Recurrent processes are a general feature of living systems, from the cell cycle to circadian day-night rhythms to hibernation and flowering cycles. During development and life, numerous recurrent processes are controlled by genetic oscillators, a specific class of genetic regulatory networks that generates oscillations in the level of gene products. A vital mechanism controlled by genetic oscillators is the rhythmic and sequential segmentation of the elongating body axis of vertebrate embryos. During this process, a large collection of coupled genetic oscillators gives rise to spatio-temporal wave patterns of oscillating gene expression at tissue level, forming a dynamic prepattern for the precursors of the vertebrae. While such systems of genetic oscillators have been studied extensively over the past years, many fundamental questions about their collective behavior remain unanswered. In this thesis, we study the behavior and the properties of genetic oscillators from the single oscillator scale to the complex pattern forming system involved in vertebrate segmentation. Genetic oscillators are subject to fluctuations because of the stochastic nature of gene expression. To study the effects of noisy biochemical coupling on genetic oscillators, we propose a theory in which both the internal dynamics of the oscillators as well as the coupling process are inherently stochastic. We find that stochastic coupling of oscillators profoundly affects their precision and synchronization properties, key features for their viability as biological pacemakers. Moreover, stochasticity introduces phenomena not known from deterministic systems, such as stochastic switching between different modes of synchrony. During vertebrate segmentation, genetic oscillators play a key role in establishing a segmental prepattern on tissue scale. We study the spatio-temporal patterns of oscillating gene expression using a continuum theory of coupled phase oscillators. We investigate the effects of different biologically relevant factors such as delayed coupling due to complex signaling processes, local tissue growth, and tissue shortening on pattern formation and segmentation. We find that the decreasing tissue length induces a Doppler effect that contributes to the rate of segment formation in a hitherto unanticipated way. Comparison of our theoretical findings with experimental data reveals the occurrence of such a Doppler effect in vivo. To this end, we develop quantification methods for the spatio-temporal patterns of gene expression in developing zebrafish embryos. On a cellular level, tissues have a discrete structure. To study the interplay of cellular processes like cell division and random cell movement with pattern formation, we go beyond the coarse-grained continuum theories and develop a three-dimensional cell-based model of vertebrate segmentation, in which the dynamics of the segmenting tissue emerges from the collective behavior of individual cells. We show that this model is able to describe tissue formation and segmentation in a self-organized way. It provides the first step of theoretically describing pattern formation and tissue dynamics during vertebrate segmentation in a unified framework involving a three-dimensional tissue with cells as distinct mechanical entities. Finally, we study the synchronization dynamics of generic oscillator systems whose coupling is subject to phase shifts and time delays. Such phase shifts and time delays are induced by complex signaling processes as found, e.g., between genetic oscillators. We show how phase shifts and coupling delays can alter the synchronization dynamics while leaving the collective frequency of the synchronized oscillators invariant. We find that in globally coupled systems, fastest synchronization occurs for non-vanishing coupling delays while in spatially extended systems, fastest synchronization can occur on length scales larger than the coupling range, giving rise to novel synchronization scenarios. Beyond their potential relevance for biological systems, these results have implications for general oscillator systems, e.g., in physics and engineering. In summary, we use discrete and continuous theories of genetic oscillators to study their dynamic behavior, comparing our theoretical results to experimental data where available. We cover a wide range of different topics, contributing to the general understanding of genetic oscillators and synchronization and revealing a hitherto unknown mechanism regulating the timing of embryonic pattern formation.

Recurrent processes are a general feature of living systems, from the cell cycle to circadian day-night rhythms to hibernation and flowering cycles. During development and life, numerous recurrent processes are controlled by genetic oscillators, a specific class of genetic regulatory networks that generates oscillations in the level of gene products. A vital mechanism controlled by genetic oscillators is the rhythmic and sequential segmentation of the elongating body axis of vertebrate embryos. During this process, a large collection of coupled genetic oscillators gives rise to spatio-temporal wave patterns of oscillating gene expression at tissue level, forming a dynamic prepattern for the precursors of the vertebrae. While such systems of genetic oscillators have been studied extensively over the past years, many fundamental questions about their collective behavior remain unanswered. In this thesis, we study the behavior and the properties of genetic oscillators from the single oscillator scale to the complex pattern forming system involved in vertebrate segmentation. Genetic oscillators are subject to fluctuations because of the stochastic nature of gene expression. To study the effects of noisy biochemical coupling on genetic oscillators, we propose a theory in which both the internal dynamics of the oscillators as well as the coupling process are inherently stochastic. We find that stochastic coupling of oscillators profoundly affects their precision and synchronization properties, key features for their viability as biological pacemakers. Moreover, stochasticity introduces phenomena not known from deterministic systems, such as stochastic switching between different modes of synchrony. During vertebrate segmentation, genetic oscillators play a key role in establishing a segmental prepattern on tissue scale. We study the spatio-temporal patterns of oscillating gene expression using a continuum theory of coupled phase oscillators. We investigate the effects of different biologically relevant factors such as delayed coupling due to complex signaling processes, local tissue growth, and tissue shortening on pattern formation and segmentation. We find that the decreasing tissue length induces a Doppler effect that contributes to the rate of segment formation in a hitherto unanticipated way. Comparison of our theoretical findings with experimental data reveals the occurrence of such a Doppler effect in vivo. To this end, we develop quantification methods for the spatio-temporal patterns of gene expression in developing zebrafish embryos. On a cellular level, tissues have a discrete structure. To study the interplay of cellular processes like cell division and random cell movement with pattern formation, we go beyond the coarse-grained continuum theories and develop a three-dimensional cell-based model of vertebrate segmentation, in which the dynamics of the segmenting tissue emerges from the collective behavior of individual cells. We show that this model is able to describe tissue formation and segmentation in a self-organized way. It provides the first step of theoretically describing pattern formation and tissue dynamics during vertebrate segmentation in a unified framework involving a three-dimensional tissue with cells as distinct mechanical entities. Finally, we study the synchronization dynamics of generic oscillator systems whose coupling is subject to phase shifts and time delays. Such phase shifts and time delays are induced by complex signaling processes as found, e.g., between genetic oscillators. We show how phase shifts and coupling delays can alter the synchronization dynamics while leaving the collective frequency of the synchronized oscillators invariant. We find that in globally coupled systems, fastest synchronization occurs for non-vanishing coupling delays while in spatially extended systems, fastest synchronization can occur on length scales larger than the coupling range, giving rise to novel synchronization scenarios. Beyond their potential relevance for biological systems, these results have implications for general oscillator systems, e.g., in physics and engineering. In summary, we use discrete and continuous theories of genetic oscillators to study their dynamic behavior, comparing our theoretical results to experimental data where available. We cover a wide range of different topics, contributing to the general understanding of genetic oscillators and synchronization and revealing a hitherto unknown mechanism regulating the timing of embryonic pattern formation.