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The chemical master equation (CME) represents the accepted stochastic description of chemical reaction kinetics in mesoscopic systems. As its exact solution – which gives the corresponding probability density function – is possible only in very simple cases, there is a clear need for approximation techniques. Here, we propose a novel perturbative three-step approach which draws heavily on graph theory: (i) we expand the eigenvalues of the transition state matrix in the CME as a series in a non-dimensional parameter that depends on the reaction rates and the reaction volume; (ii) we derive an analogous series for the corresponding eigenvectors via a graph-based algorithm; (iii) we combine the resulting expansions into an approximate solution to the CME. We illustrate our approach by applying it to a reversible dimerization reaction; then, we formulate a set of conditions, which ensure its applicability to more general reaction networks. We follow attempting to apply the results to a more complicated system, namely push-pull, but the problem reveals too complex for a complete solution. Finally, we discuss the limitations of the methodology.

Quantitative descriptions of reaction kinetics formulated at the stochastic mesoscopic level are frequently used to study various aspects of regulation and control in models of cellular control systems. For this type of systems, numerical simulation offers a variety of challenges caused by the high dimensionality of the problem and the multiscale properties often displayed by the biochemical model. In this thesis I have studied several aspects of stochastic simulation of both well-stirred and spatially heterogenous systems. In the well-stirred case, a hybrid method is proposed that reduces the dimension and stiffness of a model. We also demonstrate how both a high performance implementation and a variance reduction technique based on quasi-Monte Carlo can reduce the computational cost to estimate the probability density of the system. In the spatially dependent case, the use of unstructured, tetrahedral meshes to sample realizations of the stochastic process is proposed. Using such meshes, we then extend the reaction-diffusion framework to incorporate active transport of cellular cargo in a seamless manner. Finally, two multilevel methods for spatial stochastic simulation are considered. One of them is a space-time adaptive method combining exact stochastic, approximate stochastic and macroscopic modeling levels to reduce the simualation cost. The other method blends together mesoscale and microscale simulation methods to locally increase modeling resolution. / eSSENCE

Experimental studies have shown that the protein abundance in living cells varies from few tens to several thousands molecules per species. Molecular fluctuations roughly scale as the inverse square root of the number of molecules due to the random timing of reactions. It is hence expected that intrinsic noise plays an important role in the dynamics of biochemical networks. The Chemical Master Equation is the accepted description of these systems under well-mixed conditions. Because analytical solutions to this equation are available only for simple systems, one often has to resort to approximation methods. A popular technique is an expansion in the inverse volume to which the reactants are confined, called van Kampen's system size expansion. Its leading order terms are given by the phenomenological rate equations and the linear noise approximation that quantify the mean concentrations and the Gaussian fluctuations about them, respectively. While these approximations are valid in the limit of large molecule numbers, it is known that physiological conditions often imply low molecule numbers. We here develop systematic approximation methods based on higher terms in the system size expansion for general biochemical networks. We present an asymptotic series for the moments of the Chemical Master Equation that can be computed to arbitrary precision in the system size expansion. We then derive an analytical approximation of the corresponding time-dependent probability distribution. Finally, we devise a diagrammatic technique based on the path-integral method that allows to compute time-correlation functions. We show through the use of biological examples that the first few terms of the expansion yield accurate approximations even for low number of molecules. The theory is hence expected to closely resemble the outcomes of single cell experiments.

The chemical master equation, formulated on the Markov assumption of underlying chemical kinetics, offers an accurate stochastic description of general chemical reaction systems on the mesoscopic scale. The chemical master equation is especially useful when formulating mathematical models of gene regulatory networks and protein-protein interaction networks, where the numbers of molecules of most species are around tens or hundreds. However, solving the master equation directly suffers from the so called "curse of dimensionality" issue. This thesis first tries to study the numerical properties of the master equation using existing numerical methods and parallel machines. Next, approximation algorithms, namely the adaptive aggregation method and the radial basis function collocation method, are proposed as new paths to resolve the "curse of dimensionality". Several numerical results are presented to illustrate the promises and potential problems of these new algorithms. Comparisons with other numerical methods like Monte Carlo methods are also included. Development and analysis of the linear Shepard algorithm and its variants, all of which could be used for high dimensional scattered data interpolation problems, are also included here, as a candidate to help solve the master equation by building surrogate models in high dimensions. / Ph. D.

Parameter inference for stochastic systems is considered as one of the fundamental classical problems in the domain of computational systems biology. The problem becomes challenging and often analytically intractable with the large number of uncertain parameters. In this scenario, Markov Chain Monte Carlo (MCMC) algorithms have been proved to be highly effective. For a stochastic system, the most accurate description of the kinetics is given by the Chemical Master Equation (CME). Unfortunately, analytical solution of CME is often intractable even for considerably small amount of chemically reacting species due to its super exponential state space complexity. As a solution, Stochastic Simulation Algorithm (SSA) using Monte Carlo approach was introduced to simulate the chemical process defined by the CME. SSA is an exact stochastic method to simulate CME but it also suffers from high time complexity due to simulation of every reaction. Therefore computation of likelihood function (based on exact CME) in MCMC becomes expensive which alternately makes the rejection step expensive. In this generic work, we introduce different approximations of CME as a pre-conditioning step to the full MCMC to make rejection cheaper. The goal is to avoid expensive computation of exact CME as far as possible. We show that, with effective pre-conditioning scheme, one can save a considerable amount of exact CME computations maintaining similar convergence characteristics. Additionally, we investigate three different sampling schemes (dense sampling, longer sampling and i.i.d sampling) under which convergence for MCMC using exact CME for parameter estimation can be analyzed. We find that under i.i.d sampling scheme, better convergence can be achieved than that of dense sampling of the same process or sampling the same process for longer time. We verify our theoretical findings for two different processes: linear birth-death and dimerization.Apart from providing a framework for parameter inference using CME, this work also provides us the reasons behind avoiding CME (in general) as a parameter estimation technique for so long years after its formulation

The finite state projection algorithm provides modelers a new way of directly solving the chemical master equation. The algorithm utilizes the matrix exponential function, and so the algorithm’s performance suffers when it is applied to large problems. Other work has been done to reduce the size of the exponentiation through mathematical simplifications, but efficiently exponentiating a large matrix has not been explored. This work explores implementing the finite state projection algorithm on several different high-performance computing platforms as a means of efficiently calculating the matrix exponential function for large systems. This work finds that general purpose graphics processing can accelerate the finite state projection algorithm by several orders of magnitude. Specific biological models and modeling techniques are discussed as a demonstration of the algorithm implemented on a general purpose graphics processor. The results of this work show that general purpose graphics processing will be a key factor in modeling more complex biological systems.

This study is concerned with the numerical solution of certain stochastic models of chemical reactions. Such descriptions have been shown to be useful tools when studying biochemical processes inside living cells where classical deterministic rate equations fail to reproduce actual behavior. The main contribution of this thesis lies in its theoretical and practical investigation of different methods for obtaining numerical solutions to such descriptions. In a preliminary study, a simple but often quite effective approach to the moment closure problem is examined. A more advanced program is then developed for obtaining a consistent representation of the high dimensional probability density of the solution. The proposed method gains efficiency by utilizing a rapidly converging representation of certain functions defined over the semi-infinite integer lattice. Another contribution of this study, where the focus instead is on the spatially distributed case, is a suggestion for how to obtain a consistent stochastic reaction-diffusion model over an unstructured grid. Here it is also shown how to efficiently collect samples from the resulting model by making use of a hybrid method. In a final study, a time-parallel stochastic simulation algorithm is suggested and analyzed. Efficiency is here achieved by moving parts of the solution phase into the deterministic regime given that a parallel architecture is available. Necessary background material is developed in three chapters in this summary. An introductory chapter on an accessible level motivates the purpose of considering stochastic models in applied physics. In a second chapter the actual stochastic models considered are developed in a multi-faceted way. Finally, the current state-of-the-art in numerical solution methods is summarized and commented upon.

Quantification of stochastic or quantum systems by a joint probability density or wave function is a notoriously difficult computational problem, since the solution depends on all possible states (or realizations) of the system. Due to this combinatorial flavor, even a system containing as few as ten particles may yield as many as $10^{10}$ discretized states. None of even modern supercomputers are capable to cope with this curse of dimensionality straightforwardly, when the amount of quantum particles, for example, grows up to more or less interesting order of hundreds. A traditional approach for a long time was to avoid models formulated in terms of probabilistic functions, and simulate particular system realizations in a randomized process. Since different times in different communities, data-sparse methods came into play. Generally, they aim to define all data points indirectly, by a map from a low amount of representers, and recast all operations (e.g. linear system solution) from the initial data to the effective parameters. The most advanced techniques can be applied (at least, tried) to any given array, and do not rely explicitly on its origin. The current work contributes further progress to this area in the particular direction: tensor product methods for separation of variables. The separation of variables has a long history, and is based on the following elementary concept: a function of many variables may be expanded as a product of univariate functions. On the discrete level, a function is encoded by an array of its values, or a tensor. Therefore, instead of a huge initial array, the separation of variables allows to work with univariate factors with much less efforts. The dissertation contains a short overview of existing tensor representations: canonical PARAFAC, Hierarchical Tucker, Tensor Train (TT) formats, as well as the artificial tensorisation, resulting in the Quantized Tensor Train (QTT) approximation method. The contribution of the dissertation consists in both theoretical constructions and practical numerical algorithms for high-dimensional models, illustrated on the examples of the Fokker-Planck and the chemical master equations. Both arise from stochastic dynamical processes in multiconfigurational systems, and govern the evolution of the probability function in time. A special focus is put on time propagation schemes and their properties related to tensor product methods. We show that these applications yield large-scale systems of linear equations, and prove analytical separable representations of the involved functions and operators. We propose a new combined tensor format (QTT-Tucker), which descends from the TT format (hence TT algorithms may be generalized smoothly), but provides complexity reduction by an order of magnitude. We develop a robust iterative solution algorithm, constituting most advantageous properties of the classical iterative methods from numerical analysis and alternating density matrix renormalization group (DMRG) techniques from quantum physics. Numerical experiments confirm that the new method is preferable to DMRG algorithms. It is as fast as the simplest alternating schemes, but as reliable and accurate as the Krylov methods in linear algebra.

Information spielt in der Natur eine zentrale Rolle. Als intrinsischer Teil des genetischen Codes ist sie das Grundgerüst jeder Struktur und ihrer Entwicklung. Im Speziellen dient sie auch Organismen, ihre Umgebung wahrzunehmen und sich daran anzupassen. Die Grundvoraussetzung dafür ist, dass sie Information ihrer Umgebung sowohl messen als auch interpretieren können, wozu Zellen komplexe Signaltransduktionswege entwickelt haben. In dieser Arbeit konzentrieren wir uns auf Signalprozesse in S.cerevisiae die von osmotischem Stress (High Osmolarity Glycerol (HOG) Signalweg) und der Stimulation mit α-Faktor (Pheromon Signalweg) angesprochen werden. Wir wenden stochastische Modelle an, die das intrinsische Rauschen biologischer Prozesse darstellen können, um verstehen zu können wie Signalwege die ihnen zur Verfügung stehende Information umsetzen. Informationsübertragung wird dabei mit einem Ansatz aus Shannons Informationstheorie gemessen, indem wir sie als einen Kanal in diesem Sinne auffassen. Wir verwenden das Maß der Kanalkapazität, um die Genauigkeit des Phosphorelays einschränken zu können. In diesem Modell, simuliert mit dem Gillespie Algorithmus, können wir durch die Analyse des Signalverhaltens den Parameterraum zusätzlich stark einschränken. Eine weitere Herangehensweise der Signalverarbeitung beschäftigt sich mit dem “Crosstalk” zwischen HOG und Pheromon Signalweg. Wir zeigen, dass die Kontrolle der Signalspezifizität vor allem bei Scaffold-Proteinen liegt, die Komponenten der Signalkaskade binden. Diese konservierten Motive zellulärer Signaltransduktion besitzen eine geeignete Struktur, um Information getreu übertragen zu können. Im letzten Teil der Arbeit untersuchen wir potentielle Gründe für die evolutionäre Selektion von Scaffolds. Wir zeigen, dass ihnen bereits durch die Struktur des Mechanismus möglich ist, Informationsgenauigkeit zu verbessern und einer verteilten Informationsweiterleitung sowohl dadurch als auch durch ihre Robustheit überlegen sind. / Information plays a ubiquitous role in nature. It provides the basis for structure and development, as it is inherent part of the genetic code. It also enables organisms to make sense of their environments and react accordingly. For this, a cellular interpretation of information is needed. Cells have developed sophisticated signaling mechanisms to fulfill this task and integrate many different external cues with their help. Here we focus on signaling that senses osmotic stress (High Osmolarity Glycerol (HOG) pathway) as well as α-factor stimulation (pheromone pathway) in S.cerevisiae. We employ stochastic modeling to simulates the inherent noisy nature of biological processes to assess how systems process the information they receive. This information transmission is evaluated with an information theoretic approach by interpreting signal transduction as a transmission channel in the sense of Shannon. We use channel capacity to both constrain as well as quantify the fidelity in the phosphorelay system of the HOG pathway. In this model, simulated with the Gillespie Algorithm, the analysis of signaling behavior allows us to constrain the possible parameter sets for the system severely. A further approach to signal processing is concerned with the mechanisms that conduct crosstalk between the HOG and the pheromone pathway. We find that the control for signal specificity lies especially with the scaffold proteins that tether signaling components and facilitate signaling by trans-location to the membrane and shielding against miss-activation. As conserved motifs of cellular signal transmission, these scaffold proteins show a particularly well suited structure for accurate information transmission. In the last part of this thesis, we examine the potential reasons for an evolutionary selection of the scaffolding structure. We show that due to its structure, scaffolds are increasing information transmission fidelity and outperform a distributed signal in this regard.