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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
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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)
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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.
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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.
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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
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