The Two Pool Model

One of the earliest models for IP3-dependent Ca2+ release assumes the existence of two distinct internal Ca2+ stores, one of which is sensitive to IP3, the other of which is sensitive to Ca2+ (Kuba and Takeshita, 1981; Goldbeter et al., 1990; Goldbeter, 1996). Agonist stimulation leads to the production of IP3, which releases Ca2+ from the IP3-sensitive store through IP3 receptors. The Ca2+ that is thereby released stimulates the release of further Ca2+ from the Ca2+-sensitive store, possibly via ryanodine receptors. A crucial assumption of the model is that the concentration of Ca2+ in the IP3-sensitive store remains constant, as the store is quickly refilled from the extracellular medium. A schematic diagram of the model is given in Fig. 5.3.

Recent work by Dupont and Goldbeter (1993, 1994) has shown that the model does not depend on the existence of two separate pools of Ca2+; the model equations can equally well be used to describe the release of Ca2+ from a single pool, with the release modulated by both IP3 and Ca2+. Nevertheless, for convenience, we persist in calling this the two-pool model.

CCh10^M

CCh10^M

GnRH 1nM

GnRH 1nM

1min

1min

25^M

200^M

50s

Figure 5.2 Typical calcium oscillations from a variety of cell types. A: Hepatoctyes stimulated with vasopressin (VP). B: Rat parotid gland stimulated with carbachol (CCh). C: Gonadotropes stimulated with gonadotropin-releasing hormone (GnRH). D: Hamster eggs after fertilization. The time of fertilization is denoted by the arrow. E and F: Insulinoma cells stimulated with two different concentrations of carbachol. (Berridge and Galione, 1988, Fig. 2.)

Let c denote the concentration of Ca2+ in the cytoplasm, and cs the concentration of Ca2+ in the Ca2+-sensitive pool. We assume that IP3 causes a steady flux r of Ca2+ into the cytosol, and that Ca2+ is pumped out of the cytoplasm at the rate -kc. Then dc = r - kc - f (c, cs), at

Outside the cell extrusion (kc)

Outside the cell extrusion (kc)

Action Potential Steps

IP3-dependent release (r)

Inside the cell (c)

Ca2+-induced Ca2+ release (Please)

Figure 5.3 Schematic diagram of the two-pool model of Ca2+ oscillations

IP3-dependent release (r)

Inside the cell (c)

Ca2+-induced Ca2+ release (Please)

Figure 5.3 Schematic diagram of the two-pool model of Ca2+ oscillations dcs ~ — — f (C, Cs), dT

'uptake v1cn

Km cp

and t denotes time. The flux of Ca2+ from the cytoplasm into the Ca2+-sensitive pool is given by f; Juptake is the rate at which Ca2+ is pumped from the cytosol into the Ca2+-sensitive pool by an active process, and /release is the rate at which Ca2+ is released from the Ca2+-sensitive pool. Note that as c increases, so does Jrelease. Thus, Ca2+ stimulates its own release through positive feedback, usually called Ca2+-induced Ca2+ release, or CICR (Endo et al., 1970; Fabiato, 1983). It is this positive feedback that is central to the model's behavior. Finally, the rate at which Ca2+ leaks from the Ca2+-sensitive pool into the cytosol is kfcs. In the model, r is constant for constant [IP3 ] and is treated as a control parameter. Thus, the behavior of the model at different constant IP3 concentrations can be studied by varying r.

Table 5.1 Typical parameter values for the two-pool model of Ca2+ oscillations. (Goldbeter et al., 1990.)

k

= 10 s-1

K1

= 1 ^M

K2

= 2 ^M

K3

= 0.9 ^M

Vi

= 65 ^Ms-1

V2

= 500 ^Ms-1

kf

= 1 s-1

m

=2

n

=2

P

= 4

For convenience we nondimensionalize the model equations. Let u = c/K1, t = xk, v = cs/K2, a = K3/K1, p = V1/V2, Y = K2/K1, 5 = kfK2/V2, z = r/(kK1), and e = kK2/V2, to get du Y r/

— = -f (u,v), (5.7) dt e f(u,v) = ^-u^) -(- 5v. (5.8)

If the exchange of Ca2+ between the cytosol and the Ca2+-sensitive pool is fast (i.e., V1 and V2 are large), then e is a small parameter. A table of typical parameter values in the model is given in Table 5.1. For these values, e ~ 0.04.

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