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From Deterministic Boolean Networks to Stochastic Continuous Models ______ _______________________________ From deterministic Boolean networks to stochastic continuous models _______________________________ Master Thesis by Sebastian Winkler Technische Universität München (TUM) Department of Mathematics M.sc Mathematics in Bioscience Supervisor: Prof. Dr. Dr. Fabian Theis Advisor: Dr. Christiane Fuchs Submission date: 28. August 2014 0 1 Technische Universität München Fakultät für Mathematik Titel der Masterarbeit, englisch: From discrete Boolean networks to stochastic continuous models Titel der Masterarbeit, deutsch: Von diskreten Boolschen Netzwerken zu stochastischen stetigen Modellen Verfasser: Sebastian Winkler (Matrikelnummer 03608811) Aufgabensteller: Prof. Dr. Dr. Fabian Theis Betreuerin: Dr. Christiane Fuchs Abgabedatum: 28.08.2014 2 3 _________________________________________ Ich erkläre hiermit, dass ich die vorliegende Masterarbeit selbstständig und nur mit den angegebenen Hilfsmitteln angefertigt habe. München, den 28. August 2014, ______________________ Sebastian Winkler _________________________________________ 4 5 Danksagung Ich möchte folgenden Personen, Lebewesen und/oder sonstigen Entitäten meinen Dank aussprechen: 1. Meiner Familie und auch allen anderen, insbesondere aber meinen Hunden Schnitzlon von Schnitzly (R.I.P.) und Zwirni (Zwirniratzis) Zwirnodopoulos. 2. Frau Dr. Christiane Fuchs für die angenehme und hilfreiche Betreuung. 3. Herrn Prof. Dr. Dr. Theis und Frau Dr. Fuchs für die Möglichkeit diese Arbeit am ICB im Rahmen der dort vorliegenden Gesamtsituation verfassen zu können. 6 7 Abstract The overall topic of this thesis is the relationship between various models for biochemical systems in general and for gene regulatory networks in particular. According to classical dipolar characteristics like continuity vs. discreteness of the state space, continuity vs. discreteness of time, spatial homogeneity vs. heterogeneity or determinism vs. stochasticity, different respectively suitable model classes can be applied in order to model systems which are thought to be best modeled with a particular modeling framework. Boolean network models are chosen as a point of departure for the exploration of the outlined issue and are thus covered in slightly more detail than other modeling approaches. In particular, part of the main part of the thesis addresses the question of parameter inference in specific Boolean models. Another focus lies on methods which extend and enrich the basic state- discrete, time-discrete, spatially homogeneous and non-stochastic framework of Boolean networks. Hence the title of the thesis: From discrete Boolean networks to stochastic continuous models. While the first chapter provides some biological background and outlines basic graph-based modeling approaches, the second chapter deals with two classical modeling approaches: deterministic models based on ordinary differential equations on the one hand and stochastic chemical kinetics and its extensions and approximations on the other hand. The third chapter then contains a description of Boolean modeling including the indicated extensions with respect to the respective nature of states, time, space and stochasticity where a particular focus in laid on so called generalized kinetic models (GKL). Chapter 4 proposes a simple estimation procedure for GKL networks with exponentially distributed time delays. While these are mathematically convenient it is argued that exponential distributions are not particularly well-suited for the situations modeled with GKL networks and hence Weibull distributed time delays are motivated and implemented. Chapter 5 deals with models which incorporate discrete and continuous characteristics into a single modeling framework. These are generically called hybrid models. Chapter 5 especially considers stochastic hybrid models and explores a simple example of this model class by means of example. 8 9 Zusammenfassung Die vorliegende Arbeit beschäftigt sich mit dem Zusammenhang zwischen verschiedenen Ansätzen zur Modellierung biochemischer Netzwerke im Allgemeinen und von Genregulationsnetzwerken im Besonderen. Modellierungsansätze für biochemische Netzwerke können gemäß klassischer dipolarer Merkmale wie etwa Stetigkeit vs. Diskretheit des Zustandsraumes, Stetigkeit vs. Diskretheit der Zeit, räumliche Homogenität vs. Heterogenität oder Determinismus vs. Stochastizität klassifiziert werden und haben jeweils spezifische Anwendungen abhängig von der angenommenen Adäquatheit eines jeweiligen Ansatzes für ein konkretes biologisches System. Boolsche Netzwerke bilden im skizzierten Gesamtzusammenhang einen weiteren Schwerpunkt der Arbeit und werden daher ein wenig ausführlicher behandelt als andere Ansätze. Ein Teil der Arbeit beschäftigt sich mit der Parameterschätzung in bestimmten Boolschen Modellen, während ein weiterer Teil sich detailliert mit verschiedenen Ansätzen befasst, die zum Ziel haben, die klassischerweise Zustands-diskreten, Zeit-diskreten, Raum-homogenen und deterministischen Boolschen Netzwerk Modelle entsprechend zu erweitern und idealerweise zu bereichern. Daher der Titel der Arbeit: Von diskreten Boolschen Netzwerken zu stochastischen stetigen Modellen. Nachdem im ersten Kapitel einige biologische Hintergrundaspekte und grundlegende Graphen-basierte Modellierungsansätze besprochen wurden, behandelt Kapitel zwei die klassischen Ansätze der Modellierung mit gewöhnlichen Differentialgleichungen einerseits und jenen der stochastischen chemischen Kinetik andererseits, Erweiterungen und Annährungen zu letzterer eingeschlossen. Das dritte Kapitel schließlich behandelt Boolsche Netzwerke und die angedeuteten Erweiterungen hinsichtlich Zustandsraum, Charakter der Zeit, Räumlichkeit und Stochastizität. Einen speziellen Fokus innerhalb dieses Kapitels erfahren die sogenannten verallgemeinerten Modelle der logischen Kinetik (engl.: generalized kinetic logic, abgekürzt GKL). Daran anknüpfend, studiert Kapitel vier eine einfache Methode zur Parameterschätzung in GKL-Modellen mit exponential-verteilten Verzögerungszeiten. Dieser Ansatz ist zwar mathematisch bequem, jedoch erscheinen seine Annahmen für die gegebene zu modellierende Situation eines biochemischen Netzwerkes inadäquat und es wird demgemäß eine Modifikation vorgeschlagen und implementiert, welche letztlich durch Weibull-verteilte Verzögerungszeiten charakterisiert ist. Kapitel 5 untersucht anhand von eines speziellen Beispiels das Zusammenspiel von diskreten und stetigen Variablen im allgemeineren Gesamtzusammenhang der stochastischen Hybridmodelle. 10 11 Contents _________________________________________ 1 Biological background: biochemical networks …………………………………………………. 18 1.1 Signal transduction networks ……………………………………………………………………… 18 1.2 Metabolic networks ……………………………………………………………………………………. 22 1.3 Gene regulatory networks (GRNs) ..…………………………………………………………….. 24 1.3.1 Gene expression: the central dogma (and beyond) …………………………….. 24 1.3.2 Transcription factors (TFs) and TF-binding sites…………………………………… 28 1.3.3 TF-DNA interactions: GRNs………………………………………………………………….. 29 1.3.4 Gene regulation functions ………………………………………………………………….. 30 1.3.5 Example 1: -phage infection of E.coli ………………………………………………… 30 1.3.6 Example 2: the synthetic repressilator…………………………………………………. 31 1.4 Interconnections between network types ……………………………………………………. 33 1.5 Hypergraphs and Systems Biology ……………………………………………………………….. 38 1.5.1 Graph-based models: hypergraphs………………………………………………………. 38 1.5.4 Top-down “vs.” bottom-up modeling ………………………………………………….. 43 2 Modeling approaches for biochemical networks ……………………………………………….. 46 2.1 Reaction rate equations: ODE models…………………………………………………………… 46 2.2 Stochastic models for biochemical networks…………………………………………………. 49 2.3.1 Stochasticity in biomolecular systems ……………………………………………….. 50 2.3.2 Stochastic chemical kinetics: the chemical master equation (CME) ..…. 51 2.3.3 Exact stochastic simulation: the Gillespie algorithm…………………………… 53 2.3.4 Approximate stochastic simulation: -leaping ………………………………….. 56 2.3.5 Diffusion approximation: the chemical Langevin equation (CLE)………… 57 2.3.6 Parameter estimation for stochastic biochemical models…………………… 58 3 From discrete Boolean networks to stochastic continuous models for biochemical networks ……………………………………………………………………………………………………………. 60 3.1 Boolean models: representation, update regimes, attractors………………………. 61 3.2 Random Boolean networks (RBNs): the ensemble approach……………………….. 65 3.2.1 Biologically meaningful update rules………………………………………………… 66 3.3 Probabilistic Boolean networks (PBNs)………………………………………………………… 67 12 3.4 Stochasticity and continuous time…………………………………………………………….. 68 3.5 Generalized kinetic logic (Thomas formalism)……………………………………………. 71 3.5.1 Semi-informal description and motivation of GKL…………………………… 72 3.5.2 Formal definitions for GKL networks………………………………………………. 80 3.6 Piecewise linear differential equations (PLDEs)…………………………………………. 85 3.6.1 Relation to logical models and qualitative simulation…………………….. 88 3.7 Petri nets………………………………………………………………………………………………….. 89 3.8 Fuzzy logic, SQUAD and Odefy………………………………………………………………….. 90 3.8.1 Fuzzy logical models………………………………………………………………………. 91 3.8.2 Standardized qualitative dynamical systems (SQUAD)……………………. 92 3.8.3 Multivariate polynomial interpolation: Odefy………………………………… 93 3.9 Boolean models for apoptosis…………………………………………………………………... 93 4 Parameter estimation for GKL networks with probabilistic time delays…………..98 4.1 Model philosophy and specification………………………………………………………….. 98 4.2 Parameter estimation based on absorption
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