Model of the Fertilization Calcium Wave

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When mature Xenopus laevis oocytes (eggs) are loaded with an indicator dye (e.g., Ca2+-green dextran) and stimulated by the fusion of sperm, a propagating wave of intracellular Ca2+ release can be observed by backcalculating the free Ca2+ concentration ([Ca2+]i(x, t)) from a time-dependent fluorescence signal (F(x,t)) according to (8.3) or (8.5). This fertilization Ca2+ wave is an important step in early development. It triggers

Fertilization Model

Figure 8.2 Schematic diagram of the fertilization Ca2+ wave model. Ca2+ enters the cytosol from the ER via a passive leak and the IP3R, which is activated by both Ca2+ and IP3 on a fast time scale and inhibited by Ca2+ on a slow time scale, all at the cytoplasmic face. The ER is refilled by a SERCA-type Ca2+-ATPase pump. Diffusion of Ca2+ in both the cytosol and ER is accounted for using effective diffusion coefficients (constant) that account for interactions with Ca2+ buffers (not shown) and the volume fractions of both compartments. Reprinted from [Jafri and Keizer, 1994].

Figure 8.2 Schematic diagram of the fertilization Ca2+ wave model. Ca2+ enters the cytosol from the ER via a passive leak and the IP3R, which is activated by both Ca2+ and IP3 on a fast time scale and inhibited by Ca2+ on a slow time scale, all at the cytoplasmic face. The ER is refilled by a SERCA-type Ca2+-ATPase pump. Diffusion of Ca2+ in both the cytosol and ER is accounted for using effective diffusion coefficients (constant) that account for interactions with Ca2+ buffers (not shown) and the volume fractions of both compartments. Reprinted from [Jafri and Keizer, 1994].

the fusion of cortical granules (vesicles) with the plasma membrane, a process that initiates the raising of the viteline envelope and a long-lasting block to polyspermy. The cell divisions that initiate development of Xenopus begin only after the fertilization Ca2+ wave has propagated throughout the entire cell. The eggs of many species, from starfish to mammals, exhibit propagating Ca2+ waves upon fertilization.

Because fertilization Ca2+ waves such as those shown in Figure 8.1 still occur when the extracellular medium is nominally Ca2+ free, the phenomenon appears to be largely independent of Ca2+ influx. On the other hand, the fertilization Ca2+ wave absolutely requires functional IP3 receptors. When IP3-mediated Ca2+ release is blocked by any one of several experimental manipulations (e.g., upon introduction of heparin into the cytosol), the fertilization Ca2+ wave is not observed [Nuccitelli et al., 1993].

We expect Ca2+ diffusion to play an important role in the fertilization Ca2+ wave because Xenopus laevis eggs are large (approximately 1.2 mm in diameter). Indeed, Figure 8.1 indicates that the Ca2+ concentration within the egg depends very much on both spatial position and time. For this reason a whole cell model of this phenomenon would be deficient, and instead researchers mathematically describe the fertilization Ca2+ wave in the Xenopus egg and other cell types using a combination of spatial modeling and whole—cell modeling approaches [Atri et al., 1993, Girard et al., 1992, Dupont and Goldbeter, 1994, Jafri and Keizer, 1994, Wagner et al., 1998].

Here we introduce such a spatial whole—cell model of the fertilization Ca2+ wave by recalling that the Li—Rinzel reduction of the DeYoung—Keizer model of the IP3R can be combined with sigmoidal SERCA pump kinetics and a passive ER leak to create a whole—cell model of Ca2+ handling in pituitary gonadotrophs (Section 5.3). Using a variation of this model to represent Ca2+-induced Ca2+ release (CICR) and reuptake by the endoplasmic reticulum (ER), we follow Section 7.5.2 and account for Ca2+ diffusion in both the cytosol and ER by writing the following reaction—diffusion system:

m = fi [D;v2[Ca2+]; + jlp^R + jLEAK — jSERCA] , (8.6)

d[cdt+ ]er = /er [DERV2[Ca2+]er - (Vi/Ver) (j!p3r - jleak + Jserca)] , (8.8)

where jleak = «leak ([Ca2+]ER - [Ca2+]i), j^r Rm^w3 ([Ca2+]ER - [Ca2+]i), jSERCA = vSERCA [Ca2+]i2/ ([Ca2+]i2 + KsercO, and V's are the volumes as before. In these equations, w is the fraction of IP3Rs not inactivated and the open probability of the IP3Rs is given by PO = m^w3 where the fraction of activated IP3Rs (m^) is assumed to be an instantaneous function of Ca2+ and IP3 concentrations,

Although the Li—Rinzel reduction of the DeYoung—Kiezer model gives a time constant of IP3R inactivation that is dependent on Ca2+ and IP3 concentration, t ([Ca2+]i, [IP3]), for simplicity we will assume that t is constant (2 sec). A diagram of the fertilization Ca2+ wave model components and fluxes is presented in Figure 8.2.

Note that in (8.6)—(8.8) the maximum conductance of IP3-mediated Ca2+ release (vIP3R), passive leak rate (vLEAK), and maximum rate of ATP-dependent reuptake (vSERCA) are constants. Thus, the model assumes homogeneous Ca2+ release and re-uptake dynamics, that is, a uniform and high density of inositol 1,4,5-trisphosphate (IP3) receptors and sarco-endoplasmic reticulum Ca2+-ATPases. Also note that (8.7) includes no diffusion term for the simple reason that IP3Rs (and thus the gating variable representing their inactivation) has a fixed spatial location and does not diffuse. Throughout this chapter we will assume that [IP3] is uniform and constant (but see [Wagner et al., 1998]).

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