I will briefly discuss modeling from my group of the lactose and tryptophan operons as well as the lysis/lysogeny switch in phage lambda. Then a summary of recent modeling of the gal regulon in yeast and zebrafish somitogenesis. The emphasis will be on the utility of mathematical models in either suggesting further experimental work, or in highlighting situations in which there are unknown aspects of the biology.

Fluctuations in gene expression give cells access to a spectrum of phenotypes that can temporarily increase drug resistance and serve as a transient basis for natural selection. To take full advantage of this noise-induced drug resistance, cells may rely on mechanisms that (1) increase cell-to-cell variability in gene expression, and (2) enable inheritance of beneficial gene expression states from one generation to the next. The ATP-binding cassette (ABC) protein family is a large group of membrane transporters that are conserved from bacteria to humans, and facilitate drug resistance by driving the secretion of substances from the cell interior. In budding yeast, the expression of the ABC transporter Pdr5p, and of several other pleiotropic drug response (PDR) pumps, is controlled by a coherent feedforward loop involving the partially redundant transcriptional regulators Pdr1p and Pdr3p. Here, we demonstrate that transcriptional regulation by coherent feedforward loops can enhance drug resistance by increasing cell-to-cell variability in gene expression, and by enabling prolonged activation of gene expression in response to transient signals. Our results highlight how mechanisms enabling transient, non-genetic inheritance may play important roles in defining the effectiveness of drug treatment.

Genetic activity is partially regulated by a complicated network of proteins called transcription factors. I will describe a mathematical framework to relate the structure and dynamics of these genetic networks. The underlying idea is to capture the topology and logic of the network interactions by a Boolean network, and to then embed the logical network into continuous piecewise linear differential equations. The equations can be analyzed using methods from discrete mathematics and nonlinear dynamics. By changing the logical structure randomly, it is possible to evolve the networks in an effort to identify networks that display rare dynamics - e.g. networks with long stable cycles or with a high level of topological entropy. I also consider the concept of robustness in the context of these equations and argue that robustness should be a key feature of genetic networks underlying important biological functions.

One of the the current challenge for cell biology is understanding of the system level cellular behavior from the knowledge of a network of the individual sub-cellular agents. We address a question of how the model selection affects the predicted dynamic behavior of a gene network. In particular, for a fixed network structure, we compare protein-only models with models in which each transcriptional activation is represented both by mRNA and protein concentrations. We compare linear behavior near equilibria for both cyclic feedback systems and a general system. We show that, in general, explicit inclusion of the mRNA in the model weakens the stability of equilibria. We also study numerically dynamics of a particular gene network and show significant differences in global dynamics between the two types of models.