Mathematical Modeling of Neurons and Endocrine Cells

National Institute Of Diabetes And Digestive And Kidney Diseases

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Abstract

The endocrine cells of the pituitary have many similarities with pancreatic beta-cells and thus form an interesting group for comparative study of mechanisms. We have shown previously that some pituitary cells (somatotrophs, lactotrophs, corticotrophs) exhibit bursting activity (alternating silent, low voltage, and spiking, high voltage states) similar to pancreatic beta-cells but on much faster time scales (seconds vs. tens of seconds or minutes). This is partly the result of a smaller contribution of the endoplasmic reticulum calcium store in modulating and slowing the electrical calcium oscillations and partly due to the lack of an internal metabolic oscillator or an ion channel (eg. K(ATP)) that can link the electrical and metabolic oscillators. Under certain conditions (eg. exposure to acetylcholine or isolation of single cells from the normal environment in the islets of Langherhans), beta-cells exhibit very fast oscillations that closely resemble pituitary bursting and are similary brief. Even in this case, the models suggest that there may be differences in the mathematical structure underlying the oscillations. In the classic beta-cell type burster, the spikes are themselves fast oscillations that would persist if the calcium concentration were frozen, whereas in the pituitar-type bursters the spikes, which are smaller in amplitude, appear to be only transient, damped oscillations that would approach a plateau voltage if calcium were frozen. We have previously shown that the two mechanisms could be distinguished experimentally by attempting to reset from silent to active phase using brief depolarizations. We have now shown that the two classes of models can be unified by variation of a single parameter, the voltage at which the calcium currents activate. This may thus be under regulatory control or subject to experimental manipulation. We have further mathematically explored the differences in the two classes of models as a function of the rate of change of calcium in response to voltage changes, which is in turn dependent on the extent to which free calcium is buffered by cytosolic proteins. This is also possibly under regulatory or developmental control and is subject to experimental manipulation. This study also opens up deep questions in the general mathematical theory of bursting oscillations that our colleagues in Bristol are interested in pursuing. A paper is in review. Synaptic transmitter release is a cooperative biochemical process, in which 4 - 5 calcium ions need to bind to a vesicle protein, probably synaptotagmin, in order to trigger release. This intrinsic biochemical cooperativity can be measured experimentally by varying the global cytosolic calcium concentration using caged calcium or approximately and more simply by varying extracellular calcium, which changes the calcium concentration under each open calcium channel. Another way to probe the synaptic mechanism is by varying the number of open calcium channels using channel blockers or voltage clamp. This calcium current cooperativity was shown many years ago by Zucker and Fogelson to be a good assay for the extent to which the calcium domains of individual calcium channels overlap;the cooperativity increases as channels overlap, approaching the intrinsic biochemical cooperativity as a limit. Such experiments have been used to infer how many calcium channels typically need to open in order to release a single vesicle, a number that varies with experimental conditions and among different types of synapses. However, we have found that this is a distinct, third form of cooperativity that we call calcium channel cooperativity. We sought to clarify the relationships between these three types of cooperativity, intrinsic, calcium current, and calcium channel, by considering simplified scenarios in which explicit formulas can be derived (Ref. # 1). For example, current cooperativity, which can be measured directly, generally underestimates channel cooperativity, which is often the quantity but cannot be measured, at least with current technology. Even though channel cooperativity cannot be measured, it is important to keep the two concepts straight in order to appreciate the limitations of those measures tha tare feasible.