Mathematical Modeling of Neurons and Endocrine Cells

垂体内分泌细胞与胰腺β细胞有许多相似之处，因此形成了一个有趣的群体，可以进行机制的比较研究。我们之前已经表明，一些垂体细胞（生长滋养细胞、乳养细胞、促皮质细胞）表现出与胰腺β细胞相似的爆发活动（交替沉默、低电压和尖峰、高电压状态），但时间尺度要快得多（秒vs几十秒或几分钟）。这部分是由于内质网钙储存在调节和减缓钙电振荡方面的贡献较小，部分是由于缺乏内部代谢振荡器或离子通道(如：K(ATP))可以连接电和代谢振荡器。在某些条件下(如：暴露于乙酰胆碱或从朗格汉斯胰岛的正常环境中分离单细胞)，β细胞表现出非常快的振荡，与垂体破裂非常相似，并且同样短暂。即使在这种情况下，模型表明振荡背后的数学结构可能存在差异。在经典的β细胞型爆发中，尖峰本身是快速振荡，如果钙浓度被冻结，它会持续存在，而在垂体型爆发中，尖峰的振幅较小，似乎只是短暂的，阻尼振荡，如果钙被冻结，它会接近平台电压。我们之前已经表明，这两种机制可以通过实验来区分，尝试使用短暂的去极化从静止阶段重置到活跃阶段。我们现在已经证明，这两类模型可以通过改变一个参数来统一，即钙电流激活时的电压。因此，这可能受到监管控制或实验操纵。我们在数学上进一步探讨了两类模型之间的差异，作为响应电压变化的钙变化率的函数，这反过来又取决于胞质蛋白缓冲游离钙的程度。这也可能受到监管或发展控制，并受到实验操纵。这项研究也为我们布里斯托尔的同事感兴趣的爆炸振荡的一般数学理论提出了深刻的问题。一篇论文正在审稿中。

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.