Tailored parameter optimization methods for ordinary differential equation models with steady-state constraints

Background: Ordinary differential equation (ODE) models are widely used to describe (bio-)chemical and biological processes. To enhance the predictive power of these models, their unknown parameters are estimated from experimental data. These experimental data are mostly collected in perturbation experiments, in which the processes are pushed out of steady state by applying a stimulus. The information that the initial condition is a steady state of the unperturbed process provides valuable information, as it restricts the dynamics of the process and thereby the parameters. However, implementing steady-state constraints in the optimization often results in convergence problems. Results: In this manuscript, we propose two new methods for solving optimization problems with steady-state constraints. The first method exploits ideas from optimization algorithms on manifolds and introduces a retraction operator, essentially reducing the dimension of the optimization problem. The second method is based on the continuous analogue of the optimization problem. This continuous analogue is an ODE whose equilibrium points are the optima of the constrained optimization problem. This equivalence enables the use of adaptive numerical methods for solving optimization problems with steady-state constraints. Both methods are tailored to the problem structure and exploit the local geometry of the steady-state manifold and its stability properties. A parameterization of the steady-state manifold is not required. The efficiency and reliability of the proposed methods is evaluated using one toy example and two applications. The first application example uses published data while the second uses a novel dataset for Raf/MEK/ERK signaling. The proposed methods demonstrated better convergence properties than state-of-the-art methods employed in systems and computational biology. Furthermore, the average computation time per converged start is significantly lower. In addition to the theoretical results, the analysis of the dataset for Raf/MEK/ERK signaling provides novel biological insights regarding the existence of feedback regulation. Conclusion: Many optimization problems considered in systems and computational biology are subject to steady-state constraints. While most optimization methods have convergence problems if these steady-state constraints are highly nonlinear, the methods presented recover the convergence properties of optimizers which can exploit an analytical expression for the parameter-dependent steady state. This renders them an excellent alternative to methods which are currently employed in systems and computational biology

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

Discovery of a Cellular Mechanism Regulating Transcriptional Noise

Stochastic fluctuations in gene expression (‘noise’) are often considered detrimental but, in other fields, fluctuations are harnessed for benefit (e.g., ‘dither’ or amplification of thermal fluctuations to accelerate chemical reactions). Here, we find that DNA base-excision repair amplifies transcriptional noise, generating increased cellular plasticity and facilitating reprogramming. The DNA-repair protein Apex1 recognizes modified nucleoside substrates to amplify expression noise—while homeostatically maintaining mean levels of expression—for virtually all genes across the transcriptome. This noise amplification occurs for both naturally occurring base modifications and unnatural base analogs. Single-molecule imaging shows amplified noise originates from shorter, but more intense, transcriptional bursts that occur via increased DNA supercoiling which first impedes and then accelerates transcription, thereby maintaining mean levels. Strikingly, homeostatic noise amplification potentiates fate-conversion signals during cellular reprogramming. These data suggest a functional role for the observed occurrence of modified bases within DNA in embryonic development and disease

Mathematical Modeling of Multi-Level Behavior of the Embryonic Stem Cell System during Self-Renewal and Differentiation

Embryonic stem cells (ESC) are pluripotent cells derived from the inner cell mass of the blastocyst. These cells have the unique properties of unlimited self-renewal and differentiation capability. ESC therefore hold huge potential for use in therapeutic applications in regenerative medicine. This potential has been demonstrated in vitro by directing differentiation of ESC to various cell types by modulating the soluble and insoluble cues to which the cells are exposed. Despite their great potential, current differentiation methods are still limited in the yield and functionality of the ESC-derived mature phenotype. We hypothesize the lack of mechanistic understanding of the complex differentiation process to be the primary reason behind their restricted success. Mathematical models, coupled to experimental data, can aid in this understanding. While the past several decades have seen advances in the mathematical analysis of biological systems, mathematical approaches to the ESC system have received limited attention. Furthermore, variability of ESC restricts direct application of deterministic approaches towards drawing mechanistic insight. The goal of the current work is to obtain a more thorough mechanistic understanding of the ESC system through mathematical modeling. In ESC, extracellular cues guide single cell behavior in a non-deterministic fashion, giving rise to heterogeneous populations. Therefore, in this work we focus on modeling three levels of the ESC system: intracellular, extracellular, and population. We first developed an optimization framework to identify intracellular gene regulatory interactions from time series data. We show that incorporation of the bootstrapping technique into the formulism allows for accurate prediction of robust interactions from noisy data. A regression approach was then utilized to identify extracellular substrate features influential to cellular behavior. We apply this model to identify fibrin microstructural features which guide differentiation of mESC. Finally, we developed a stochastic model to capture heterogeneous population dynamics of hESC. We demonstrate the usefulness of the model to obtain mechanistic information of cell cycle transition and lineage commitment during differentiation. Through development and utilization of different mathematical approaches to analyze multilevel behavior and variability of ESC self-renewal and differentiation, we demonstrate the applicability of mathematical models in extracting mechanistic information from the ESC system

Stochastic simulation of droplet breakup in turbulence

This study investigates single droplet breakup from a theoretical perspective and addresses whether breakup in turbulent flows can be studied using highly-resolved simulations. Transient and three-dimensional turbulent flow simulations are performed to investigate if the apparent stochastic outcome from the droplet breakup can be predicted. For a given turbulent dissipation rate the breakup events were simulated for various detailed turbulence realizations. For this purpose, a well-characterized system widely used for kernel development is utilized to validate the simulations with respect to the key characteristics of stochastic breakup, including droplet deformation time, the number of fragments, and the specific breakup rate. The statistical validations show very good agreement with all the quantitative properties relevant to the breakup dynamics. Necklace breakup is also observed in line with patterns found in experiments. Evidence is found that the rate of energy transfer is positively correlated with higher order fragmentation. This can allow development of more accurate breakup kernels compared to the ones that only relies on the maximum amount of energy transfer. It is concluded that the simulation method provides new data on the stochastic characteristics of breakup. The method also provides a means to extract more details than experimentally possible since the analysis allows better spatial and temporal resolutions, and 3D analysis of energy transfer which provides better accuracy compared to experimental 2D data

Properties of cell death models calibrated and compared using Bayesian approaches

Using models to simulate and analyze biological networks requires principled approaches to parameter estimation and model discrimination. We use Bayesian and Monte Carlo methods to recover the full probability distributions of free parameters (initial protein concentrations and rate constants) for mass-action models of receptor-mediated cell death. The width of the individual parameter distributions is largely determined by non-identifiability but covariation among parameters, even those that are poorly determined, encodes essential information. Knowledge of joint parameter distributions makes it possible to compute the uncertainty of model-based predictions whereas ignoring it (e.g., by treating parameters as a simple list of values and variances) yields nonsensical predictions. Computing the Bayes factor from joint distributions yields the odds ratio (~20-fold) for competing ‘direct’ and ‘indirect’ apoptosis models having different numbers of parameters. Our results illustrate how Bayesian approaches to model calibration and discrimination combined with single-cell data represent a generally useful and rigorous approach to discriminate between competing hypotheses in the face of parametric and topological uncertainty

Computational Investigations of Biomolecular Mechanisms in Genomic Replication, Repair and Transcription

High fidelity maintenance of the genome is imperative to ensuring stability and proliferation of cells. The genetic material (DNA) of a cell faces a constant barrage of metabolic and environmental assaults throughout the its lifetime, ultimately leading to DNA damage. Left unchecked, DNA damage can result in genomic instability, inviting a cascade of mutations that initiate cancer and other aging disorders. Thus, a large area of focus has been dedicated to understanding how DNA is damaged, repaired, expressed and replicated. At the heart of these processes lie complex macromolecular dynamics coupled with intricate protein-DNA interactions. Through advanced computational techniques it has become possible to probe these mechanisms at the atomic level, providing a physical basis to describe biomolecular phenomena. To this end, we have performed studies aimed at elucidating the dynamics and interactions intrinsic to the functionality of biomolecules critical to maintaining genomic integrity: modeling the DNA editing mechanism of DNA polymerase III, uncovering the DNA damage recognition/repair mechanism of thymine DNA glycosylase and linking genetic disease to the functional dynamics of the pre-initiation complex transcription machinery. Collectively, our results elucidate the dynamic interplay between proteins and DNA, further broadening our understanding of these complex processes involved with genomic maintenance