### Mini-workshop: Quantitative mathematical modeling of gene regulatory networks: Abstracts and Lecture Materials

Toward and understand of the design and evolution of the NCR-circuit
Erik Boczko, Department of Biomedical Informatics, Vanderbilt University

The goal of this talk is to describe our ongoing efforts at understanding the dynamical and physiological properties and evolutionary design principles of a network of five genes that ultimately regulate nitrogen catabolite repression in the yeast Saccharomyces Cerevisiae.

Regulation at the intersection of carbon and nitrogen metabolism in Saccharomyces cerevisiae
Terry Cooper, Department of Molecular Sciences, University of Tennessee

Significant progress has been made in understanding the integrated regulation of nitrogen and carbon metabolism in Saccharomyces cerevisiae. Identification of four GATA-family transcription factors (Gln3, Gat1, Dal80, Deh1) that competitively regulate nitrogen cata-bolic gene expression as well as the influence of Tor1/2 on their intracellular localization form the bases of current investigations into the biochemical mechanisms responsible for this regulation. However, nitrogen catabolic genes respond not only to environmental signals of nitrogen availability, but also to the levels of carbon metabolites required for nitrogen assimilation. One of the pivotal carbon metabolites is -ketoglutarate which is synthesized by the gene products of the retrograde pathway. Like the GATA-factors, Gln3 and Gat1, phosphorylation and intracellular localization of the retrograde transcrip-tion factors, Rtg1/3, have been shown to respond to both the nitrogen source provided in the culture medium and to the Tor1/2 signal transduction pathway. We will discuss recent progress contributing to our understanding of Gln3/Rtg3-mediated transcription and mechanisms responsible for the regulation of nitrogen catabolism and its integration with carbon metabolism at the level of a-ketoglutarate.

Understanding Bistability in Compex Enzyme-Driven Reaction Networks
Martin Feinberg, Chemical Engineering & Mathematics, The Ohio State University

In nature there are millions of distinct networks of chemical reactions that might present themselves for study at one time or another. Each network gives rise to its own system of differential equations. These are usually large and almost always nonlinear. Nevertheless, the reaction network induces the corresponding differential equations (up to parameter values) in a precise way. This raises the possibility that qualitative properties of the induced differential equations might be tied directly to reaction network structure.

Chemical reaction network theory has as its goal the development of powerful but readily implementable tools for connecting reaction network structure to the qualitative capacity for certain phenomena. The theory goes back at least to the 1970s*. It has not been specific to biology, but, for obvious reasons, there is now growing interest in biological applications. Very recent work (with Gheorghe Craciun) has been dedicated specifically to biochemical networks driven by enzyme-catalyzed reactions. In particular, it is now known that there are remarkable and quite subtle connections between properties of reaction diagrams of the kind that biochemists normally draw and the capacity for biochemical switching. My aim in this talk will be to explain, for an audience unfamiliar with chemical reaction network theory, those tools that have recently become available.

*For some early work, see M. Feinberg, Lectures on Chemical Reaction Networks, University of Wisconsin Mathematics Research Center, 1979, available at http://www.che.eng.ohio-state.edu/~feinberg/LecturesOnReactionNetworks/

A model of nitrogen catabolite repression gene regulatory circuit
Tomas Gedeon, Department of Mathematical Sciences, Montana State University

We consider a model for the dynamics of a gene circuit involved in nitrogen catabolite repression (NCR) in yeast. The model consists of a large system of periodicaly forced system of delay-differential equations, where delay accounts for transcriptional and translational delays and the periodical forcing describes cell cycle.

We analyze a GLN3-URE2 subcircuit, whose role is to switch the circuit on and off. Using monotone system theory we conclude that the behavior of this subcircuit is generically simple, and almost all solution converge to a periodic orbit, whose orbit is a multiple of the cell cycle.

Dynamics in Genetic Networks
Leon Glass, Department of Physiology, McGill University

Genetic activity is partially regulated by a complicated network of proteins called transcription factors. I will describe a mathematical framework that can be used to relate the structure and dynamics of these genetic networks. The networks are represented by differential equations with switchlike nonlinearities. These equations are represented schematically using a directed graph on an hypercube. There are many advantages to these equations. Because of the discrete representation of the continuous dynamics, the numbers of different networks with N model genes can be counted and classified. The methods are helpful in identifying networks that have certain types of dynamic behaviors such as stable fixed points, stable cycles, and chaotic dynamics. These methods can be used to help design in vitro genetic networks that show oscillation and multistability. They can also be used to determine gene network structure based on the patterns of activation of genes (1). Finally, the framework offers novel ways to study the evolution of rhythmic patterns in model equations and also in electronic circuits that simulate the differential equations (2).

1. T. J. Perkins, M. T. Hallett, L. Glass. Inferring models of gene expression dynamics. Journal of Theoretical Biology 230, 289-299 (2004).
2. J. P. Mason, P. S. Linsay, J. J. Collins, L. Glass. Evolving complex dynamics in electronic models of genetic networks. Chaos 14, 707-715 (2004).

Coupled Cell Systems: Theory and Examples
Martin Golubitsky, Department of Mathematics, University of Houston

A coupled cell system is a collection of interacting dynamical systems. Coupled cell models assume that the output from each cell is important and that signals from two or more cells can be compared so that patterns of synchrony can emerge. We ask: How much of the qualitative dynamics observed in coupled systems is the product of network architecture and how much depends on the specific equations?

We will discuss necessary and sufficient conditions for (robust) synchrony and some surprising aspects of synchrony-breaking bifurcations.

Dissecting the Molecular Mechanics of Eukaryotic Translation Initiation
Jon Lorsch, Department of Biophysics and Biophysical Chemistry, Johns Hopkins University School of Medicine

Our lab is trying to elucidate the detailed molecular mechanics underlying a complex biological process - the initiation of protein synthesis in eukaryotic organisms. This process requires at least 24 different protein factors, two ribosomal subunits, a special kind of transfer RNA (tRNA), a messenger RNA (mRNA) and both ATP and GTP. We would like to understand how all of these components work together to facilitate and coordinate the assembly of a ribosomal complex at the appropriate place on an mRNA in a form ready to begin synthesizing the encoded protein. Our approach has been to reconstitute this system in vitro using purified components and then use biochemical and biophysical techniques to measure the rate and equilibrium constants governing each step and interaction in the pathway. Because we have reconstituted yeast translation initiation, we can synergistically couple the awesome power of yeast genetics to detailed quantitative and molecular analyses possible only in vitro. By studying in our system mutant components that produce interesting phenotypes when expressed in living yeast cells, we are learning about what individual chemical elements within the components of the system do and are thus building up a picture of how the dynamic interactions and rearrangements among these components lower the appropriate energy barriers and stabilize the necessary states to allow this stunningly complex molecular machinery to function.

One of the steps we have been focusing on recently is the identification of the correct first (initiation) codon in the mRNA. This is arguably the most important reading of the genetic code during gene expression because if it is not done correctly the wrong protein will be made from the mRNA. It remains largely mysterious, however, how a subset of the protein synthesis machinery recognizes the initiation codon and how the signal is sent to the rest of the machinery that it has been found and the process should continue. We have recently made substantial advances in our understanding of this step and these will be presented.

Insights into the evolution of the GATA family of transcription factors
Jason Lowry, School of Molecular and Microbial Biosciences, University of Sydney

Since the article detailing the completion of the S. cerevisiae genomic sequence was published in Science (Goffeau et al, 1996), numerous other eukaryotic organisms' genomes have also been completed, thereby enabling comparative genome analysis. In the last decade, the genome sequencing centers have produced an enormous amount of information, particularly with regard to fungal genomes as eight are now considered to be completely sequenced. Utilizing this information, we take a closer look into the evolution of GATA genes within fungi (particularly Gln3, Gat1, Dal80, and Gzf3), as well as among the other eukaryotes.

Mathematical Models of Genetic Control in Bacteria
Joseph M. Mahaffy, Department of Math Sciences, San Diego State University

Regulation of metabolic pathways is very important in bacteria. Cells employ both positive and negative control mechanisms to maintain fairly stable amounts of key compounds, such as nutrients in the cytoplasm. Mathematical models have been developed using biochemical kinetics, which results in systems of differential equations. Multi-step processes can be modeled with a delay. These systems of delay-differential equations can be analyzed for their behavior. Several bacterial metabolic pathways and the control of the replication cycle in Escherichia coli are examined using some kinetic models. Analysis provides insight into the key regulatory elements in the pathway.

Limiting Profiles of Solutions of State-Dependent Delay-Differential Equations
John Mallet-Paret, Division of Applied Mathematics, Brown University

We study some idealized models of state-dependent delay-differential equations. Introducing a singular perturbation parameter , we see that as $\epsilon$ tends to zero, the differential equation limits to a difference equation. A fundamental issue is to understand the relation between solutions of the (infinite dimensional) differential-delay equation for small $\epsilon$, and the (finite dimensional) difference equation. Among our results is one that shows, in a precise sense, that solutions of state-dependent equations have simpler profiles (limiting graphs) than do solutions of equations with constant delays.

The Nitrogen Regulatory Circuit of the Filamentous Fungi, Neurospora and Aspergillus
George A. Marzluf, Department of Biochemistry, The Ohio State University

The filamentous fungi, Neurospora crassa and Aspergillus nidulans, utilize ammonia, glutamine or glutamate as nitrogen sources preferentially, but when these favored sources are not available, they are capable of using many other secondary nitrogen-containing compounds, such as purines, nitrate, amino acids and proteins. Nitrogen repression prevents the expression of the spectrum of genes encoding enzymes for catabolism of secondary nitrogen sources. Repression occurs by glutamine but the identity of the element which actually senses this metabolite remains unknown. Upon conditions of derepression, the expression of an entire set of structural genes which allow catabolism of many varied alternative nitrogen sources is controlled by a complex regulatory circuit. The synthesis of the permeases and enzymes of a particular catabolic pathway often requires two distinct positive signals: First, a global signal indicating nitrotgen derepression, and second, a pathway-specific signal which indicates the presence of a substrate or intermediate of the pathway. This two-step requirement permits the selective expression of just the enzymes of a specific catabolic pathway. Positive-acting global nitrogen regulatory proteins, NIT2 in Neurospora and AREA in Aspergillus, are GATA factors, DNA binding proteins which possess a single Cys2/Cys2 zinc finger. A negative-acting protein, NMR, binds to two regions of NIT2, preventing its function during conditions of nitrogen repression. Although similar, significant differences occur in the mechanism by which NIT2 and AREA control gene expression. The utilization of inorganic nitrate requires the presence of a pathway-specific factor, NIT4 in Neurospora, a DNA binding protein with a Gal4-like Cys6/Zn2 motif. Optimal expression of nit-3, which encodes nitrate reductase, is dependent upon a specific protein interaction between NIT2 and NIT4. The synthesis of enzymes required to use purines, e.g., xanthine or uric acid, requires nitrogen derepression and induction, mediated by a pathway specific factor PCO1, a DNA binding protein with a Cys6/Zn2 motif. However, expression of the purine catabolic genes does not require NIT2, nor is there any interaction between NIT2 and PCO1, features which demonstrate additional complexity in nitrogen regulation.

On noisy networks
Sebastian Schreiber, Department of Mathematics, The College of William & Mary

Gene networks often involve interacting cellular components each of which consist of a small numbers (tens to hundreds) of molecules. These small numbers combined with random timings of degradation, transcription, transportation, and translation events result in an intrinsic stochasticity that may substantially influence network dynamics. To understand how intrinsic noise influences network dynamics, I review two mathematical approaches (linearization'' and the large deviation theory of Freidlin and Wentzell) to studying the behavior of continuous time Markov chain models of gene networks. Using these approaches, I illustrate how the details of the network architecture can suppress or amplify noise and result in attractor selection in systems whose deterministic counterparts have multiple attractors.

Qualitative/quantitative analysis of a class of biological networks
Eduardo Sontag, Department of Mathematics & BioMaPS Institute for Quantitative Biology, Rutgers University

The analysis of signaling networks constitutes one of the central questions in systems biology: there is an pressing need for powerful mathematical tools to help understand, quantify, and conceptualize their information processing and dynamic properties. Approaches based upon detailed modeling and simulation are hampered by the fact that is virtually impossible to experimentally validate the form of the nonlinearities used in reaction terms, or, even when such forms are known, to accurately estimate coefficients (parameters). In this presentation, we show how some signaling systems may be profitably studied by first decomposing them into several subsystems, each of which is endowed with certain "qualitative" mathematical properties. These properties, in conjunction with a relatively small amount of "quantitative" data, allow the behavior of the entire, reconstituted system, to be deduced from the behavior of its parts. This novel approach emerged originally from our study of possible multi-stability or oscillations in feedback loops in cell signal transduction modeling, but turns out to be of more general applicability. The mathematical techniques rely heavily on the theory of monotone systems. (Most of the work reported in this talk was caried out in collaboration with D. Angeli, and parts of it with J. Ferrell, G. Enciso, and P. de Leenheer.)

Inference of gene regulation parameters in a mathematically well-posed model of nitrogen source selection in yeast
Natal A.W. van Riel, Control Systems group, Depts. of Biomedical Engineering and Electrical Engineering Eindhoven University of Technology

The 'systems biology' concept has gained increasing interest in the last years, especially because large-scale approaches, such as genomics, confront (molecular) biologists with genes and gene products integrated in a functional network. The physical properties of fundamental cellular processes, such as transcription, protein-protein interactions and cell signaling, make the area well-suited for models that combine continuous dynamics and discrete events, so-called hybrid systems.

It will be shown how system identification methods can be used to derive a model that describes the genetic control and substrate fluxes which enable baker's yeast (Saccharomyces cerevisiae) to optimally respond to changes in nitrogen availability. In addition, it will be discussed how a reduced, piecewise-linear model can be derived that is biologically meaningful and can be used for the interpretation of experimental data.

First, a continuous, nonlinear model will be proposed. By including a priori information in the identification criterion, it is possible to obtain estimates of parameter values and their variance with a relatively limited experimental dataset. Based on the continuous model realization, a hybrid model can be identified. In the hybrid model, there is a small approximation error in the description of the regulation, but this has no effect on the predicted fluxes. The presented approach is useful to derive mathematically well-posed models of biomolecular networks. It is illustrated how the identified models yield insight in the biological system, hereby contributing to an ongoing discussion between cell biologists on which signals trigger the regulation to select the best nitrogen substrate.