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Electrical activity and insulin release in pancreatic beta cells
Electrical Activity and Insulin Release in Pancreatic Beta Cells JOEL
KEIZER
Institute of Theoretical Dynamics and Department University of California, Davis, California 95616 Received
of Chemistv,
I June 1987: revised I9 October I987
ABSTRACT The pancreatic beta cell, which is responsible for insulin secretion from the islets of Langerhans, exhibits a complex pattern of membrane-potential oscillations called bursting. These oscillations are induced by glucose and are strongly correlated with the release of insulin. Recently a number of the ion channels which are responsible for carrying membrane currents in the beta cell have been uncovered. This has led to several hypothesized mechanisms for bursting and a good deal of mathematical modeling. In this paper the progress in understanding bursting in the beta cell is reviewed, including the impact of the discovery of the ATP-inactivated, ADP-modulated channel on modeling and the effect of single-channel
events on electrical
activity.
INTRODUCTION Of the many remarkable organs in the mammalian body, the pancreas is one of the simplest. Lying next to the stomach and below the liver, it occupies a central position in the control and regulation of digestion. The great bulk of the cells in the pancreas are acinar cells, which synthesize and secrete digestive enzymes into the pancreatic ducts for eventual use in the duodenum. Imbedded among these exocrine cells, so called because their secretion is mediated by ducts, are the islets of Langerhans. Islets make up only a few percent of the pancreas and contain three distinct kinds of cells, called alpha, beta, and delta cells. All three types of cells are involved in secretion of hormones, including somatostatin, glucagon, and insulin. Islet cells secrete directly into a veinous system that carries the hormones to the liver. For this reason they are classified as endocrine cells. It is the beta cells in the islets of Langerhans that are responsible for insulin release, and it is the electrical properties of these cells that interest us here. Microscopic examination of islets with fluorescent dyes sensitive to insulin show that beta cells make up about 75% of a typical islet, a percentage which increases to over 90% in certain strains of mice. For this MATHEMATICAL
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90:127-138
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(1988)
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reason it has been easier to study the beta cell than the other islet cells, and a great deal has been learned about their microstructure from electron microscopy [26]. A typical beta cell is 16-20 pm in diameter and in the islet is connected to neighboring cells by gap junctions. Using various types of electron-dense labels, Orci and coworkers have established that insulin is packaged in granules within the beta cell that are surrounded by a lipid bilayer. Like secretory granules in other cells, the insulin granules are synthesized by the Golgi apparatus and occupy a large fraction (about 30%) of the cell volume. In addition to the granules, the beta cell contains the usual complement of internal organelles, e.g., a nucleus, mitocondria, and endoplasmic reticulum. Release of insulin occurs when the membrane of the granule fuses with the plasma membrane of the cell, an event which has been captured in freeze-fracture photographs. Insulin release is a slow process which is triggered by numerous secretogogues, the most important physiologically being glucose. Because islets are viable in physiological solutions, it has been possible to study insulin release in excised islets [16]. These in vitro studies have revealed that insulin is released in a biphasic process lasting for an hour or more. Although less than a percent of the insulin in any given cell is released during this period, the rate of release lags behind the addition of glucose for several minutes and then reaches a pronouced maximum within four to five minutes. This is followed by a minimum in the rate of release between eight and ten minutes and a gradual increase in rate over the remainder of the hour. In this latter phase the rate of release maintains a memory of the initial phase, and if glucose is withdrawn and then readded within ten minutes, the first phase of release is not observed. Insulin release in the beta cells first came to my attention as a phenomenon in nonlinear science in 1974. During a visit that year to Davis, Illani Atwater and her husband and colleague, Fduardo Rojas, showed me their fascinating data on the electrical behavior of the beta cell. Using microelectrodes, they had confirmed the observation of Dean and Matthews [13] which showed that the membrane potential of the beta cell, when impaled in an intact islet, underwent sustained oscillations when the islets were perfused with glucose concentrations in the range of 8-17 mM. The oscillations were of the burst type and were characterized by a silent phase with a membrane potential of about - 55 mV and a depolarized active phase of rapid potential spikes varying between - 30 and - 40 mV. Since insulin release elicited by glucose occurs in the same concentration range, it appeared that the electrical activity was of physiological importance. At that time little was known about the ionic conductance of beta cells, and it seemed fruitless to me to pursue the phenomenon theoretically despite its obvious interest. By the spring of 1981 I had almost forgotten about the beta cell, when Rojas again visited Davis and talked about recent breakthroughs in the
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electrophysiology of the cell [2]. Evidence had accumulated that the beta cell carried two major ionic currents, an outward potassium current and an inward calcium current. Moreover, these two currents appeared to be coupled by a potassium conductance that was activated by internal calcium concentration. All this seemed to fit with the growing belief that secretion was controlled by calcium [38]. Thus one could form a picture of the ion channels functioning to bring calcium into the beta cell and thereby stimulating insulin release. Just how it was that such mechanistic features could lead to bursting was not so clear, although by that time is was known that the rate of insulin release and the duration of the active phase of electrical activity were strongly correlated [23]. Several months later and just shortly before Teresa Chay was scheduled for a one-week visit, I received a bundle of reprints from Atwater and Rojas. Thus when Teresa arrived and asked what we should work on, I handed her the reprints and said, “What about this?” For several months we fumbled around with a number of ideas until one day I recalled a seminar by John Rinzel [32] in which he had discussed continuous spiking in the HodgkinHuxley model [19]. If a small, constant inward current is used to depolarize the Hodgkin-Huxley model, the membrane potential exhibits continuous action-potential spikes. It seemed that the beta cell might act in a related fashion, namely, the outward current of potassium might be reduced by a decrease of calcium (induced by uptake of calcium into the intracellular stores) lowering the calcium-activated potassium conductance. This would function like the depolarizing current in Rinzel’s constant spiking model, ultimately bringing the membrane potential into a voltage range which supported action potential spikes. Unlike the Hodgkin-Huxley model, the inward current would be calcium, so that after a suitable interval enough calcium might accmulate to reactivate the calcium-activated potassium conductance and thus hyperpolarize the cell into the silent phase. As we later discovered, my colleague at Davis, Richard Plant, had used related ideas to explain bursting in Aplysia neurons. Plant’s work [29, 301 turned out to be most helpful in converting these ideas into a quantitative model. THE MINIMAL
MODEL
To translate the potassium and calcium conductances into a mathematical model we used the standard device [31] of treating ionic channels as elements of electrical resistance in parallel with the capacitance of the plasma membrane. Using Kirchoff’s laws, this leads to the following equation for the time rate of change of the membrane potential, defined as V = &, - $0,,t:
dV c,=-I,,
(1)
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JOEL KEIZER
where 1s is the resistive current, which can be expressed as a sum of currents from the various types of channels as
with g, the conductance of the ith kind of channel potential, given by the expression
and r/; the Nemst
v=~,n[ilOUt ’
nF
[iii,
’
with [i] either the outside or the inside concentration of the ion transported by the channel. Written this way, the conductances might depend on the voltage or other variables, such as the internal calcium concentration, c = [Cali,, or state variables for the channels. Equations (l)-(3) provide a general deterministic description of the time rate of change of the membrane potential. They need to be supplemented with specific expressions for conductances as well as differential equations describing the time course of other variables, e.g. the calcium concentration. The first model of this sort, developed by Chay and me [8], was dubbed the minimal model, since it focused on explaining only the electrical behavior that developed in the second phase of insulin release (so-called steady state bursting). The model is also minimal in that only those channels which were suspected [2] at that time to be involved in the regulation of the beta cell were used. These were (1) voltage-sensitive potassium channels, (2) voltage-sensitive calcium channels, and (3) calcium-activated potassium channels. At that time the mechanisms by which internal calcium was handled in the beta cell were largely unknown, although evidence suggested that glucose stimulated uptake of calcium into organelles, especially the mitochondria [17]. To complete the minimal model we assumed that the potassium and calcium channels had kinetics comparable to the HodgkinHuxley channels and thus wrote for their conductances [19] g,,,
=
&,vn4y
k,,
.m3h.
v = ik,
The variables n, m, and h are channel state variables equations suggested by Hodgkin and Huxley, e.g., dm -z=where both the characteristic on voltage. The characteristic
which satisfy linear
m-mm(V) cl(O
’
time r,,, and the steady-state value mm depend times are the order of l-10 ms. The steady-state
ELECTRICAL
ACTIVITY IN PANCREATIC BETA CELLS
131
values of m and n increase with increasing voltage and are referred to as activation terms, while h decreases with increasing voltage and is called an inactivafion. The calcium-activated potassium channels, which serve as the switch between the silent and spiking phases of the bursts, were assumed to have a conductance that depended on the binding of calcium of the form [301 _ gK,Ca= gK,Ca1
c/K +
c/K
with K = 1 PM, although power-law dependences of the same form worked just as well. If R is the radius of the beta cell and F is Faraday’s constant, then the time rate of change of internal calcium can be written dc f -Ia
-_ % = 8?rR3F
kcac,
where the factor f represents the fraction of calcium that is free in the cytosol. The rate constant kc, determines the rate of loss of calcium from the cytoplasm and is supposed to be an increasing function of the concentration of glucose. Thus the final term in Equation (6) represents the uptake of calcium into internal organelles and its loss via ion pumps through the plasma membrane. Using judicious choices of parameter values, which nonetheless fall within the physiological domain, the minimal model exhibits a rich nonlinear dynamics. Setting the temperature in the Hodgkin-Huxley model to 20°C and the glucose sensitive parameter kc, to 0.04 mss’ produces burst oscillations in numerical solutions of the differential equations that have the correct period (about 15 s) and a silent and active phase that closely resemble experimental measurements [8]. Furthermore, the effect of decreasing or mcreasmg kc,, i.e., decreasing or increasing glucose, has the correct physiological effect, namely, to lengthen or shorten the silent phase [3]. In fact, if kc, is increased to 0.06 mss’, bursting no longer occurs and only continuous spiking remains, just as is seen experimentally at concentrations above 20 mM glucose. Because internal calcium is an explicit variable in the minimal model, the time course of the internal calcium during the bursts can be followed. According to Equation (7), the parameter f provides a natural time scale for the calcium, and it turns out that on this time scale calcium is a slow variable whose values range from about 0.3 to 1.3 pM during the course of a period [8]. During the spiking phase one can see small, but regular ripples in the calcium concentration corresponding to the inward current of the voltagesensitive calcium channels during a spike. As yet, know one has been able to measure oscillating calcium concentrations in islets, which is complicated, even with the newest fluorescent dyes, by the presence of the islet matrix.
132
JOEL KEIZER
Using the fact that calcium varies on a much slower time scale than the voltage, Rinzel [33] has developed an elegant mathematical picture of how the minimal model works. His two-variable reduction of the minimal model consists in setting the activation and inactivation terms in Equation (5) equal to their steady-state values. Thus only the voltage and calcium concentration remain as independent variables satisfying
&
=
%ka,“mm3hm 8aR3F
(ka-
V-&c.
Since calcium is a slow variable, one can treat it as a parameter in the voltage equation in (8) and solve that equation as a function of the voltage alone. Doing this at low calcium, one finds a single stable steady state for V in the range of - 30 mV, while at high calcium there is a single stable steady state at about - 55 mV. For intermediate calcium concentations three steady states are possible, the one at intermediate voltages being unstable, In Rinzel’s explanation, the oscillations correspond to this region of bistability when the value of k,, is adjusted so that the null cline of the calcium equation intersects the unstable state. This gives rise to a limit cycle in which the calcium concentration slowly increases as it traverses the depolarized steady states (now only pseudosteady) of the voltage equation near - 30 mV until it reaches the critical point of that curve, at which point the voltage rapidly jumps to the hyperpolarized pseudosteady states near -55 mV. Because of the shape of the calcium null cline, the calcium now decreases on this branch of voltage steady states until the lower critical point is achieved and the voltage rapidly returns to the manifold of depolarized pseudosteady states. The bifurcation diagram for the voltage at fixed calcium is greatly enriched if the activation n is reincluded as an independent variable (Chay’s three-variable model [5]) or if the full minimal model is used. In this case one finds that the depolarized steady states near the lower critical point are no longer stable but are now surrounded by a stable limit cycle with an amplitude of about 15 mV and a period of about 20 ms. These are the action-potential-like spikes seen in the bursting of the beta cell and correspond to the fast dynamics of the voltage. The silent phase, on the other hand, corresponds to the hyperpolarized steady states, which are triggered when the calcium concentration is sufficient to destabilize the spiking by causing an increase in the potassium conductance. When k,, is increased sufficiently, the calcium null cline then intersects the manifold of depolarized states, which are still not stable. This establishes a maximum value
ELECTRICAL
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BETA CELLS
133
beyond which calcium cannot increase and leads to continuous spiking such as is observed experimentally. The minimal model has been used to describe other experiments [9, 61, including oscillations in potassium and calcium which have been measured with ion specific electrodes in the extracellular space around the beta cell [27, 281. By including these extracellular concentrations as variables, the local increase in external potassium (due to the outward current) and the local decrease in external calcium (due to the inward current) have been shown to agree with experiment both in magnitude and in phase relationships with the voltage oscillation. Finally, on adjusting the temperature to 17-18°C the minimal model exhibits period doubling bifurcations and deterministic chaos when either of the parameters gk,c. or k,, is adjusted appropriately [5, lo]. It is thought that this behavior may be related to irregular spiking observed in some collections of laboratory animals [20].
VARIANTS
OF THE MINIMAL
MODELS
In the past two years experimental evidence has come to light suggesting that channels other than those which appear in the minimal model may be important in the electrical activity in the beta cell. Judicious use of the patch-clamp technique, using either whole cells or isolated patches, has uncovered a third potassium channel in the beta cell [l, 11, 35, 241. This channel is inactivated by ATP, which in the absence of ADP binds to the channel so tightly that at physiological concentrations it blocks these channels completely. Himmel and Chay [18] have modeled this behavior, and their calculations, along with the 55-pS conductance and high density of these channels, have given rise to the idea that they are responsible for setting the resting potential of the beta cell in the absence of glucose [34]. Careful work by Rorsman and Trube [36] has provided a characterization of the voltage-activated potassium and calcium channels in the beta cell. Using their work, it has been possible to construct an accurate model of these channels, which no longer relies on the Hodgkin-Huxley model [37]. When the parameters,appropriate to these channels are used in the minimal model, the characteristic burst oscillations are still observed. Several groups have found high-conductance calcium-activated potassium channels in the beta cell [12, 151. These channels are strongly voltage-dependent but are almost completely inactivated in the voltage region of the oscillations. The relevance of these channels for busting cannot be completely ruled out, however, since they are present in great abundance (perhaps 200-500 per cell) and have a conductivity of 150-250 pS. Thus the presence of even a single open channel per cell would affect the potassium conductance significantly. We will return to the stochastic aspects of this situation in the final section of this paper.
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JOEL KEIZER
Chay has recently investigated a variant of the minimal model involving only two kinds of ion channels [6]. This model relies on calcium-activated potassium channels as the sole source of potassium current and voltageactivated calcium channels for the calcium current. The parameters for the calcium-activated potassium channel are based on patch-clamp experiments [15], and for a range of the other parameter values in the model, Chay finds reasonable burst oscillations. However, the neglect of the voltage-activated potassium channels found by Rorsman and Trube [36], which implies that the calcium-activated potassium channels carry all the outward current, is difficult to justify, and it seems best to consider this model as an illustration of the robustness of bursting to changes in the description of the currents. Because the calcium-activated potassium channels appear to be so inactive at physiological voltages, it has been important to explore other mechanisms which might lead to bursting. Using a model which is based on the work of Rorsman and Trube [36], Chay [7] has obtained bursting using only voltage-activated potassium and calcium channels. To achieve this it was necessary to assume that the calcium current is inactivated by internal calcium. Thus instead of activating an outward current, internal calcium is assigned the role of inactivating an inward current. While this model is different from the minimal model in important ways, it still utilizes internal calcium to regulate its own uptake. It is, of course, possible that calcium activates a potassium current indirectly. Recent observations of the competitive effect of ADP on the ATP-inactivated potassium channel suggest a possible mechanism for this [39]. ADP has been shown to shift the inactivation curve for ATP to higher concentrations just as it would if ADP competed for the same binding site without blocking the channel. The shift is significant enough that by modulation of the ADP concentration in the physiological range the cell should be able to modulate the outward current through these channels, If increased calcium increases the ADP levels in the cell, then calcium could indirectly activate these channels. One possible mechanism for this involves the uptake of calcium by the mitochondria, which is known to interfere with oxidative phosphorylation of ADP [25]. Since ATP is constantly being converted to ADP by celluar metabolism, an increase in internal calcium, therefore, would have the desired effect. We have examined this possibility [22] by eliminating the calciumactivated potassium current from the minimal model and substituting in its place an ATP-inactivated, ADP-modulated channel with a conductance of the form qK,AT’P=
&ATP1 +
1-k D/K, D/K,
+
(A
_ D)/K,
’
ELECTRICAL
ACTIVITY IN PANCREATIC BETA CELLS
135
where D represents the concentration of ADP and A represents the total concentration of ATP and ADP, which is assumed fixed. Calcium is assumed to affect the ADP by the mitochondrial mechanism given above, and the concentration of ADP satisfies the following differential equation: dD -z- --k,D+k(A-D), with the rate parameter
(10)
having the form k,=kexp{a[l$-V,(c)]}.
(11)
The exponential factor represents the interference of calcium on the rate of oxidative phosphorylation. In this model ADP will be a second slow variable, dynamically coupled to calcium, unless the constant k in Equations (10) and (11) is sufficiently large. In that case ADP is slavishly tied to the calcium concentration through the condition dD/dt = 0, i.e.,
(14 When this condition holds, the conductance in Equation (9) is really just a function of the calcium concentration, as it is in the minimal model. Thus it may not be surprising that even when k is as small as 0.1 s- ‘, so that the calcium and ADP are not tightly coupled, reasonable choices of the other parameter values in this model give rise to burst oscillations [22]. CRITIQUE
OF PRESENT
MODELS
OF THE BETA CELL
Despite the fact that the minimal model and its variants give rise to burst oscillations comparable to those measured in intact islets, none of the models discussed above provide an accurate description of the electrical behavior of an isolated beta cell. Indeed, the electrical activity of isolated cells [36] is stimulated by glucose, but the time course is quite irregular and appears to be dominated by the openings and closings of individual channels [4]. On the other hand, the minimal model does do a good job of describing the properties of an individual beta cell clustered together with other cells, either in an islet or in a petri dish. In fact, beta cells are known to be in close electrical contact [14] with their neighbors and to communicate intercellularly via gap junctions [21]. How important this coupling is to the functioning of the islet is not presently known, despite the fact that coupling radically modifies the electrical properties of a single cell. Since present models of the beta cell work well for cells in islets, it has been generally assumed that these models represent the average behavior of
136
JOEL KEIZER
a single cell in a tightly coupled cluster. One advantage of tight coupling is that channels are shared among the larger aggregate of cells, so that the current caused by the opening of a single channel on one cell can be shared by all the cells. If the cluster is large enough, then the effect of single-channel openings will not be apparent and the cluster will act determinstically. This would not be the case for a single cell or even a small cluster, especially if the channels had large conductances and only a small number of channels were open at a time. Recently Sherman, Rinzel, and I [37] have used this idea in an attempt to explain the irregular electrical behavior of single cells and small clusters. For simplicity we have used the minimal model, with the improved RorsmanTrube [36] description of the voltage-activated potassium and calcium channels. Since there are several hundred of these channels per cell and their open probabilities are large, these channels were treated deterministically using their average conductances. The calcium-activated potassium channels, however, are seldom open despite their large numbers [15]. Thus the stochastic openings and closings of these channels, either in a single cell or in tightly coupled clusters of cells, were used to calculate the instantaneous values of the calcium-activated potassium current. Numerical calculations using this procedure produce regular burst patterns only when the number of cells in the cluster is sufficiently large. For small clusters or single cells the electrical activity is quite irregular, and one often sees incipient bursts with no spikes and bursts of spikes of varying duration that are reminiscent of what one finds experimentally. While the minimal model of electrical activity in the beta cell has proved useful for understanding many aspects of bursting, it is not yet clear whether the mechanism of direct activation of a potassium current by internal calcium is actually important in the beta cell. Certainly, more detailed information about the ion channels in the beta cell will be required to convincingly establish the molecular mechanism by which calcium regulates bursting. Nonetheless, it seems likely that calcium regulates its own entry into the beta cell, either directly or indirectly, by activation of a potassium current. How calcium entry stimulates insulin release and how these processes are related to other mechanisms of calcium handling in the beta cell remain much bigger mysteries, which will serve as stating points for future work.
This work was supported by NSF grant CHE 8618647, a John Simon Guggenheim Memorial Fellowship to the author, and a sabbatical leave from the University of California. It is a pleasure to acknowledge many fruitful conversations with John Rinzel, Artie Sherman, and Illani Atwater at the NIH during the past year and with Teresa Chay over the past six years. I especially want to thank John Rinzel and Peter Mazur for their hospitality at the Mathematical
ELECTRICAL
ACTIVITY
IN PANCREATIC
Research Branch of the NIH LAden during my sabbatical.
137
BETA CELLS
and the Instituut-Lorentz
of the University of
REFERENCES 1
2
F. M. Ashcroft, single potassium (1984). I. Atwater, oscillatory
C. M. Dawson, A. Scott, G. Eddlestone, and E. Rojas, The nature of the behavior in electrical activity for the pancreatic beta cell. in Biochemisfq
and Bmphysics 3
4
D. E. Harrison, and S. J. H. Ashcroft, Glucose induces closure of a channel in isolated rat pancreatic beta cells, Nature 312:446-448
of the Pancreatic
Bera Cell, Georg
Thieme,
New
pp. 353-362. J. A. Bangham, activity,
P. A. Smith,
in Biophysics
and P. C. Croghan,
of Pancreutic
5
Eds.), 1986, pp. 265-278. T. Chay, Chaos in a three-variable
6
(1985). T. Chay,
Modelling
Beta Cells (I. Atwater, model
On the effect of intracellular
of an excitable
calcium-sensitive
cell, Phys. potassium
I 8
T. Chay
9
beta cell, Biophys. J. 42:181-190 (1983). T. Chay and J. Keizer, Theory of the effect of extracellular
and J. Keizer,
in the pancreatic
Minimal
model
T. Chay and J. Rinzel, Bursting,
11
Biophys. J. 471357-366 (1985). D. L. Cook and C. N. Hales, Intracellular
14 15
16 17
for membrane
beta cell, Biophys. J. 48:815-827
10
13
1980,
pp.
the beta cell electrical E. Rojas,
bursting pancreatic beta cell, Biophys. J. 50:756-777 (1986). T. Chay, The effect of inactivation of calcium channels by intracellular the bursting pancreatic beta cells, Cc// Biphys., Dec. 1987.
12
York,
100-107. I. Atwater and J. Rinzel, The beta cell bursting pattern and intracellular calcium, in Ionic Channels in Cells and Model Systems (R. Latorre, Ed.), Plenum, New York, 1986.
beating,
oscillations
and B. Soria, D 16:233-242 channels
in the
calcium
ions in
in the pancreatic
potassium
on oscillations
(1985).
and chaos in an excitable ATP directly
membrane
blocks potassium
model,
channels
in
pancreatic beta cells, Nature 311:271-273 (1984). D. L. Cook, M. Ikeuchi, and W. Y. Fujimoto, Lowering of pH inhibits calciumactivated potassium channels in isolated rat pancreatic islet cells, Nature 311:269-271 (1984). P. M. Dean and E. K. Matthews, Glucose-induced electrical activity in pancreatic islet cells, J. Physiol. (London) 210:255-264 (1970). G. T. Eddlestone, A. Goncalves, J. A. Bangham, and E. Rojas, Electrical coupling between cells in islets of Langerhans from mouse, J. Membrane Biol. 77:1-S (1984). I. Findlay, M. J. Dunne, and 0. H. Petersen, High conductance potassium channel in pancreatic islet cells can be activated and inactivated by internal calcium, J. Membrane Biol. 83:169-175 (1985). C. J. Hedeskov, Mechanism of glucose-induced insulin release, Physiol. Reu. 60:442-509 (1980). B. Hellman, H. Abrahamson, T. Anderson, P.-O. Berggren, P. Flatt, E. Gylfe, and H.-J. Hahn, Calcium movements in relation to glucose-stimulated isulin release, in Biochemistry und Biophysics of the Pancreatic Beta Cell, Georg Thieme, New York, 1980, pp. 122-130.
138 18 19
20
JOEL KEIZER
D. Himmel and T. Chay, Theoretical studies on the electrical activity of pancreatic beta cells as a function of glucose, Biophys. J. 51:89-107 (1987). A. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London) 177:500-544 (1952).
22
P. Lebrun pancreatic P. Meda, topography Quart. J. G. Magnus
23
H. P. Meissner,
21
Phvsiol.
24
and I. Atwater, Chaotic and irregular bursting electrical activity in mouse beta cells, Biophys. J. 48:529-531 (1985). I. Atwater, A. Goncalves, A. Bangham, L. Orci, and E. Rojas, The of electrical synchrony among beta cells in the mouse islet of Langerhans, Exp.
(Paris)
Electrical
(1984).
characteristics
12:151-761
S. Misler,
L. C. Falke,
potassium
channel
83:7119-7123
25
Physiol. 69:719-735
and J. Keizer, in preparation.
K. Gillis,
in
of the beta
cells in pancreatic
J.
islets,
(1976).
rat
and
pancreatic
M. L. McDaniel, B cells,
Proc.
A metabolite-regulated Nat.
Acad.
Sci.
U.S. A.
(1986).
D. Nicholls,
Some recent
advances
in mitochondrial
calcium
Trends
transport,
Biol.
Sci. 6:36-38
26 27
(1981). L. Orci, A portrait of the pancreatic B cell, Diabetologia 10:163-187 (1974). E. M. Perez-Armendariz, I. Atwater, and E. Rojas, Glucose induced oscillatory changes in extracellular potassium concentration in mouse islets of Langerhans, Biophys.
J. 48:741-749
(1985).
29
E. M. Perez-Armendariz and I. Atwater, Glucose invoked changes in [K+ ] and [Ca2+ ] in intercellular spaces of the mouse islet of Langerhans, in Biophysics of Pancreutic Betu C&s (I. Atwater, E. Rojas, and B. Soria, Eds.), 1986, pp. 31-51. R. Plant and M. Kim, Mathematical description of a bursting pacemaker neuron by a
30
modification R. E. Plant,
28
of the Hodgkin-Huxley equations, Biophys. J. 16:227-244 The effects of calcium on bursting neurons, Biophys.
(1976). J.
21:217-237
(1978). Bioelectric
Phenomenu,
31
R. Plonsey,
32 33
J. Rinzel, On repetitive activity J. Rinzel, Bursting oscillations Parflu/
34
Differential
36
New York, 1969.
(B. D. Sleeman
and R. J. Jarvis,
Ids.)
Springer,
and
New
York, 1985, pp. 304-316. J. Rinzel, T. R. Chay, D. Himmel, and I. Atwater, Prediction of the glucose-induced changes in membrane ionic permeability and cytosolic calcium by mathematical modeling,
35
Equations
McGraw-Hill,
in nerve, Fed. Proc. 37:2793-2802 (1978). in an excitable membrane model, in Ordinmy
in Biophysics of Pancreatic
Betu Cells (I. Atwater,
E. Rojas,
and B. Soria,
Eds.), 1986, pp. 241-263. P. Rorsman and G. Trube, Glucose dependent potassium channels in pancreatic beta cells are regulated by intracellular ATP, Pf/i&ers Arch. 405:305-309 (1985). P. Rorsman and G. Trube, Calcium and delayed potassium currents in mouse pancreatic beta cells under voltage-clamp conditions, J. Physiof. (London) 374:531-550 (1986).
31 38
A. Sherman, J. Rinzel, and J. Keizer, Emergence of organized bursting in clusters of pancreatic P-cells by channel sharing, Biophys. J., to appear. C. B. Wollheim and G. W. G. Sharp, Regulation of insulin release by calcium, Physiol. Rev. 61:914-974
39
(1981).
M. Kakei, P. Kelly, S. J. H. Ashcroft, and F. M. Ashcroft, The ATP-sensitivity of K+ channels in rat pancreatic p-cells is modulated by ADP, Fed. Eur. Biochem. Sot. 208:63-66
(1986).
KEIZER
Institute of Theoretical Dynamics and Department University of California, Davis, California 95616 Received
of Chemistv,
I June 1987: revised I9 October I987
ABSTRACT The pancreatic beta cell, which is responsible for insulin secretion from the islets of Langerhans, exhibits a complex pattern of membrane-potential oscillations called bursting. These oscillations are induced by glucose and are strongly correlated with the release of insulin. Recently a number of the ion channels which are responsible for carrying membrane currents in the beta cell have been uncovered. This has led to several hypothesized mechanisms for bursting and a good deal of mathematical modeling. In this paper the progress in understanding bursting in the beta cell is reviewed, including the impact of the discovery of the ATP-inactivated, ADP-modulated channel on modeling and the effect of single-channel
events on electrical
activity.
INTRODUCTION Of the many remarkable organs in the mammalian body, the pancreas is one of the simplest. Lying next to the stomach and below the liver, it occupies a central position in the control and regulation of digestion. The great bulk of the cells in the pancreas are acinar cells, which synthesize and secrete digestive enzymes into the pancreatic ducts for eventual use in the duodenum. Imbedded among these exocrine cells, so called because their secretion is mediated by ducts, are the islets of Langerhans. Islets make up only a few percent of the pancreas and contain three distinct kinds of cells, called alpha, beta, and delta cells. All three types of cells are involved in secretion of hormones, including somatostatin, glucagon, and insulin. Islet cells secrete directly into a veinous system that carries the hormones to the liver. For this reason they are classified as endocrine cells. It is the beta cells in the islets of Langerhans that are responsible for insulin release, and it is the electrical properties of these cells that interest us here. Microscopic examination of islets with fluorescent dyes sensitive to insulin show that beta cells make up about 75% of a typical islet, a percentage which increases to over 90% in certain strains of mice. For this MATHEMATICAL
BIOSCIENCES
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(1988)
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reason it has been easier to study the beta cell than the other islet cells, and a great deal has been learned about their microstructure from electron microscopy [26]. A typical beta cell is 16-20 pm in diameter and in the islet is connected to neighboring cells by gap junctions. Using various types of electron-dense labels, Orci and coworkers have established that insulin is packaged in granules within the beta cell that are surrounded by a lipid bilayer. Like secretory granules in other cells, the insulin granules are synthesized by the Golgi apparatus and occupy a large fraction (about 30%) of the cell volume. In addition to the granules, the beta cell contains the usual complement of internal organelles, e.g., a nucleus, mitocondria, and endoplasmic reticulum. Release of insulin occurs when the membrane of the granule fuses with the plasma membrane of the cell, an event which has been captured in freeze-fracture photographs. Insulin release is a slow process which is triggered by numerous secretogogues, the most important physiologically being glucose. Because islets are viable in physiological solutions, it has been possible to study insulin release in excised islets [16]. These in vitro studies have revealed that insulin is released in a biphasic process lasting for an hour or more. Although less than a percent of the insulin in any given cell is released during this period, the rate of release lags behind the addition of glucose for several minutes and then reaches a pronouced maximum within four to five minutes. This is followed by a minimum in the rate of release between eight and ten minutes and a gradual increase in rate over the remainder of the hour. In this latter phase the rate of release maintains a memory of the initial phase, and if glucose is withdrawn and then readded within ten minutes, the first phase of release is not observed. Insulin release in the beta cells first came to my attention as a phenomenon in nonlinear science in 1974. During a visit that year to Davis, Illani Atwater and her husband and colleague, Fduardo Rojas, showed me their fascinating data on the electrical behavior of the beta cell. Using microelectrodes, they had confirmed the observation of Dean and Matthews [13] which showed that the membrane potential of the beta cell, when impaled in an intact islet, underwent sustained oscillations when the islets were perfused with glucose concentrations in the range of 8-17 mM. The oscillations were of the burst type and were characterized by a silent phase with a membrane potential of about - 55 mV and a depolarized active phase of rapid potential spikes varying between - 30 and - 40 mV. Since insulin release elicited by glucose occurs in the same concentration range, it appeared that the electrical activity was of physiological importance. At that time little was known about the ionic conductance of beta cells, and it seemed fruitless to me to pursue the phenomenon theoretically despite its obvious interest. By the spring of 1981 I had almost forgotten about the beta cell, when Rojas again visited Davis and talked about recent breakthroughs in the
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electrophysiology of the cell [2]. Evidence had accumulated that the beta cell carried two major ionic currents, an outward potassium current and an inward calcium current. Moreover, these two currents appeared to be coupled by a potassium conductance that was activated by internal calcium concentration. All this seemed to fit with the growing belief that secretion was controlled by calcium [38]. Thus one could form a picture of the ion channels functioning to bring calcium into the beta cell and thereby stimulating insulin release. Just how it was that such mechanistic features could lead to bursting was not so clear, although by that time is was known that the rate of insulin release and the duration of the active phase of electrical activity were strongly correlated [23]. Several months later and just shortly before Teresa Chay was scheduled for a one-week visit, I received a bundle of reprints from Atwater and Rojas. Thus when Teresa arrived and asked what we should work on, I handed her the reprints and said, “What about this?” For several months we fumbled around with a number of ideas until one day I recalled a seminar by John Rinzel [32] in which he had discussed continuous spiking in the HodgkinHuxley model [19]. If a small, constant inward current is used to depolarize the Hodgkin-Huxley model, the membrane potential exhibits continuous action-potential spikes. It seemed that the beta cell might act in a related fashion, namely, the outward current of potassium might be reduced by a decrease of calcium (induced by uptake of calcium into the intracellular stores) lowering the calcium-activated potassium conductance. This would function like the depolarizing current in Rinzel’s constant spiking model, ultimately bringing the membrane potential into a voltage range which supported action potential spikes. Unlike the Hodgkin-Huxley model, the inward current would be calcium, so that after a suitable interval enough calcium might accmulate to reactivate the calcium-activated potassium conductance and thus hyperpolarize the cell into the silent phase. As we later discovered, my colleague at Davis, Richard Plant, had used related ideas to explain bursting in Aplysia neurons. Plant’s work [29, 301 turned out to be most helpful in converting these ideas into a quantitative model. THE MINIMAL
MODEL
To translate the potassium and calcium conductances into a mathematical model we used the standard device [31] of treating ionic channels as elements of electrical resistance in parallel with the capacitance of the plasma membrane. Using Kirchoff’s laws, this leads to the following equation for the time rate of change of the membrane potential, defined as V = &, - $0,,t:
dV c,=-I,,
(1)
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JOEL KEIZER
where 1s is the resistive current, which can be expressed as a sum of currents from the various types of channels as
with g, the conductance of the ith kind of channel potential, given by the expression
and r/; the Nemst
v=~,n[ilOUt ’
nF
[iii,
’
with [i] either the outside or the inside concentration of the ion transported by the channel. Written this way, the conductances might depend on the voltage or other variables, such as the internal calcium concentration, c = [Cali,, or state variables for the channels. Equations (l)-(3) provide a general deterministic description of the time rate of change of the membrane potential. They need to be supplemented with specific expressions for conductances as well as differential equations describing the time course of other variables, e.g. the calcium concentration. The first model of this sort, developed by Chay and me [8], was dubbed the minimal model, since it focused on explaining only the electrical behavior that developed in the second phase of insulin release (so-called steady state bursting). The model is also minimal in that only those channels which were suspected [2] at that time to be involved in the regulation of the beta cell were used. These were (1) voltage-sensitive potassium channels, (2) voltage-sensitive calcium channels, and (3) calcium-activated potassium channels. At that time the mechanisms by which internal calcium was handled in the beta cell were largely unknown, although evidence suggested that glucose stimulated uptake of calcium into organelles, especially the mitochondria [17]. To complete the minimal model we assumed that the potassium and calcium channels had kinetics comparable to the HodgkinHuxley channels and thus wrote for their conductances [19] g,,,
=
&,vn4y
k,,
.m3h.
v = ik,
The variables n, m, and h are channel state variables equations suggested by Hodgkin and Huxley, e.g., dm -z=where both the characteristic on voltage. The characteristic
which satisfy linear
m-mm(V) cl(O
’
time r,,, and the steady-state value mm depend times are the order of l-10 ms. The steady-state
ELECTRICAL
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131
values of m and n increase with increasing voltage and are referred to as activation terms, while h decreases with increasing voltage and is called an inactivafion. The calcium-activated potassium channels, which serve as the switch between the silent and spiking phases of the bursts, were assumed to have a conductance that depended on the binding of calcium of the form [301 _ gK,Ca= gK,Ca1
c/K +
c/K
with K = 1 PM, although power-law dependences of the same form worked just as well. If R is the radius of the beta cell and F is Faraday’s constant, then the time rate of change of internal calcium can be written dc f -Ia
-_ % = 8?rR3F
kcac,
where the factor f represents the fraction of calcium that is free in the cytosol. The rate constant kc, determines the rate of loss of calcium from the cytoplasm and is supposed to be an increasing function of the concentration of glucose. Thus the final term in Equation (6) represents the uptake of calcium into internal organelles and its loss via ion pumps through the plasma membrane. Using judicious choices of parameter values, which nonetheless fall within the physiological domain, the minimal model exhibits a rich nonlinear dynamics. Setting the temperature in the Hodgkin-Huxley model to 20°C and the glucose sensitive parameter kc, to 0.04 mss’ produces burst oscillations in numerical solutions of the differential equations that have the correct period (about 15 s) and a silent and active phase that closely resemble experimental measurements [8]. Furthermore, the effect of decreasing or mcreasmg kc,, i.e., decreasing or increasing glucose, has the correct physiological effect, namely, to lengthen or shorten the silent phase [3]. In fact, if kc, is increased to 0.06 mss’, bursting no longer occurs and only continuous spiking remains, just as is seen experimentally at concentrations above 20 mM glucose. Because internal calcium is an explicit variable in the minimal model, the time course of the internal calcium during the bursts can be followed. According to Equation (7), the parameter f provides a natural time scale for the calcium, and it turns out that on this time scale calcium is a slow variable whose values range from about 0.3 to 1.3 pM during the course of a period [8]. During the spiking phase one can see small, but regular ripples in the calcium concentration corresponding to the inward current of the voltagesensitive calcium channels during a spike. As yet, know one has been able to measure oscillating calcium concentrations in islets, which is complicated, even with the newest fluorescent dyes, by the presence of the islet matrix.
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Using the fact that calcium varies on a much slower time scale than the voltage, Rinzel [33] has developed an elegant mathematical picture of how the minimal model works. His two-variable reduction of the minimal model consists in setting the activation and inactivation terms in Equation (5) equal to their steady-state values. Thus only the voltage and calcium concentration remain as independent variables satisfying
&
=
%ka,“mm3hm 8aR3F
(ka-
V-&c.
Since calcium is a slow variable, one can treat it as a parameter in the voltage equation in (8) and solve that equation as a function of the voltage alone. Doing this at low calcium, one finds a single stable steady state for V in the range of - 30 mV, while at high calcium there is a single stable steady state at about - 55 mV. For intermediate calcium concentations three steady states are possible, the one at intermediate voltages being unstable, In Rinzel’s explanation, the oscillations correspond to this region of bistability when the value of k,, is adjusted so that the null cline of the calcium equation intersects the unstable state. This gives rise to a limit cycle in which the calcium concentration slowly increases as it traverses the depolarized steady states (now only pseudosteady) of the voltage equation near - 30 mV until it reaches the critical point of that curve, at which point the voltage rapidly jumps to the hyperpolarized pseudosteady states near -55 mV. Because of the shape of the calcium null cline, the calcium now decreases on this branch of voltage steady states until the lower critical point is achieved and the voltage rapidly returns to the manifold of depolarized pseudosteady states. The bifurcation diagram for the voltage at fixed calcium is greatly enriched if the activation n is reincluded as an independent variable (Chay’s three-variable model [5]) or if the full minimal model is used. In this case one finds that the depolarized steady states near the lower critical point are no longer stable but are now surrounded by a stable limit cycle with an amplitude of about 15 mV and a period of about 20 ms. These are the action-potential-like spikes seen in the bursting of the beta cell and correspond to the fast dynamics of the voltage. The silent phase, on the other hand, corresponds to the hyperpolarized steady states, which are triggered when the calcium concentration is sufficient to destabilize the spiking by causing an increase in the potassium conductance. When k,, is increased sufficiently, the calcium null cline then intersects the manifold of depolarized states, which are still not stable. This establishes a maximum value
ELECTRICAL
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BETA CELLS
133
beyond which calcium cannot increase and leads to continuous spiking such as is observed experimentally. The minimal model has been used to describe other experiments [9, 61, including oscillations in potassium and calcium which have been measured with ion specific electrodes in the extracellular space around the beta cell [27, 281. By including these extracellular concentrations as variables, the local increase in external potassium (due to the outward current) and the local decrease in external calcium (due to the inward current) have been shown to agree with experiment both in magnitude and in phase relationships with the voltage oscillation. Finally, on adjusting the temperature to 17-18°C the minimal model exhibits period doubling bifurcations and deterministic chaos when either of the parameters gk,c. or k,, is adjusted appropriately [5, lo]. It is thought that this behavior may be related to irregular spiking observed in some collections of laboratory animals [20].
VARIANTS
OF THE MINIMAL
MODELS
In the past two years experimental evidence has come to light suggesting that channels other than those which appear in the minimal model may be important in the electrical activity in the beta cell. Judicious use of the patch-clamp technique, using either whole cells or isolated patches, has uncovered a third potassium channel in the beta cell [l, 11, 35, 241. This channel is inactivated by ATP, which in the absence of ADP binds to the channel so tightly that at physiological concentrations it blocks these channels completely. Himmel and Chay [18] have modeled this behavior, and their calculations, along with the 55-pS conductance and high density of these channels, have given rise to the idea that they are responsible for setting the resting potential of the beta cell in the absence of glucose [34]. Careful work by Rorsman and Trube [36] has provided a characterization of the voltage-activated potassium and calcium channels in the beta cell. Using their work, it has been possible to construct an accurate model of these channels, which no longer relies on the Hodgkin-Huxley model [37]. When the parameters,appropriate to these channels are used in the minimal model, the characteristic burst oscillations are still observed. Several groups have found high-conductance calcium-activated potassium channels in the beta cell [12, 151. These channels are strongly voltage-dependent but are almost completely inactivated in the voltage region of the oscillations. The relevance of these channels for busting cannot be completely ruled out, however, since they are present in great abundance (perhaps 200-500 per cell) and have a conductivity of 150-250 pS. Thus the presence of even a single open channel per cell would affect the potassium conductance significantly. We will return to the stochastic aspects of this situation in the final section of this paper.
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JOEL KEIZER
Chay has recently investigated a variant of the minimal model involving only two kinds of ion channels [6]. This model relies on calcium-activated potassium channels as the sole source of potassium current and voltageactivated calcium channels for the calcium current. The parameters for the calcium-activated potassium channel are based on patch-clamp experiments [15], and for a range of the other parameter values in the model, Chay finds reasonable burst oscillations. However, the neglect of the voltage-activated potassium channels found by Rorsman and Trube [36], which implies that the calcium-activated potassium channels carry all the outward current, is difficult to justify, and it seems best to consider this model as an illustration of the robustness of bursting to changes in the description of the currents. Because the calcium-activated potassium channels appear to be so inactive at physiological voltages, it has been important to explore other mechanisms which might lead to bursting. Using a model which is based on the work of Rorsman and Trube [36], Chay [7] has obtained bursting using only voltage-activated potassium and calcium channels. To achieve this it was necessary to assume that the calcium current is inactivated by internal calcium. Thus instead of activating an outward current, internal calcium is assigned the role of inactivating an inward current. While this model is different from the minimal model in important ways, it still utilizes internal calcium to regulate its own uptake. It is, of course, possible that calcium activates a potassium current indirectly. Recent observations of the competitive effect of ADP on the ATP-inactivated potassium channel suggest a possible mechanism for this [39]. ADP has been shown to shift the inactivation curve for ATP to higher concentrations just as it would if ADP competed for the same binding site without blocking the channel. The shift is significant enough that by modulation of the ADP concentration in the physiological range the cell should be able to modulate the outward current through these channels, If increased calcium increases the ADP levels in the cell, then calcium could indirectly activate these channels. One possible mechanism for this involves the uptake of calcium by the mitochondria, which is known to interfere with oxidative phosphorylation of ADP [25]. Since ATP is constantly being converted to ADP by celluar metabolism, an increase in internal calcium, therefore, would have the desired effect. We have examined this possibility [22] by eliminating the calciumactivated potassium current from the minimal model and substituting in its place an ATP-inactivated, ADP-modulated channel with a conductance of the form qK,AT’P=
&ATP1 +
1-k D/K, D/K,
+
(A
_ D)/K,
’
ELECTRICAL
ACTIVITY IN PANCREATIC BETA CELLS
135
where D represents the concentration of ADP and A represents the total concentration of ATP and ADP, which is assumed fixed. Calcium is assumed to affect the ADP by the mitochondrial mechanism given above, and the concentration of ADP satisfies the following differential equation: dD -z- --k,D+k(A-D), with the rate parameter
(10)
having the form k,=kexp{a[l$-V,(c)]}.
(11)
The exponential factor represents the interference of calcium on the rate of oxidative phosphorylation. In this model ADP will be a second slow variable, dynamically coupled to calcium, unless the constant k in Equations (10) and (11) is sufficiently large. In that case ADP is slavishly tied to the calcium concentration through the condition dD/dt = 0, i.e.,
(14 When this condition holds, the conductance in Equation (9) is really just a function of the calcium concentration, as it is in the minimal model. Thus it may not be surprising that even when k is as small as 0.1 s- ‘, so that the calcium and ADP are not tightly coupled, reasonable choices of the other parameter values in this model give rise to burst oscillations [22]. CRITIQUE
OF PRESENT
MODELS
OF THE BETA CELL
Despite the fact that the minimal model and its variants give rise to burst oscillations comparable to those measured in intact islets, none of the models discussed above provide an accurate description of the electrical behavior of an isolated beta cell. Indeed, the electrical activity of isolated cells [36] is stimulated by glucose, but the time course is quite irregular and appears to be dominated by the openings and closings of individual channels [4]. On the other hand, the minimal model does do a good job of describing the properties of an individual beta cell clustered together with other cells, either in an islet or in a petri dish. In fact, beta cells are known to be in close electrical contact [14] with their neighbors and to communicate intercellularly via gap junctions [21]. How important this coupling is to the functioning of the islet is not presently known, despite the fact that coupling radically modifies the electrical properties of a single cell. Since present models of the beta cell work well for cells in islets, it has been generally assumed that these models represent the average behavior of
136
JOEL KEIZER
a single cell in a tightly coupled cluster. One advantage of tight coupling is that channels are shared among the larger aggregate of cells, so that the current caused by the opening of a single channel on one cell can be shared by all the cells. If the cluster is large enough, then the effect of single-channel openings will not be apparent and the cluster will act determinstically. This would not be the case for a single cell or even a small cluster, especially if the channels had large conductances and only a small number of channels were open at a time. Recently Sherman, Rinzel, and I [37] have used this idea in an attempt to explain the irregular electrical behavior of single cells and small clusters. For simplicity we have used the minimal model, with the improved RorsmanTrube [36] description of the voltage-activated potassium and calcium channels. Since there are several hundred of these channels per cell and their open probabilities are large, these channels were treated deterministically using their average conductances. The calcium-activated potassium channels, however, are seldom open despite their large numbers [15]. Thus the stochastic openings and closings of these channels, either in a single cell or in tightly coupled clusters of cells, were used to calculate the instantaneous values of the calcium-activated potassium current. Numerical calculations using this procedure produce regular burst patterns only when the number of cells in the cluster is sufficiently large. For small clusters or single cells the electrical activity is quite irregular, and one often sees incipient bursts with no spikes and bursts of spikes of varying duration that are reminiscent of what one finds experimentally. While the minimal model of electrical activity in the beta cell has proved useful for understanding many aspects of bursting, it is not yet clear whether the mechanism of direct activation of a potassium current by internal calcium is actually important in the beta cell. Certainly, more detailed information about the ion channels in the beta cell will be required to convincingly establish the molecular mechanism by which calcium regulates bursting. Nonetheless, it seems likely that calcium regulates its own entry into the beta cell, either directly or indirectly, by activation of a potassium current. How calcium entry stimulates insulin release and how these processes are related to other mechanisms of calcium handling in the beta cell remain much bigger mysteries, which will serve as stating points for future work.
This work was supported by NSF grant CHE 8618647, a John Simon Guggenheim Memorial Fellowship to the author, and a sabbatical leave from the University of California. It is a pleasure to acknowledge many fruitful conversations with John Rinzel, Artie Sherman, and Illani Atwater at the NIH during the past year and with Teresa Chay over the past six years. I especially want to thank John Rinzel and Peter Mazur for their hospitality at the Mathematical
ELECTRICAL
ACTIVITY
IN PANCREATIC
Research Branch of the NIH LAden during my sabbatical.
137
BETA CELLS
and the Instituut-Lorentz
of the University of
REFERENCES 1
2
F. M. Ashcroft, single potassium (1984). I. Atwater, oscillatory
C. M. Dawson, A. Scott, G. Eddlestone, and E. Rojas, The nature of the behavior in electrical activity for the pancreatic beta cell. in Biochemisfq
and Bmphysics 3
4
D. E. Harrison, and S. J. H. Ashcroft, Glucose induces closure of a channel in isolated rat pancreatic beta cells, Nature 312:446-448
of the Pancreatic
Bera Cell, Georg
Thieme,
New
pp. 353-362. J. A. Bangham, activity,
P. A. Smith,
in Biophysics
and P. C. Croghan,
of Pancreutic
5
Eds.), 1986, pp. 265-278. T. Chay, Chaos in a three-variable
6
(1985). T. Chay,
Modelling
Beta Cells (I. Atwater, model
On the effect of intracellular
of an excitable
calcium-sensitive
cell, Phys. potassium
I 8
T. Chay
9
beta cell, Biophys. J. 42:181-190 (1983). T. Chay and J. Keizer, Theory of the effect of extracellular
and J. Keizer,
in the pancreatic
Minimal
model
T. Chay and J. Rinzel, Bursting,
11
Biophys. J. 471357-366 (1985). D. L. Cook and C. N. Hales, Intracellular
14 15
16 17
for membrane
beta cell, Biophys. J. 48:815-827
10
13
1980,
pp.
the beta cell electrical E. Rojas,
bursting pancreatic beta cell, Biophys. J. 50:756-777 (1986). T. Chay, The effect of inactivation of calcium channels by intracellular the bursting pancreatic beta cells, Cc// Biphys., Dec. 1987.
12
York,
100-107. I. Atwater and J. Rinzel, The beta cell bursting pattern and intracellular calcium, in Ionic Channels in Cells and Model Systems (R. Latorre, Ed.), Plenum, New York, 1986.
beating,
oscillations
and B. Soria, D 16:233-242 channels
in the
calcium
ions in
in the pancreatic
potassium
on oscillations
(1985).
and chaos in an excitable ATP directly
membrane
blocks potassium
model,
channels
in
pancreatic beta cells, Nature 311:271-273 (1984). D. L. Cook, M. Ikeuchi, and W. Y. Fujimoto, Lowering of pH inhibits calciumactivated potassium channels in isolated rat pancreatic islet cells, Nature 311:269-271 (1984). P. M. Dean and E. K. Matthews, Glucose-induced electrical activity in pancreatic islet cells, J. Physiol. (London) 210:255-264 (1970). G. T. Eddlestone, A. Goncalves, J. A. Bangham, and E. Rojas, Electrical coupling between cells in islets of Langerhans from mouse, J. Membrane Biol. 77:1-S (1984). I. Findlay, M. J. Dunne, and 0. H. Petersen, High conductance potassium channel in pancreatic islet cells can be activated and inactivated by internal calcium, J. Membrane Biol. 83:169-175 (1985). C. J. Hedeskov, Mechanism of glucose-induced insulin release, Physiol. Reu. 60:442-509 (1980). B. Hellman, H. Abrahamson, T. Anderson, P.-O. Berggren, P. Flatt, E. Gylfe, and H.-J. Hahn, Calcium movements in relation to glucose-stimulated isulin release, in Biochemistry und Biophysics of the Pancreatic Beta Cell, Georg Thieme, New York, 1980, pp. 122-130.
138 18 19
20
JOEL KEIZER
D. Himmel and T. Chay, Theoretical studies on the electrical activity of pancreatic beta cells as a function of glucose, Biophys. J. 51:89-107 (1987). A. Hodgkin and A. F. Huxley, A quantitative description of membrane current and its application to conduction and excitation in nerve, J. Physiol. (London) 177:500-544 (1952).
22
P. Lebrun pancreatic P. Meda, topography Quart. J. G. Magnus
23
H. P. Meissner,
21
Phvsiol.
24
and I. Atwater, Chaotic and irregular bursting electrical activity in mouse beta cells, Biophys. J. 48:529-531 (1985). I. Atwater, A. Goncalves, A. Bangham, L. Orci, and E. Rojas, The of electrical synchrony among beta cells in the mouse islet of Langerhans, Exp.
(Paris)
Electrical
(1984).
characteristics
12:151-761
S. Misler,
L. C. Falke,
potassium
channel
83:7119-7123
25
Physiol. 69:719-735
and J. Keizer, in preparation.
K. Gillis,
in
of the beta
cells in pancreatic
J.
islets,
(1976).
rat
and
pancreatic
M. L. McDaniel, B cells,
Proc.
A metabolite-regulated Nat.
Acad.
Sci.
U.S. A.
(1986).
D. Nicholls,
Some recent
advances
in mitochondrial
calcium
Trends
transport,
Biol.
Sci. 6:36-38
26 27
(1981). L. Orci, A portrait of the pancreatic B cell, Diabetologia 10:163-187 (1974). E. M. Perez-Armendariz, I. Atwater, and E. Rojas, Glucose induced oscillatory changes in extracellular potassium concentration in mouse islets of Langerhans, Biophys.
J. 48:741-749
(1985).
29
E. M. Perez-Armendariz and I. Atwater, Glucose invoked changes in [K+ ] and [Ca2+ ] in intercellular spaces of the mouse islet of Langerhans, in Biophysics of Pancreutic Betu C&s (I. Atwater, E. Rojas, and B. Soria, Eds.), 1986, pp. 31-51. R. Plant and M. Kim, Mathematical description of a bursting pacemaker neuron by a
30
modification R. E. Plant,
28
of the Hodgkin-Huxley equations, Biophys. J. 16:227-244 The effects of calcium on bursting neurons, Biophys.
(1976). J.
21:217-237
(1978). Bioelectric
Phenomenu,
31
R. Plonsey,
32 33
J. Rinzel, On repetitive activity J. Rinzel, Bursting oscillations Parflu/
34
Differential
36
New York, 1969.
(B. D. Sleeman
and R. J. Jarvis,
Ids.)
Springer,
and
New
York, 1985, pp. 304-316. J. Rinzel, T. R. Chay, D. Himmel, and I. Atwater, Prediction of the glucose-induced changes in membrane ionic permeability and cytosolic calcium by mathematical modeling,
35
Equations
McGraw-Hill,
in nerve, Fed. Proc. 37:2793-2802 (1978). in an excitable membrane model, in Ordinmy
in Biophysics of Pancreatic
Betu Cells (I. Atwater,
E. Rojas,
and B. Soria,
Eds.), 1986, pp. 241-263. P. Rorsman and G. Trube, Glucose dependent potassium channels in pancreatic beta cells are regulated by intracellular ATP, Pf/i&ers Arch. 405:305-309 (1985). P. Rorsman and G. Trube, Calcium and delayed potassium currents in mouse pancreatic beta cells under voltage-clamp conditions, J. Physiof. (London) 374:531-550 (1986).
31 38
A. Sherman, J. Rinzel, and J. Keizer, Emergence of organized bursting in clusters of pancreatic P-cells by channel sharing, Biophys. J., to appear. C. B. Wollheim and G. W. G. Sharp, Regulation of insulin release by calcium, Physiol. Rev. 61:914-974
39
(1981).
M. Kakei, P. Kelly, S. J. H. Ashcroft, and F. M. Ashcroft, The ATP-sensitivity of K+ channels in rat pancreatic p-cells is modulated by ADP, Fed. Eur. Biochem. Sot. 208:63-66
(1986).