********************************************************************************* HANS OTHMER, Department of Mathematics, University of Utah Title: Signal Transduction and Second Messenger Systems Topics to be covered will include the following: Models of G-protein mediated signal transduction in mammalian systems Mechanisms for adaptation Frequency encoding in excitable systems Calcium dynamics ********************************************************************************* JOHN TYSON, Department of Biology, Virginia State Title: The Eukaryotic Cell Cycle: Molecules, Mechanisms and Mathematical Models The cell cycle is the sequence of events that mark the passage of a growing cell from birth to division. The major events are DNA replication, mitosis, and cell division. The timing of these events is controlled by a complex biochemical mechanism based on cyclin-dependent kinases (Cdk's). Cdk activity is regulated by synthesis and degradation of cyclin subunits, by phosphorylation of the kinase subunit, and by association with various inhibitory and protective proteins. The mechanism of Cdk regulation is known in such detail that it is now impossible to understand its operation comprehensively by schematic diagrams and informal verbal reasoning. Mathematical modeling provides an indispensable tool for investigating the mechanisms of cell cycle regulation with precision and confidence. The first lectures will summarize the basic physiology, genetics and molecular biology of cell cycle control, followed by some general principles of modeling Cdk networks. Then, as time permits, we will pursue "industrial strength" models of the three primary experimental systems in this field: frog egg extracts, fission yeast cells and budding yeast cells. References: Tyson, PNAS 88:7328-7332 (1991) Novak & Tyson, J Cell Sci 106:1153-1168 (1993) Novak & Tyson, J Theor Biol 173:283-305 (1995) ********************************************************************************* MICHAEL MACKEY, Department of Physiology, McGill University Title: Delays and their role in determining the dynamics of cellular replication These talks will focus on the role of nonlinearities and delays in feedback in determining the dynamics of cellular replication processes. Using examples from the control of red and white blood cell and platelet production, as well as the regulation of the hematopoietic stem cell, I will illustrate how nonlinear dynamics is capable of giving insight into a variety of hematological diseases that display a periodic dynamic behaviour. The concept of "dynamic diseases" will also be discussed. ********************************************************************************* JOHN MILTON, University of Chicago In the lectures I will talk mainly about differential delay equations (DDE) with an emphasis on the interplay between theory and experiment in trying to understand what is going on. I'll start with a first-order DDE model and then extend it to a second-order DDE. I thought that I would, at least at the beginning, focus on the pupil light reflex since direct comparisons between theory and observation are often possible. Particular emphasis will be on the clamping paradigm. Matching the model's predictions to the data lead naturally to questions of the effects of noise. A curious property of 2nd-order DDEs is the co-existence of multiple attractors (multistability). I will show how multistability arises in simple neural networks having delayed recurrent loops. I have a little electronic circuit set-up which illustrates very well the phenomena of multistability (and whose properties can be completely understood analytically !). The clamping paradigm has other applications: dynamic clamping of neurons, closed-loop drug delivery systems and control of chaos. Thus I thought that I would discuss these applications from the point of view of the properties of DDEs. From a mathematical point of view the lectures will be self-contained. ********************************************************************************* ARTHUR SHERMAN, National Institute of Health Title: Calcium and Membrane Potential Oscillations with Applications to Insulin-Secreting Pancreatic Beta-Cells Lecture I. Basic Neural Oscillations 1. The Hodgkin-Huxley equations 2. Excitability, phase planes. 3. Relaxation oscillators 4. Hopf bifurcation 5. Homoclinic orbits: Type I and Type II thresholds (ie. with and without homoclinic) 6. HH analogy to calcium oscillators: beta-cells and pituitary gonadotrophs (Keizer-De Young and Li-Rinzel models) Lecture II. Bursting Mechanisms 1. Fast-Slow decomposition 2. Type I - relaxation-oscillator-like (beta-cell) 3. Type II - parabolic bursters (R-15 neuron) 4. Type III - sub-critical Hopf bifurcation based (Lobster cardiac ganglion) 5. Imperfect bursting - when the slow variable is not very slow. - application to muscarinic bursting in beta-cells 6. Chaotic bursting (Terman) Lecture III. Coupled Oscillators and Bursters 1. Weak coupling: H-functions, Anti-phase, In-phase oscillations 2. Coupling near a Hopf bifurcation: Asymmetric, Quasi-periodic oscillations 3. Application to coupled bursters: Emergent bursting 4. Strong coupling/Heterogeneous: strong coupling limit. 5. Fast/slow synapses: Wang-Rinzel; Frankel-Rinzel models Lecture IV. Stochastic Channel Models 1. Two-state Markov channel model - HH as deterministic limit (DeFelice) 2. Diffusion approximations (Fox, Keizer) 3. Application: channel sharing in pancreatic islets. Misc: 1. Domain theory of inactivation *********************************************************************************