The auxiliary region method: A hybrid method for coupling PDE- and Brownian-based dynamics for reaction-diffusion systems

Reaction-diffusion systems are used to represent many biological and physical phenomena. They model the random motion of particles (diffusion) and interactions between them (reactions). Such systems can be modelled at multiple scales with varying degrees of accuracy and computational efficiency. When representing genuinely multiscale phenomena, fine-scale models can be prohibitively expensive, whereas coarser models, although cheaper, often lack sufficient detail to accurately represent the phenomenon at hand. Spatial hybrid methods couple two or more of these representations in order to improve efficiency without compromising accuracy. In this paper, we present a novel spatial hybrid method, which we call the auxiliary region method (ARM), which couples PDE and Brownian-based representations of reaction-diffusion systems. Numerical PDE solutions on one side of an interface are coupled to Brownian-based dynamics on the other side using compartment-based "auxiliary regions". We demonstrate that the hybrid method is able to simulate reaction-diffusion dynamics for a number of different test problems with high accuracy. Further, we undertake error analysis on the ARM which demonstrates that it is robust to changes in the free parameters in the model, where previous coupling algorithms are not. In particular, we envisage that the method will be applicable for a wide range of spatial multi-scales problems including, filopodial dynamics, intracellular signalling, embryogenesis and travelling wave phenomena.Comment: 29 pages, 14 figures, 2 table

3D Organization of Eukaryotic and Prokaryotic Genomes

There is a complex mutual interplay between three-dimensional (3D) genome organization and cellular activities in bacteria and eukaryotes. The aim of this thesis is to investigate such structure-function relationships. A main part of this thesis deals with the study of the three-dimensional genome organization using novel techniques for detecting genome-wide contacts using next-generation sequencing. These so called chromatin conformation capture-based methods, such as 5C and Hi-C, give deep insights into the architecture of the genome inside the nucleus, even on a small scale. We shed light on the question how the vastly increasing Hi-C data can generate new insights about the way the genome is organized in 3D. To this end, we first present the typical Hi-C data processing workflow to obtain Hi-C contact maps and show potential pitfalls in the interpretation of such contact maps using our own data pipeline and publicly available Hi-C data sets. Subsequently, we focus on approaches to modeling 3D genome organization based on contact maps. In this context, a computational tool was developed which interactively visualizes contact maps alongside complementary genomic data tracks. Inspired by machine learning with the help of probabilistic graphical models, we developed a tool that detects the compartmentalization structure within contact maps on multiple scales. In a further project, we propose and test one possible mechanism for the observed compartmentalization within contact maps of genomes across multiple species: Dynamic formation of loops within domains. In the context of 3D organization of bacterial chromosomes, we present the first direct evidence for global restructuring by long-range interactions of a DNA binding protein. Using Hi-C and live cell imaging of DNA loci, we show that the DNA binding protein Rok forms insulator-like complexes looping the B. subtilis genome over large distances. This biological mechanism agrees with our model based on dynamic formation of loops affecting domain formation in eukaryotic genomes. We further investigate the spatial segregation of the E. coli chromosome during cell division. In particular, we are interested in the positioning of the chromosomal replication origin region based on its interaction with the protein complex MukBEF. We tackle the problem using a combined approach of stochastic and polymer simulations. Last but not least, we develop a completely new methodology to analyze single molecule localization microscopy images based on topological data analysis. By using this new approach in the analysis of irradiated cells, we are able to show that the topology of repair foci can be categorized depending the distance to heterochromatin

Stochastic reaction-diffusion models in biology

Every cell contains several millions of diffusing and reacting biological molecules. The interactions between these molecules ultimately manifest themselves in all aspects of life, from the smallest bacterium to the largest whale. One of the greatest open scientific challenges is to understand how the microscopic chemistry determines the macroscopic biology. Key to this challenge is the development of mathematical and computational models of biochemistry with molecule-level detail, but which are sufficiently coarse to enable the study of large systems at the cell or organism scale. Two such models are in common usage: the reaction-diffusion master equation, and Brownian dynamics. These models are utterly different in both their history and in their approaches to chemical reactions and diffusion, but they both seek to address the same reaction-diffusion question. Here we make an in-depth study into the physical validity of these models under various biological conditions, determining when they can reliably be used. Taking each model in turn, we propose modifications to the models to better model the realities of the cellular environment, and to enable more efficient computational implementations. We use the models to make predictions about how and why cells behave the way they do, from mechanisms of self-organisation to noise reduction. We conclude that both models are extremely powerful tools for clarifying the details of the mysterious relationship between chemistry and biology