Predicting how interactions between transcription factors and regulatory DNA sequence dictate rates of transcription and, ultimately, drive developmental outcomes remains an open challenge in quantitative biology. Indeed, despite decades of biochemical and genetic studies that have established a reasonably complete "parts list" of the molecular components driving eukaryotic transcription, the field nonetheless lacks a satisfactory understanding of how interactions between these molecular components unfold across space and time to give rise to gene regulatory logic. Recently, technical advancements have begun to provide glimpses into this molecular black box, dramatically improving our ability to trace how molecular pieces move, interact, and assemble. However, if we are to fully realize the immense potential of these exciting new technologies, then our ideas need to catch up with our experiments.

In this thesis, we argue that quantitative models have a central role to play in synthesizing the ever-increasing array of cutting-edge experimental measurements into a coherent theory for the molecular basis of transcriptional control. To this end, we seek to develop conceptual, theoretical and computational frameworks for dissecting how molecular reactions at individual gene loci give rise to the formation of dynamic patterns of gene expression and facilitate cellular decision-making. Chapters 2 and 3 describe previously published works that combine live imaging, statistical inference, and simple quantitative models to probe how transcription factor proteins regulate the dynamics of transcriptional bursting at target gene loci to give rise to stripes of gene expression early on in fruit fly development. Chapter 4 describes a series of analyses following-up on various results from these works. We also use this chapter to break new ground, however, examining how 2 spot experiments, which track the output of two identical gene loci in each cell, can be used to estimate rates of information transmission at individual gene loci from live imaging data.

In Chapters 5 and 6, we connect phenomenological models of transcriptional bursting employed in the preceding sections to truly molecular models that seek to understand how key transcriptional behaviors emerge from molecular interactions at the gene locus. Chapter 5 examines a puzzling gap revealed by recent live imaging studies between the rapid timescale (seconds) of transcription factor binding and the slow timescale (minutes to hours) of transcriptional bursts, and proposes two simple theoretical frameworks for bridging this gap. Chapter 6 investigates how the presence of energy-dissipating processes within the eukaryotic transcriptional cycle can open the door to new kinds of gene regulatory logic that increase the rate at which gene loci transmit information. Finally, Chapter 7 describes ongoing work to develop a novel Bayesian framework, burstMCMC, that uses efficient inference techniques to examine how transcription factor proteins regulate transcriptional burst dynamics.