Improved estimations of stochastic chemical kinetics by finite state expansion

Stochastic reaction networks are a fundamental model to describe interactions between species where random fluctuations are relevant. The master equation provides the evolution of the probability distribution across the discrete state space consisting of vectors of population counts for each species. However, since its exact solution is often elusive, several analytical approximations have been proposed. The deterministic rate equation (DRE) gives a macroscopic approximation as a compact system of differential equations that estimate the average populations for each species, but it may be inaccurate in the case of nonlinear interaction dynamics. Here we propose finite state expansion (FSE), an analytical method mediating between the microscopic and the macroscopic interpretations of a stochastic reaction network by coupling the master equation dynamics of a chosen subset of the discrete state space with the mean population dynamics of the DRE. An algorithm translates a network into an expanded one where each discrete state is represented as a further distinct species. This translation exactly preserves the stochastic dynamics, but the DRE of the expanded network can be interpreted as a correction to the original one. The effectiveness of FSE is demonstrated in models that challenge state-of-the-art techniques due to intrinsic noise, multi-scale populations, and multi-stability.Comment: 33 pages, 9 figure

Stochastic Nonlinear Model Predictive Control with Efficient Sample Approximation of Chance Constraints

This paper presents a stochastic model predictive control approach for nonlinear systems subject to time-invariant probabilistic uncertainties in model parameters and initial conditions. The stochastic optimal control problem entails a cost function in terms of expected values and higher moments of the states, and chance constraints that ensure probabilistic constraint satisfaction. The generalized polynomial chaos framework is used to propagate the time-invariant stochastic uncertainties through the nonlinear system dynamics, and to efficiently sample from the probability densities of the states to approximate the satisfaction probability of the chance constraints. To increase computational efficiency by avoiding excessive sampling, a statistical analysis is proposed to systematically determine a-priori the least conservative constraint tightening required at a given sample size to guarantee a desired feasibility probability of the sample-approximated chance constraint optimization problem. In addition, a method is presented for sample-based approximation of the analytic gradients of the chance constraints, which increases the optimization efficiency significantly. The proposed stochastic nonlinear model predictive control approach is applicable to a broad class of nonlinear systems with the sufficient condition that each term is analytic with respect to the states, and separable with respect to the inputs, states and parameters. The closed-loop performance of the proposed approach is evaluated using the Williams-Otto reactor with seven states, and ten uncertain parameters and initial conditions. The results demonstrate the efficiency of the approach for real-time stochastic model predictive control and its capability to systematically account for probabilistic uncertainties in contrast to a nonlinear model predictive control approaches.Comment: Submitted to Journal of Process Contro

Stochastic spatial modelling of DNA methylation patterns and moment-based parameter estimation

In the first part of this thesis, we introduce and analyze spatial stochastic models for DNA methylation, an epigenetic mark with an important role in development. The underlying mechanisms controlling methylation are only partly understood. Several mechanistic models of enzyme activities responsible for methylation have been proposed. Here, we extend existing hidden Markov models (HMMs) for DNA methylation by describing the occurrence of spatial methylation patterns with stochastic automata networks. We perform numerical analysis of the HMMs applied to (non-)hairpin bisulfite sequencing KO data and accurately predict the wild-type data from these results. We find evidence that the activities of Dnmt3a/b responsible for de novo methylation depend on the left but not on the right CpG neighbors. The second part focuses on parameter estimation in chemical reaction networks (CRNs). We propose a generalized method of moments (GMM) approach for inferring the parameters of CRNs based on a sophisticated matching of the statistical moments of the stochastic model and the sample moments of population snapshot data. The proposed parameter estimation method exploits recently developed moment-based approximations and provides estimators with desirable statistical properties when many samples are available. The GMM provides accurate and fast estimations of unknown parameters of CRNs. The accuracy increases and the variance decreases when higher-order moments are considered.Im ersten Teil der Arbeit führen wir eine Analyse für spatielle stochastische Modelle der DNA Methylierung, ein wichtiger epigenetischer Marker in der Entwicklung, durch. Die zugrunde liegenden Mechanismen der Methylierung werden noch nicht vollständig verstanden. Mechanistische Modelle beschreiben die Aktivität der Methylierungsenzyme. Wir erweitern bestehende Hidden Markov Models (HMMs) zur DNA Methylierung durch eine Stochastic Automata Networks Beschreibung von spatiellen Methylierungsmustern. Wir führen eine numerische Analyse der HMMs auf bisulfit-sequenzierten KO Datens¨atzen aus und nutzen die Resultate, um die Wildtyp-Daten erfolgreich vorherzusagen. Unsere Ergebnisse deuten an, dass die Aktivitäten von Dnmt3a/b, die überwiegend für die de novo Methylierung verantwortlich sind, nur vom Methylierungsstatus des linken, nicht aber vom rechten CpG Nachbarn abhängen. Der zweite Teil befasst sich mit Parameterschätzung in chemischen Reaktionsnetzwerken (CRNs). Wir führen eine Verallgemeinerte Momentenmethode (GMM) ein, die die statistischen Momente des stochastischen Modells an die Momente von Stichproben geschickt anpasst. Die GMM nutzt hier kürzlich entwickelte, momentenbasierte Näherungen, liefert Schätzer mit wünschenswerten statistischen Eigenschaften, wenn genügend Stichproben verfügbar sind, mit schnellen und genauen Schätzungen der unbekannten Parameter in CRNs. Momente höherer Ordnung steigern die Genauigkeit des Schätzers, während die Varianz sinkt

A Brownian Model for Crystal Nucleation

In this work a phenomenological stochastic differential equation is proposed to model the time evolution of the radius of a pre-critical molecular cluster during nucleation (the classical order parameter). Such a stochastic differential equation constitutes the basis for the calculation of the (nucleation) induction time under Kramers' theory of thermally activated escape processes. Considering the nucleation stage as a Poisson rare-event, analytical expressions for the induction time statistics are deduced for both steady and unsteady conditions, the latter assuming the semiadiabatic limit. These expressions can be used to identify the underlying mechanism of molecular cluster formation (distinguishing between homogeneous or heterogeneous nucleation from the nucleation statistics is possible) as well as to predict induction times and induction time distributions. The predictions of this model are in good agreement with experimentally measured induction times at constant temperature, unlike the values obtained from the classical equation, but agreement is not so good for induction time statistics. Stochastic simulations truncated to the maximum waiting time of the experiments confirm that this fact is due to the time constraints imposed by experiments. Correcting for this effect, the experimental and predicted curves fit remarkably well. Thus, the proposed model seems to be a versatile tool to predict cluster size distributions, nucleation rates, (nucleation) induction time and induction time statistics for a wide range of conditions (e.g. time-dependent temperature, supersaturation, pH, etc.) where classical nucleation theory is of limited applicability.Comment: 20 pages, 3 figure

Data-driven modelling of biological multi-scale processes

Biological processes involve a variety of spatial and temporal scales. A holistic understanding of many biological processes therefore requires multi-scale models which capture the relevant properties on all these scales. In this manuscript we review mathematical modelling approaches used to describe the individual spatial scales and how they are integrated into holistic models. We discuss the relation between spatial and temporal scales and the implication of that on multi-scale modelling. Based upon this overview over state-of-the-art modelling approaches, we formulate key challenges in mathematical and computational modelling of biological multi-scale and multi-physics processes. In particular, we considered the availability of analysis tools for multi-scale models and model-based multi-scale data integration. We provide a compact review of methods for model-based data integration and model-based hypothesis testing. Furthermore, novel approaches and recent trends are discussed, including computation time reduction using reduced order and surrogate models, which contribute to the solution of inference problems. We conclude the manuscript by providing a few ideas for the development of tailored multi-scale inference methods.Comment: This manuscript will appear in the Journal of Coupled Systems and Multiscale Dynamics (American Scientific Publishers

Aging concrete structures: a review of mechanics and concepts

The safe and cost-efficient management of our built infrastructure is a challenging task considering the expected service life of at least 50 years. In spite of time-dependent changes in material properties, deterioration processes and changing demand by society, the structures need to satisfy many technical requirements related to serviceability, durability, sustainability and bearing capacity. This review paper summarizes the challenges associated with the safe design and maintenance of aging concrete structures and gives an overview of some concepts and approaches that are being developed to address these challenges

Biyokimyasal sistemlerin stokastik modellemesi ile filtreleme ve yumuşatma algoritmaları.

Deterministic modeling approach is the traditional way of analyzing the dynamical behavior of a reaction network. However, this approach ignores the discrete and stochastic nature of biochemical processes. In this study, modeling approaches, stochastic simulation algorithms and their relationships to each other are investigated. Then, stochastic and deterministic modeling approaches are applied to biological systems, Lotka-Volterra prey-predator model, Michaelis-Menten enzyme kinetics and JACK-STAT signaling pathway. Also, numerical solutions for ODE system and realizations obtained through stochastic simulation algorithms are compared. In general, it is not possible to assess all elements of the state vector of biochemical systems. Hence, some statistical models are used to obtain the best estimation. Filtering and smoothing distributions can be obtained via Bayes’ rule. However, as an alternative to approximate these distributions Monte Carlo methods might be used. In the second part, bootstrap particle filter algorithm is derived and applied to birthdeath process. Estimated probability distribution functions are compared according to number of particles used.Thesis (M.S.) -- Graduate School of Applied Mathematics. Scientific Computing

Determining cluster-cluster aggregation rate kernals using inverse methods

We investigate the potential of inverse methods for retrieving adequate information about the rate kernel functions of cluster-cluster aggregation processes from mass density distribution data. Since many of the classical physical kernels have fractional order exponents the ability of an inverse method to appropriately represent such functions is a key concern. In early chapters, the properties of the Smoluchowski Coagulation Equation and its simulation using Monte Carlo techniques are introduced. Two key discoveries made using the Monte Carlo simulations are briefly reported. First, that for a range of nonlocal solutions of finite mass spectrum aggregation systems with a source of mass injection, collective oscillations of the solution can persist indefinitely despite the presence of significant noise. Second, that for similar finite mass spectrum systems with (deterministic) stable, but sensitive, nonlocal stationary solutions, the presence of noise in the system can give rise to behaviour indicative of phase-remembering, noise-driven quasicycles. The main research material on inverse methods is then presented in two subsequent chapters. The first of these chapters investigates the capacity of an existing inverse method in respect of the concerns about fractional order exponents in homogeneous kernels. The second chapter then introduces a new more powerful nonlinear inverse method, based upon a novel factorisation of homogeneous kernels, whose properties are assessed in respect of both stationary and scaling mass distribution data inputs