Model Reconstruction for Moment-based Stochastic Chemical Kinetics

Based on the theory of stochastic chemical kinetics, the inherent randomness and stochasticity of biochemical reaction networks can be accurately described by discrete-state continuous-time Markov chains. The analysis of such processes is, however, computationally expensive and sophisticated numerical methods are required. Here, we propose an analysis framework in which we integrate a number of moments of the process instead of the state probabilities. This results in a very efficient simulation of the time evolution of the process. In order to regain the state probabilities from the moment representation, we combine the fast moment-based simulation with a maximum entropy approach for the reconstruction of the underlying probability distribution. We investigate the usefulness of this combined approach in the setting of stochastic chemical kinetics and present numerical results for three reaction networks showing its efficiency and accuracy. Besides a simple dimerization system, we study a bistable switch system and a multi-attractor network with complex dynamics.Comment: 20 pages,5 figure

Model reconstruction for moment-based stochastic chemical kinetics

Based on the theory of stochastic chemical kinetics, the inherent randomness and stochasticity of biochemical reaction networks can be accurately described by discrete-state continuous-time Markov chains, where each chemical reaction corresponds to a state transition of the process. However, the analysis of such processes is computationally expensive and sophisticated numerical methods are required. The main complication comes due to the largeness problem of the state space, so that analysis techniques based on an exploration of the state space are often not feasible and the integration of the moments of the underlying probability distribution has become a very popular alternative. In this thesis we propose an analysis framework in which we integrate a number of moments of the process instead of the state probabilities. This results in a more timeefficient simulation of the time evolution of the process. In order to regain the state probabilities from the moment representation, we combine the moment-based simulation (MM) with a maximum entropy approach: the maximum entropy principle is applied to derive a distribution that fits best to a given sequence of moments. We further extend this approach by incorporating the conditional moments (MCM) which allows not only to reconstruct the distribution of the species present in high amount in the system, but also to approximate the probabilities of species with low molecular counts. For the given distribution reconstruction framework, we investigate the numerical accuracy and stability using case studies from systems biology, compare two different moment approximation methods (MM and MCM), examine if it can be used for the reaction rates estimation problem and describe the possible future applications. In this thesis we propose an analysis framework in which we integrate a number of moments of the process instead of the state probabilities. This results in a more time-efficient simulation of the time evolution of the process. In order to regain the state probabilities from the moment representation, we combine the moment-based simulation (MM) with a maximum entropy approach: the maximum entropy principle is applied to derive a distribution that fits best to a given sequence of moments. We further extend this approach by incorporating the conditional moments (MCM) which allows not only to reconstruct the distribution of the species present in high amount in the system, but also to approximate the probabilities of species with low molecular counts. For the given distribution reconstruction framework, we investigate the numerical accuracy and stability using case studies from systems biology, compare two different moment approximation methods (MM and MCM), examine if it can be used for the reaction rates estimation problem and describe the possible future applications.Basierend auf der Theorie der stochastischen chemischen Kinetiken können die inhärente Zufälligkeit und Stochastizität von biochemischen Reaktionsnetzwerken durch diskrete zeitkontinuierliche Markow-Ketten genau beschrieben werden, wobei jede chemische Reaktion einem Zustandsübergang des Prozesses entspricht. Die Analyse solcher Prozesse ist jedoch rechenaufwendig und komplexe numerische Verfahren sind erforderlich. Analysetechniken, die auf dem Abtasten des Zustandsraums basieren, sind durch dessen Größe oft nicht anwendbar. Als populäre Alternative wird heute häufig die Integration der Momente der zugrundeliegenden Wahrscheinlichkeitsverteilung genutzt. In dieser Arbeit schlagen wir einen Analyserahmen vor, in dem wir, anstatt der Zustandswahrscheinlichkeiten, zugrundeliegende Momente des Prozesses integrieren. Dies führt zu einer zeiteffizienteren Simulation der zeitlichen Entwicklung des Prozesses. Um die Zustandswahrscheinlichkeiten aus der Momentreprsentation wiederzugewinnen, kombinieren wir die momentbasierte Simulation (MM) mit Entropiemaximierung: Die Maximum- Entropie-Methode wird angewendet, um eine Verteilung abzuleiten, die am besten zu einer bestimmten Sequenz von Momenten passt. Wir erweitern diesen Ansatz durch das Einbeziehen bedingter Momente (MCM), die es nicht nur erlauben, die Verteilung der in großer Menge im System enthaltenen Spezies zu rekonstruieren, sondern es ebenso ermöglicht, sich den Wahrscheinlichkeiten von Spezies mit niedrigen Molekulargewichten anzunähern. Für das gegebene System zur Verteilungsrekonstruktion untersuchen wir die numerische Genauigkeit und Stabilität anhand von Fallstudien aus der Systembiologie, vergleichen zwei unterschiedliche Verfahren der Momentapproximation (MM und MCM), untersuchen, ob es für das Problem der Abschätzung von Reaktionsraten verwendet werden kann und beschreiben die mögliche zukünftige Anwendungen

Stochastic reaction networks with input processes: Analysis and applications to reporter gene systems

Stochastic reaction network models are widely utilized in biology and chemistry to describe the probabilistic dynamics of biochemical systems in general, and gene interaction networks in particular. Most often, statistical analysis and inference of these systems is addressed by parametric approaches, where the laws governing exogenous input processes, if present, are themselves fixed in advance. Motivated by reporter gene systems, widely utilized in biology to monitor gene activation at the individual cell level, we address the analysis of reaction networks with state-affine reaction rates and arbitrary input processes. We derive a generalization of the so-called moment equations where the dynamics of the network statistics are expressed as a function of the input process statistics. In stationary conditions, we provide a spectral analysis of the system and elaborate on connections with linear filtering. We then apply the theoretical results to develop a method for the reconstruction of input process statistics, namely the gene activation autocovariance function, from reporter gene population snapshot data, and demonstrate its performance on a simulated case study

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

A variational approach to moment-closure approximations for the kinetics of biomolecular reaction networks

Approximate solutions of the chemical master equation and the chemical Fokker-Planck equation are an important tool in the analysis of biomolecular reaction networks. Previous studies have highlighted a number of problems with the moment-closure approach used to obtain such approximations, calling it an ad-hoc method. In this article, we give a new variational derivation of moment-closure equations which provides us with an intuitive understanding of their properties and failure modes and allows us to correct some of these problems. We use mixtures of product-Poisson distributions to obtain a flexible parametric family which solves the commonly observed problem of divergences at low system sizes. We also extend the recently introduced entropic matching approach to arbitrary ansatz distributions and Markov processes, demonstrating that it is a special case of variational moment closure. This provides us with a particularly principled approximation method. Finally, we extend the above approaches to cover the approximation of multi-time joint distributions, resulting in a viable alternative to process-level approximations which are often intractable.Comment: Minor changes and clarifications; corrected some typo

Dissipation in noisy chemical networks: The role of deficiency

We study the effect of intrinsic noise on the thermodynamic balance of complex chemical networks subtending cellular metabolism and gene regulation. A topological network property called deficiency, known to determine the possibility of complex behavior such as multistability and oscillations, is shown to also characterize the entropic balance. In particular, only when deficiency is zero does the average stochastic dissipation rate equal that of the corresponding deterministic model, where correlations are disregarded. In fact, dissipation can be reduced by the effect of noise, as occurs in a toy model of metabolism that we employ to illustrate our findings. This phenomenon highlights that there is a close interplay between deficiency and the activation of new dissipative pathways at low molecule numbers.Comment: 10 Pages, 6 figure

Path mutual information for a class of biochemical reaction networks

Living cells encode and transmit information in the temporal dynamics of biochemical components. Gaining a detailed understanding of the input-output relationship in biological systems therefore requires quantitative measures that capture the interdependence between complete time trajectories of biochemical components. Mutual information provides such a measure but its calculation in the context of stochastic reaction networks is associated with mathematical challenges. Here we show how to estimate the mutual information between complete paths of two molecular species that interact with each other through biochemical reactions. We demonstrate our approach using three simple case studies.Comment: 6 pages, 2 figure

Global parameter identification of stochastic reaction networks from single trajectories

We consider the problem of inferring the unknown parameters of a stochastic biochemical network model from a single measured time-course of the concentration of some of the involved species. Such measurements are available, e.g., from live-cell fluorescence microscopy in image-based systems biology. In addition, fluctuation time-courses from, e.g., fluorescence correlation spectroscopy provide additional information about the system dynamics that can be used to more robustly infer parameters than when considering only mean concentrations. Estimating model parameters from a single experimental trajectory enables single-cell measurements and quantification of cell--cell variability. We propose a novel combination of an adaptive Monte Carlo sampler, called Gaussian Adaptation, and efficient exact stochastic simulation algorithms that allows parameter identification from single stochastic trajectories. We benchmark the proposed method on a linear and a non-linear reaction network at steady state and during transient phases. In addition, we demonstrate that the present method also provides an ellipsoidal volume estimate of the viable part of parameter space and is able to estimate the physical volume of the compartment in which the observed reactions take place.Comment: Article in print as a book chapter in Springer's "Advances in Systems Biology