## Insulin Oscillations with Intermediate Frequency

Insulin also oscillates with a period of about 10-20 minutes (Figs. 19.11 and 19.12). Because these oscillations occur in islets and the isolated pancreas, it appears that unlike the ultradian oscillations, they are caused by a mechanism intrinsic to the pancreatic islets.

These oscillations also occur in the experimental perifusion system depicted in Fig. 19.16. A thin layer of insulin-secreting ^-cells is sandwiched between beads and exposed to the flow of a solution, the perifusate, with flow rate f. By collecting the solution exiting the bottom of the perifusion system, one can determine how the rate of insulin release of the cells in the bed depends on the composition and flow rate of the influx solution.

After a step increase in perifusate glucose concentration (to synchronize the cells), regular oscillations in the rate of insulin release are seen (Fig. 19.12). The insulin oscillations are influenced by both the flow rate and glucose concentration of the perifusate. This suggests that secretions from the cells into the fluid affect the rate of secretion. If the oscillatory mechanism were confined to the interior of the cells, differing flow rates (with the same glucose concentrations) should not alter the properties of the oscillations.

The release of insulin from a ^-cell depends on the uptake of glucose by the cell. Glucose is taken up into cells by a family of glucose transporters, called GLUT-type transporters. They operate by mechanisms discussed in Chapter 2, and come in five subtypes distinguished by their affinities, capacities, and kinetic properties; the particular subtypes of concern to us here are the GLUT-1 and GLUT-2 transporters. The

GLUT-1 transporters are assumed to be activated by insulin, providing a positive feedback. Furthermore, in the presence of insulin, GLUT-1 carrier protein is recruited into the cell membrane from the cytoplasm. There is also evidence that insulin promotes the synthesis of GLUT-1 protein, thereby adding to the positive feedback effect. It is also proposed that insulin inhibits the uptake of glucose by GLUT-2 transporters, producing negative feedback.

### Model equations

A model for the release of insulin from 3-cells using these assumptions was constructed by Maki and Keizer (1995a,b). If the volume flow rate, f, is large compared to the volume of the cells in the bed, Vbed, it is reasonable to approximate the bed as a continuously stirred mixture, and thus ignore spatial dependencies in the bed (see Exercise 4). We let G and I denote the concentrations of glucose and insulin, respectively, in the efflux solution, and let the subscript 0 denote the concentrations in the influx solution. Also, we let k0 = f/Vbed. Then, the rate at which glucose flows out of the bed is k0G, and the rate at which it flows into the bed is k0G0. Hence, we have the conservation laws for glucose and insulin,

where R1 is the rate of glucose uptake by GLUT-1 receptors, R2 is the rate of glucose uptake by GLUT-2 receptors, and Rs is the rate of insulin secretion by the cells in the bed.

When the flow rate is large enough and when the concentration of insulin in the influx is small enough, both G and I may be replaced by their pseudo-steady states. For if k0G is large compared to R1 and R2, and k0I/Rs is order 1, a simple asymptotic argument shows that

Inside the cell, glucose is metabolized at the rate Rm. Thus, if Gi denotes the interior concentration of glucose, then dG

To complete the model equations we introduce a variable J, an inhibition variable (similar to h or n in the Hodgkin-Huxley context), which measures the extent to which insulin inhibits its own release. J does not correspond directly to a measured physiological process, but is a phenomenological representation of a slow negative feedback process. The variable J obeys the differential equation dJ

where

Note that decreases as the concentration of insulin increases, and thus an increase in insulin leads to a decrease in J, with a time delay related to the time constant t. In summary, the model equations are

To complete the model description, it remains to discuss the functional forms of the various rate terms. First, the rate of glucose metabolism is assumed to be an increasing function of glucose concentration. Thus,

Km + Gi for some constants Vm and Km. In a similar way, the rates of the GLUT-1 and GLUT-2 transporters are assumed to be simple increasing functions of the external concentration of glucose G0. R1 is assumed to be an increasing function of I, which models the recruitment of GLUT-1 transporters by insulin and results in positive feedback,

R2 is assumed to be an increasing function of J, and thus a decreasing function of I, at least at steady state,

A leak term LgGi describes the leak of glucose out of the cell and is appended to R2.

Finally, the rate of insulin secretion, Rs, is described by an empirical function determined by fitting to experimental data. By combining data on how Rm depends on G0 with data on how Rs depends on G0, one can determine the relationship between Rs and Rm. We are then able to express Rs in terms of Rm and hence in terms of Gi. By doing so we circumvent the inconvenient fact that although the rate of insulin secretion depends in some way on internal glucose concentrations, this relationship has not

Table 19.2 Standard parameter values of the model for intermediate insulin oscillations.

 Fixed by experiment Vm 0.24 mM/min Km 9.8 mM Vs 0.034 mM/min Ks 0.13 mM/min Vi 34.7 mM/min Ki 1.4 mM V2 32 mM/min K2 17 mM Experimentally variable ko 550/min Io 0 mM Go 8-19 mM Adjustable Kinh 1 x 10-7 mM Ki 6 x 10-8 mM T 20 min Lg 20/min

been measured directly. The result is

The factor Jn does not follow from the experimental data but is included here so that insulin exerts a direct negative feedback effect on the rate of insulin secretion.

Most of the model parameters can be determined from experimental data, and are summarized in Table 19.2. The adjustable parameters are Kinh, Ki, t, and Lg. With the exception of Lg, these are parameters associated with the various types of insulin feedback. Since this is the part of the model for which there is the least direct evidence, it is not surprising that these parameters cannot be determined directly from experimental data.

By choosing different values for m and n, it is possible to vary the type of negative feedback. If m = 0 and n = 1, insulin directly inhibits the rate of insulin secretion, whereas if m = 2 and n = 0, insulin decreases the rate of glucose uptake by GLUT-2 receptors. Since this reduces the concentration of glucose inside the cell, it indirectly decreases the rate of insulin secretion.

In the direct inhibition model (m = 0,n = 1), as the concentration of glucose in the influx solution, G0, is increased, the steady-state concentration of insulin in the efflux solution increases. This corresponds to an increase in the rate of insulin secretion. When G0 is large enough, oscillations with a period of about 16 minutes arise via a Hopf bifurcation, and as G0 increases further, the amplitude of the oscillations decreases until they disappear via another Hopf bifurcation. Although the period of the oscillations agrees well with experimental data, and oscillations occur at approximately the correct glucose concentrations, the decrease in amplitude with increasing G0 is opposite to what is observed experimentally.

In the indirect inhibition model the opposite effect is seen. As before, the oscillations appear and disappear at Hopf bifurcations, but here the amplitude of the oscillations increases as G0 increases, in better agreement with experiment. It thus appears that of the two hypotheses, indirect inhibition is the more plausible.

It remains to answer the question of why hormone secretion is pulsatile in the first place. As with many oscillatory physiological systems, there is no completely satisfactory answer to this question. However, one plausible hypothesis has been proposed by Li and Goldbeter (1989). Based on a model of a hormone receptor first constructed by Segel, Goldbeter, and their coworkers (Segel et al., 1986; Knox et al., 1986), Li and Goldbeter constructed a model of a hormone receptor that responds best to stimuli of a certain frequency, thus providing a possible reason for the importance of pulsatility.

Closely linked to this hypothesis is the phenomenon of receptor adaptation. Often, the response to a constant hormone stimulus is much smaller than the response to a time-varying stimulus. In the extreme case, the receptor responds to a time-varying input, but has no response to a steady input, regardless of the input magnitude, a phenomenon called exact adaptation. We have seen a number of examples of adaptation in this book; for example, the models of the IP3 receptor discussed in Chapter 5 show adaptation in their response to a step-function increase in Ca2+ concentration; i.e., their response is an initial peak in the Ca2+ release, followed by a decrease to a lower plateau as the receptor is slowly inactivated by Ca2+. Similarly, in Chapter 22 we will see how biochemical feedback in photoreceptors can result in a system that displays remarkably precise adaptational properties, as embodied in Weber's law. Because of the importance of adaptation in physiological systems, it is interesting to study how adaptation arises in a simple receptor model.

The key assumption is that the hormone receptor can exist in two different conformational states, R and D, and each conformational state can have hormone bound or unbound (Fig. 19.17). For simplicity we assume that the active form of the receptor has hormone bound to the receptor in state R. The addition of hormone to the receptor system causes a change in the proportion of each receptor state, but the total receptor concentration stays fixed.

Letting r,x,y, d denote [R]/RT, [RH]/RT, [DH]/RT and [D]/RT respectively, where RT is the total receptor concentration, we find the following equations for the receptor system:
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