One of our main activities over the last few years has been the development of a comprehensive model for oscillations of membrane potential and calcium on time scales ranging from seconds to minutes. These lead to corresponding oscillations of insulin secretion. The basic hypothesis of the model is that the faster (tens of seconds) oscillations stem from feedback of calcium onto ion channels, likely calcium-activated potassium (K(Ca)) channels and ATP-dependent potassium (K(ATP)) channels, whereas the slower (five minutes) oscillations stem from oscillations in metabolism. The metabolic oscillations are transduced into electrical oscillations via the K(ATP) channels. The latter, notably, are a first-line target of insulin-stimulating drugs, such as the sulfonylureas (tolbutamide, glyburide) used in the treatment of Type 2 Diabetes. The model thus consists of an electrical oscillator (EO) and a metabolic (glycolytic) oscillator (G)) and is referred to as the Dual Oscillator Model (DOM). We are currently testing this model in several ways. Last year we reported that metabolic oscillations, assayed by NAD(P)H measurements, often persist in steady calcium, indicating that calcium oscillations are not required for metabolic oscillations. The two, however, are generally found in tandem, and the calcium oscillations, as well as mean calcium level, do influence the metabolic oscillations. We have now confirmed these findings with measurements of K(ATP) channel conductance and are preparing a paper on the subject. We have written a commentary (Ref. # 1)about dynamical systems methods in physiology in order to enhance the benefit for the physiology community of two recent papers by others presenting a new, more comprehensive model for (fast) beta-cell electrical activity. Whereas we have made our models as simple as possible for the phenemona addressed, the new model includes a much wider set of mechanisms. This raises issues of how to assess the relative importance of the different mechanisms and of how cells use redundancy. The complexity of the new model and others like it also poses a challenge for understanding how the model works and what its capabilities and limitations are. The commentary describes with a minimum of mathematics how bifurcation diagrams can still be applied effectively. Such diagrams are at one level maps of the parameter regimes in which the various behaviors of the model, including steady states, spiking and bursting, are found. They also provide a way to dissect the dynamics by exploiting the fact that different processes (here, spiking and bursting) operate on different time scales (<1 sec vs. 10 - 60 sec) and can be considered as semi-independent. This reduces the collective behavior into the behavior of simpler sub-systems and greatly increases the power of analysis. Evolution may exploit such timescale separation as well, as it serves to make cell function modular - the individual subsystems can be altered with limited effect on the others. The review can be profitably read as a didactic guide to the work described in this report. A figure from the commentary was selected as the cover art for the journal's July issue. A particularly interesting application of the separation of timescales in models for bursting in beta cells is the phenomenon of resetting. An insight from the earliest beta-cell model (Chay-Keizer, 1983) is that the plateau from which spiking occurs is established by bi-stability. That is, if the slow variable calcium is fixed, the cell can sit at either a low-voltage (-60 mV) steady state or a high-voltage (-20 mV) spiking state. Consequently, brief electrical stimuli should be able to switch the cell from one state to the other. Moreover, the models predicted that the later in the low-voltage (silent) phase in which the perturbation is delivered, the shorter would be the induced high-voltage (active) phase. Experiments have confirmed that silent-active phase transitions can be induced as expected, but the duration of the induced phase does not seem to depend on when the perturbation is applied. In Ref. # 2 we show in collaboration with the Bertram group that more recent beta-cell models, with two slow variables controlling the active and silent phase durations can account for this heretofore puzzling experimental observation. Ref. # 4 addresses the issues of bistability, resettability and separation of timescales in models of bursting for both beta cells and closely related but different pituitary cells. It is discussed in detail in our report on Mathematical Modeling of Neurons and Endocrine Cells. In collaboration with Max Pietropaolo (U. Michigan) post-doctoral fellow Anmar Khadra and I began a new line of work for the lab on Type 1 Diabetes (T1D), characterized by auto-immune destruction of beta cells. Pietropaolo has a long-standing interest in use of islet autoantibodies as biomarkers of risk for progression to T1D. While differences in rate of progression have been correlated with the appearance of different autoantibodies or the number of autoantibody types, we sought to determine the underlying mechanism by developing a mathematical model for the interactions among beta cells, T cells and B cells. We identified two key parameters controlling the time to progression to T1D, the avidity of the T cells for beta cells and their killing efficiency. The model was also able to illuminate the phenomenon of avidity maturation, in which T-cell avidity increases over time, accelerating the disease process. See Ref. # 3.
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