Analysis and Simulation of Hybrid Models for Reaction Networks

The dynamics of biochemical reaction networks can be described by a variety of models, like the Reaction Rate equation (RRE), the Chemical Master equation (CME) or the Fokker-Planck equation (FPE). In this thesis, the behaviour of these different models is analysed. It is shown that the FPE can be motivated as an approximation of the CME and convergence is proven. Furthermore, two hybrid models are constructed by combining different approaches and convergence properties are proven and discussed

Central Dogma at the Single-Molecule Level in Living Cells

Gene expression originates from individual DNA molecules within living cells. Like many single-molecule processes, gene expression and regulation are stochastic, that is, sporadic in time. This leads to heterogeneity in the messenger RNA and protein copy numbers in a population of cells with identical genomes. With advanced single-cell fluorescence microscopy, it is now possible to quantify transcriptomes and proteomes with single-molecule sensitivity. Dynamic processes such as transcription factor binding, transcription and translation can be monitored in real time, providing quantitative descriptions of gene expression and regulation, and the demonstration that a single-molecule event can determine the phenotype of a cell.Chemistry and Chemical Biolog

Recycling random numbers in the stochastic simulation algorithm

The stochastic simulation algorithm (SSA) was introduced by Gillespie and in a different form by Kurtz. Since its original formulation there have been several attempts at improving the efficiency and hence the speed of the algorithm. We briefly discuss some of these methods before outlining our own simple improvement, the recycling direct method (RDM), and demonstrating that it is capable of increasing the speed of most stochastic simulations. The RDM involves the statistically acceptable recycling of random numbers in order to reduce the computational cost associated with their generation and is compatible with several of the pre-existing improvements on the original SSA. Our improvement is also sufficiently simple (one additional line of code) that we hope will be adopted by both trained mathematical modelers and experimentalists wishing to simulate their model systems

Development and Analysis of Dynamic Models of the LAC Operon

In this work, the mathematical models describing the dynamics of the gene regulatory network of the lac operon are considered. The lac operon is one of the simplest biological systems which involves the regulation network of three genes. The mathematical models of the regulatory mechanisms of the lac system, developed in the literature are based on deterministic or fully stochastic approach to the problem. The aim of the thesis is the development of two stochastic models (reduced and full) based on extension of existing deterministic models with noise terms. The two models reflect different level of complexity of the regulatory processes. The advantage of this approach is based on the realistic description of protein concentrations, protein kinetics and time delays. The research considers first order stochastic delayed differential equations (SDDEs) and their solutions. Stability properties of the stochastic models are investigated by linearization of the systems of SDDEs. New sufficient conditions of mean square stability are obtained analytically for these models using Lyapunov function. Additionally, the threshold values for SDDEs are derived. These conditions and threshold values are applied to nd analytical solutions of the two models of nonlinear SDDE. Further, numerical solutions of these equations are obtained using Euler Maruyama method. A detailed analysis of the stability regions of the models is performed, analytically and numerically. A specific attention is given to the bistable region as it reflects important biological features of the system linked to the positive regulatory mechanism. It is concluded that the stochasticity can change the boundaries of the bistable region which cannot be obtained in the case of the deterministic model of the lac operon. This thesis provides a thorough investigation of the stochastic stability of two lac operon models and demonstrates that the system behaviour is very sensitive to protein concentrations. It also provides a novel way for estimating such concentrations

Theoretical investigation of a genetic switch for metabolic adaptation

Membrane transporters carry key metabolites across the cell membrane and, from a resource standpoint, are hypothesized to be produced when necessary. The expression of membrane transporters in metabolic pathways is often upregulated by the transporter substrate. In E. coli, such systems include for example the lacY, araFGH, and xylFGH genes, which encode for lactose, arabinose, and xylose transporters, respectively. As a case study of a minimal system, we build a generalizable physical model of the xapABR genetic circuit, which features a regulatory feedback loop via membrane transport (positive feedback) and enzymatic degradation (negative feedback) of an inducer. Dynamical systems analysis and stochastic simulations show that the membrane transport makes the model system bistable in certain parameter regimes. Thus, it serves as a genetic “on-off” switch, enabling the cell to only produce a set of metabolic enzymes when the corresponding metabolite is present in large amounts. We find that the negative feedback from the degradation enzyme does not significantly disturb the positive feedback from the membrane transporter. We investigate hysteresis in the switching and discuss the role of cooperativity and multiple binding sites in the model circuit. Fundamentally, this work explores how a stable genetic switch for a set of enzymes is obtained from transcriptional auto-activation of a membrane transporter through its substrate

Optimizing information flow in small genetic networks. I

In order to survive, reproduce and (in multicellular organisms) differentiate, cells must control the concentrations of the myriad different proteins that are encoded in the genome. The precision of this control is limited by the inevitable randomness of individual molecular events. Here we explore how cells can maximize their control power in the presence of these physical limits; formally, we solve the theoretical problem of maximizing the information transferred from inputs to outputs when the number of available molecules is held fixed. We start with the simplest version of the problem, in which a single transcription factor protein controls the readout of one or more genes by binding to DNA. We further simplify by assuming that this regulatory network operates in steady state, that the noise is small relative to the available dynamic range, and that the target genes do not interact. Even in this simple limit, we find a surprisingly rich set of optimal solutions. Importantly, for each locally optimal regulatory network, all parameters are determined once the physical constraints on the number of available molecules are specified. Although we are solving an over--simplified version of the problem facing real cells, we see parallels between the structure of these optimal solutions and the behavior of actual genetic regulatory networks. Subsequent papers will discuss more complete versions of the problem

Adaptive evolution of transcription factor binding sites

The regulation of a gene depends on the binding of transcription factors to specific sites located in the regulatory region of the gene. The generation of these binding sites and of cooperativity between them are essential building blocks in the evolution of complex regulatory networks. We study a theoretical model for the sequence evolution of binding sites by point mutations. The approach is based on biophysical models for the binding of transcription factors to DNA. Hence we derive empirically grounded fitness landscapes, which enter a population genetics model including mutations, genetic drift, and selection. We show that the selection for factor binding generically leads to specific correlations between nucleotide frequencies at different positions of a binding site. We demonstrate the possibility of rapid adaptive evolution generating a new binding site for a given transcription factor by point mutations. The evolutionary time required is estimated in terms of the neutral (background) mutation rate, the selection coefficient, and the effective population size. The efficiency of binding site formation is seen to depend on two joint conditions: the binding site motif must be short enough and the promoter region must be long enough. These constraints on promoter architecture are indeed seen in eukaryotic systems. Furthermore, we analyse the adaptive evolution of genetic switches and of signal integration through binding cooperativity between different sites. Experimental tests of this picture involving the statistics of polymorphisms and phylogenies of sites are discussed.Comment: published versio