions do not occur and hence do not affect the probability of the state. In the second and third part of the dissertation, we employ stochastic operator splitting algorithm for (chemotaxis- )diffusion-reaction processes. The reaction and diffusion steps employ stochastic simulation algorithm and Brownian dynamics, respectively. Through theoretical analysis, we develop an algorithm to identify if the system is reaction-controlled, diffusion-controlled or is in an intermediate regime. The time-step size is chosen accordingly at each step of the simulation. We apply our algorithm to several examples in order to demonstrate the accuracy, efficiency and robustness of the proposed algorithm comparing with the solutions obtained from deterministic partial differential equations and Gillespie multi-particle method. The third part deals with application of the stochastic-operator splitting approach to model the chemotaxis of leukocytes as part of the inflammation process during wound healing. We analyze both chemotaxis as well as the diffusion process as a drift phenomenon. We use two dimensionless numbers, Damkohler and Peclet number, in order to analyze the system. Damkohler number determines if the system is reaction-controlled or drift controlled and Peclet number identifies which phenomenon is dominant between diffusion and chemotaxis.

Download or read book Metastability and Markov State Models in Molecular Dynamics written by Christof Schütte and published by American Mathematical Soc.. This book was released on 2013-12-26 with total page 141 pages. Available in PDF, EPUB and Kindle. Book excerpt: Applications in modern biotechnology and molecular medicine often require simulation of biomolecular systems in atomic representation with immense length and timescales that are far beyond the capacity of computer power currently available. As a consequence, there is an increasing need for reduced models that describe the relevant dynamical properties while at the same time being less complex. In this book the authors exploit the existence of metastable sets for constructing such a reduced molecular dynamics model, the so-called Markov state model (MSM), with good approximation properties on the long timescales. With its many examples and illustrations, this book is addressed to graduate students, mathematicians, and practical computational scientists wanting an overview of the mathematical background for the ever-increasing research activity on how to construct MSMs for very different molecular systems ranging from peptides to proteins, from RNA to DNA, and via molecular sensors to molecular aggregation. This book bridges the gap between mathematical research on molecular dynamics and its practical use for realistic molecular systems by providing readers with tools for performing in-depth analysis of simulation and data-analysis methods. Titles in this series are co-published with the Courant Institute of Mathematical Sciences at New York University.

Download or read book Control Theory and Systems Biology written by Pablo A. Iglesias and published by MIT Press. This book was released on 2010 with total page 359 pages. Available in PDF, EPUB and Kindle. Book excerpt: A survey of how engineering techniques from control and systems theory can be used to help biologists understand the behavior of cellular systems.

Download or read book Approximation of Large Scale Dynamical Systems written by Athanasios C. Antoulas and published by SIAM. This book was released on 2009-06-25 with total page 489 pages. Available in PDF, EPUB and Kindle. Book excerpt: Mathematical models are used to simulate, and sometimes control, the behavior of physical and artificial processes such as the weather and very large-scale integration (VLSI) circuits. The increasing need for accuracy has led to the development of highly complex models. However, in the presence of limited computational accuracy and storage capabilities model reduction (system approximation) is often necessary. Approximation of Large-Scale Dynamical Systems provides a comprehensive picture of model reduction, combining system theory with numerical linear algebra and computational considerations. It addresses the issue of model reduction and the resulting trade-offs between accuracy and complexity. Special attention is given to numerical aspects, simulation questions, and practical applications.

Download or read book Design and Analysis of Biomolecular Circuits written by Heinz Koeppl and published by Springer Science & Business Media. This book was released on 2011-05-21 with total page 407 pages. Available in PDF, EPUB and Kindle. Book excerpt: The book deals with engineering aspects of the two emerging and intertwined fields of synthetic and systems biology. Both fields hold promise to revolutionize the way molecular biology research is done, the way today’s drug discovery works and the way bio-engineering is done. Both fields stress the importance of building and characterizing small bio-molecular networks in order to synthesize incrementally and understand large complex networks inside living cells. Reminiscent of computer-aided design (CAD) of electronic circuits, abstraction is believed to be the key concept to achieve this goal. It allows hiding the overwhelming complexity of cellular processes by encapsulating network parts into abstract modules. This book provides a unique perspective on how concepts and methods from CAD of electronic circuits can be leveraged to overcome complexity barrier perceived in synthetic and systems biology.