Impacto General del Proyecto

This work presented the mathematical modelling and analysis of an engineered ge- netic oscillator inE.coli, in concert with the acquisition and analysis of experimental data on that circuit, based on a newly set up imaging assay. Genetic oscillators are the basis of biological clocks, which allow organisms to maintain stable circadian rhythms. The mathematical modelling took into account the special, heterogeneous nature of genetic regulatory systems, where variables can describe discrete finite gene states, or infinite protein numbers (or concentrations). The newly set up imaging assay for the engineered clock allowed to collect data over long time periods in an automated fashion. Following an introductory chapter (chapter one), the work was divided into three parts: Chapter two described the theoretical modelling framework and introduced a newly developed algorithmic tool for automated model construc- tion. Chapter three presented the model derivation and its analysis. Chapter four discussed the details of the imaging assay and the data collected from the clock cells.

Summary of the presented work. Models with finite, inherently discrete vari- ables in conjunction with variables describing particle numbers or concentrations require special theoretical consideration. In chapter two, a theoretical framework, previously described in [66, 69], was discussed, that allows to derive deterministic and continuous rate equations from discrete stochastic model formulations of molec- ular interactions. In order to be able to apply this theory to realistic systems, an algorithmic modelling tool was devised and implemented, that automatically con- structs Markov chain models for gene activity states based on binding dynamics of regulatory factors to enhancer and operator sites of the respective promoters. In particular, the algorithm is able to account for cooperative binding and DNA loop formation. The source code and example files can be downloaded from the website of the Compbio group at ‘lora.maths.warwick.ac.uk’. The developed algorithm was

applied to study different models for repressive DNA loop formation induced by the

lac repressor. The model analysis suggested, that loop formation based on solely dimeric repressor molecules fails to convey the noise reducing properties of looping by tetrameric rerpressors, which were reported previously in [85].

Chapter three presented the derivation and analysis of models for the engineered clock circuit. To reduce the dimension of the modelling problem, the Markov chain of the gene states of the clock model was decoupled into submodules, to be treated

separately, describing the regulation of the promoters glnAp2 and glnKp by the

transcription factor NRI, and the repression of the glnAp2 promoter by the lac

repressor, LacI. Models for the NRI dependent regulation were constructed, which took into account the cooperative binding of NRI to DNA, and whose parameters

were fitted from the literature, where the glnAp2 and glnKp promoters have been

studied in detail. The average dynamics for the clock circuit were then derived by reassembling the rate laws for the respective submodules. The transcriptional and translational processes were considered jointly, motivated by a quasi-steady-state assumption on the mRNA concentrations, to arrive at a two dimensional system of coupled ordinary differential equations.

Nullcline and linear stability analysis revealed that the occurrence of a Hopf bi- furcation requires the positiveness of the slope of the NRI nullcline. This in turn was only possible for models where theglnAp2promoter was driven by at least two enhancer sites. Further, sufficient conditions on the nullcines were derived for sus- tained ocillations to exist. Other possible behaviour of the system, like the existence of a single stable steady state, corresponding to damped oscillations in the case of a stable spiral, or bistability, were characterized in terms of corresponding nullcline scenarios. The model suggested, that a short NRI lifetime relative to a longer LacI lifetime, as well as the difference in the activation thresholds of the glnAp2 and

glnKppromoters, were important for stable oscillations.

In order to fit and optimize the model and to be able to test predictions, a new monitoring assay for the eningeered clock was set up based on luminescent transcrip- tional reporter constructs. The imaging of colonies growing on a petri dish allowed for automated data collection over long time spans and also opened the possibility to screen for clock mutants. Damped oscillations were observed after starting the clock by the removal of the lac inducer IPTG, as was expected from previously re-

ported results based on chemostat culture assays monitoring LacZ activity [8]. Also, damped oscillations were observed after cells were exposed to a metabolic shock due to colony transfer between agar plates and consequent exposure to fresh nutrients. (Colony transfer between plates was used to change the growth medium.) The ob- served period of oscillations differed with the age of the colonies. On the grounds of previous results [8], it was concluded that different average growth rates of cells within colonies of different ages may be responsible for these period variations. The esablished imaging assay constitutes a decent basis for the further testing of model predictions. For this, the assay may need to be further optimized to improve control of growth conditions. Also the imaging data could be complemented by measure- ments of mRNA and protein numbers to improve parameter fitting.

Discussion and Outlook. The Markov chain based modelling framework allows to bridge from detailed, discrete stochastic model formulations to deterministic and continuous systems, which are more tractable by mathematical analysis. The pre- sented algorithm, that automatizes the construction of the discrete Markov chain models for the described class of genetic regulatory systems, constitutes a step to- wards facilitating the application of this kind of modelling to realistic systems and making it more accessible to non-specialists. A possible route for further devel- opment would be to extend the algorithm, such that it can treat a larger class of models, or to embed the algorithm into a larger modelling package, such that the derivation of invariant measures and average dynamics can be performed routinely. This would facilitate the comparative analysis of models for gene regulation and re- lated systems, which are able to distinguish between specific molecular mechanisms, and would make such modelling available to a larger part of the multidisciplinary research community. The software package Copasi represents an example for such a modelling tool used mainly for metabolical reaction networks [37]. A comparable tool specifically designed for genetic regulatory networks appears still to be missing. The analysis of the models of the clock circuit highlighted, that apart from the overall composition of the regulatory dependencies, also the molecular details of the implementations (number of binding sites, DNA looping, cooperativity) are important for determining the qualitative dynamic properties of the system. For the studied clock circuit the analysis suggested that the cooperative nature of the auto-

activation of the activator module is an essential feature for oscillations, while the nonlinearity of other regulatory mechanisms (activation of the repressor module and repression of the activator module) seemed to be less important. The importance of the degradation rates is in accordance with studies of other genetic oscillators [33] and the difference in activation thresholds of the activator and repressor module may mimic the role of delayed negative feedback, like for example in theDrosophila

clock described in chapter one.

Monitoring the clock using the newly setup imaging assay proved feasible, even though further optimization is needed, like more homogenous growth conditions, to improve the quality of the data. Maintaining homogenous growth conditions for this colony based assay is more difficult than for chemostat cultures, for which, on the other hand, assays based on culture samples are more difficult to automatize. Future work on the established clock assay could include site-specific mutations of the clock’s gene modules to study for example the effect of different activation thresholds for the two gene modules of the clock, or increasing the degradation rate of the activator protein NRI, by tagging it with degradation signal peptides. Random mutagenesis of the chromosome of the clock strain might also yield interesting clock behaviour which could be screened for.

On a wider perspective, the current work highlighted some characteristic features of models of genetic regulatory systems, like of the heterogeneity of variables and also the complexity in the details of the corresponding molecular mechanisms. The presented results suggest, that it is important to develop models that are able to precisely distinguish between different molecular mechanisms. On the other hand, biological models found in the literature, are often not precise enough to be trans- lated into mathematical models without some ambiguity and additional assump-

tions. Making the construction and formulation of mathematically well defined

models more routine, should thus help advance the understanding of biological sys- tems.

In particular in the context of models for genetic oscillators, their possibly com- plex dynamics are sensitive to the molecular details of gene regulation. The ad- vancing ability to construct artificial genetic constructs promises to facilitate the comparison of different mechanisms by modelling and experiment. This may help elucidate the more complex properties of natural biological clocks like temperature

compensation and robustness, by aiming to engineer circuits that are able to mimic some of these advanced properties.

This thesis, being an interdisciplinary work of mathematics and biology, is also characterized by the challenge to bridge between the corresponding research cultures. While biologists typically use descriptive language in order to understand a certain biological function, mathematicians prefer more rigorous and precise formulations in order to capture more general patterns. This two-sidedness presents a challenge in the field of mathematical biology in general. But as the understanding of the cell as a complex molecular machine gains coherency, the description of its workings will naturally become more mathematical. In this respect genetic oscillators are an example for a long standing, and successfully developing field of research for mathematical biology.