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RESEARCH |
1 School of Mathematical Sciences and 2 Research Centre for Reproductive Health, Department of Obstetrics and Gynaecology, The University of Adelaide, Adelaide, 5000 SA, Australia
Correspondence should be addressed to J G Thompson; Email: jeremy.thompson{at}adelaide.edu.au
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Introduction |
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One component that appears to be important for the establishment of developmental competence in the oocyte within antral follicles is oxygen. In general, pO2 levels around 7–11% have been recorded in antral follicles of humans and pigs (Knudsen et al. 1978, Fischer et al. 1992). Several studies in humans (Van Blerkom et al. 1997, Van Blerkom 1998) indicate that perifollicular vascularity is associated with oocyte developmental competence. This was subsequently related to the dissolved O2 content (Van Blerkom 1998), although this association has been disputed (Huey et al. 1999). In contrast, there are many studies that have examined the effect of O2 concentration during in vitro maturation of oocytes across several species. Examples of such studies include hamster (Gwatkin & Haidri 1974), squirrel monkey (Yeoman et al. 1999), cow (Hashimoto et al. 2000, Minami et al. 2000), pig (Park et al. 2005) and mouse (Haidri et al. 1971, Eppig & Wigglesworth 1995, Hu et al. 2001). Such studies have not yielded any consensus over the optimal O2 concentration during maturation, as there are conflicting results, especially in regard to the benefits of a low O2 atmosphere. Of central importance to these studies is the choice of oxygen concentration used. Many have compared atmosphere (approximately 20% O2) against 5% O2 concentrations, the latter being the widely accepted optimal value for the development of mouse embryos in vitro (Quinn & Harlow 1978).
Recently we measured the uptake of key nutrients in bovine cumulus–oocyte complexes (COCs) to determine if oocyte-secreted factors influence bovine COC metabolism (Sutton et al. 2003). In contrast to the mouse (Sugiura et al. 2005), oocyte-secreted factors appear to have little influence over the metabolism of the surrounding cumulus cells. However, as with the mouse (Sugiura et al. 2005), bovine COCs have significant levels of glycolysis, with lactic acid production accounting for much of the glucose taken up, especially prior to cumulus expansion (Sutton et al. 2003). These data suggest that oxygen consumption by the bovine cumulus cell mass is minimal, thus sparing O2 for oxidative phosphorylation by the oocyte. We have therefore developed a mathematical model of the diffusion of oxygen through the COC so as to assess how the O2 concentration decreases from the follicular-fluid cumulus oophorus interface to the corona radiata oocyte interface. We have applied this model to data already described for both bovine and murine COCs.
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Materials and Methods |
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Mathematical model
The approximate spherical symmetry of the COC and our interest in the very early stage of development justifies our use of a spherically symmetric steady-state model. The rate of oxygen consumption by both oocyte and cumulus is expected to vary with the concentration of oxygen. Thus we solve the equation
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subject to suitable boundary conditions, for the oxygen concentration C as a function of radial position r from the centre of the COC, where D is the diffusion coefficient and M(C) is the rate of oxygen consumption per unit volume of cells as a function of the concentration C.
As the oxygen consumption rate of the oocyte is considerably different to that of the cumulus cells, the COC has been modelled as two spherically symmetric regions. The oxygen consumption rates in the cumulus and oocyte regions are taken to be M1(C1) and M2(C2) respectively where C1, C2 are the O2 concentrations in each of the regions. The cumulus mass is surrounded by follicular fluid that is assumed to have a known oxygen concentration Cf.
Thus, the oxygen concentration in the COC is found by solving two spherically symmetric diffusion equations, one for the cumulus mass yielding C1 and the other for the oocyte yielding C2:
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where:
At the boundary between the oocyte and the cumulus mass (r = a) we must have continuity of both mass flux and concentration, that is,
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Also, the concentration must remain bounded at the centre of the COC (as r 
0), or, equivalently for spherical symmetry,
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A further condition, on the boundary between the cumulus mass and the follicular fluid (r = b), governs the transport of oxygen into the COC. Here we assume that the flux of oxygen into the cumulus mass is proportional to the difference in oxygen concentration between the cumulus and the surrounding follicular fluid, that is,
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where h is the transfer coefficient for this problem.
A common approach to estimating oxygen concentration gradients in the biological literature (e.g. Gosden & Byatt-Smith 1986, Byatt-Smith et al. 1991) is to solve a diffusion-equation model with the rate of oxygen consumption assumed to be constant or proportional to the oxygen concentration. This assumption allows analytical solution of the governing equations. However, these solutions may be quite inaccurate, and even non-physical and hence invalid, in some regions of the parameter space, so that care must be exercised when using them. To overcome such difficulties, we have used a functional description of the consumption term that we believe more accurately represents the dependence of oxygen consumption rate on concentration, as we now describe.
In practice, cells consume oxygen at a maximal (or near to maximal) level, until oxygen concentration itself becomes rate limiting. The rate of consumption then falls with concentration, approaching zero as the O2 concentration approaches zero. This behaviour is seen in the curves given in Jones (1984), which show oxygen concentration (O2) versus cytochrome-c oxidation (as a measure of consumption rate) in hepatocytes.
A function that captures the general shape of these curves is the Arrhenius law from chemistry,
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where 
and ß are positive constants. We use this function for the consumption terms in our equations, to relate the rate of oxygen consumption M to the concentration C. As C 0, e–/c 0 and, hence, M(C) 0, while as C 
, e–/C 1 and, hence M(C) ß, i.e. ß is the maximum rate of consumption. The constants and ß are chosen to fit appropriate experimental data. This function has the benefit of handling the changing oxygen consumption rate with concentration in a consistent manner for both the cumulus mass and oocyte, but also means that the equations cannot be solved analytically.
Our equations for oxygen transport in the COC are now
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where , ß, 
and 
are positive constants. We have allowed that the constants in the consumption term may differ between the cumulus mass (, ß) and the oocyte (, ). While not solvable analytically, this system of equations, together with the associated boundary conditions described earlier, is readily solved numerically using, for example, MATLAB (Version 6.5, 2002, The Mathworks Inc.).
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). In particular, using data on the size and total oxygen consumption by oocyte and COC, the maximum rate of oxygen consumption in moles (mol) per litre (l) of tissue per second (s) was computed thus: for a maximum oxygen uptake U picolitres per hour (pl/h) by a cell volume V cubic metres, the maximum oxygen consumption rate in mol per litre of tissue per second is given by
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Here we have assumed oxygen to be an ideal gas so that the volume of a mole of oxygen at 37 ° C is 25.4 litres. For the oocyte the uptake is as given in Table 1; for the cumulus it is the difference between that for the COC and that for the oocyte, i.e. 1873–240 = 1633 pl/h. The volume of the oocyte is given by V = 4
a3/3, where a is its radius in metres; the volume of the cumulus is V = 4(b3–a3)/3, where b is the radius of the COC in metres. The results of these calculations are shown in the bottom two rows of Table 1. It can be seen that a bovine oocyte consumes 3.5 times as much O2 per unit volume of tissue per unit time than the surrounding cumulus cell mass. In murine COCs this is reversed, with the cumulus cell mass consuming 2.4 times as much as the oocyte (per unit volume per unit time).
We set ß to be the maximum rate of oxygen consumption by the cumulus mass (per unit volume per unit time), as given in Table 1 for each of cow and mouse. Since the consumption in a bovine (murine) oocyte is approximately 3.5 (0.4) times that in the cumulus (see Table 1), we take = 3.5ß and = 0.4ß for each of cow and mouse respectively.
The parameters and in the consumption-rate terms determine the dependence of consumption rate on oxygen concentration, i.e. they are effectively measures of the half-maximal oxidation P50 value in the cumulus and oocyte respectively. We obtain estimates for these parameters by assuming, in the absence of specific data for the oocyte and cumulus, that they behave similarly to hepatocyte cells, and fitting the Arrhenius consumption-rate term to experimental data given in Jones (1984) which relates the rate of oxygen consumption to oxygen concentration for both untreated and digitonin-treated hepatocytes; digitonin-treated hepatocytes are shown to be less O2-dependent than untreated (intact) cells, with a P50 of about 2 µmol L–1 compared to 6 µmol L–1. The acellular oocyte can be expected to have similar O2 dependence to digitonin-treated hepatocytes. Physiologically, the cumulus is likely to have a lower respiratory rate and, consequently, lower O2 dependence than (untreated) hepatocytes; however, due to its cellular structure, the cumulus mass is likely to have greater O2 dependence than digitonin-treated hepatocytes. Thus, the correct relation for the cumulus is expected to lie between these two curves, which may therefore be considered as limiting cases; intermediate cases may be obtained by varying the P50 value between that for digitonin-treated and that for untreated hepatocytes. In fact, the oxygen concentrations in the oocyte are lowest under the assumption that the cumulus has an O2 dependence equal to digitonin-treated hepatocytes; as the P50 increases towards that for untreated hepatocytes, the concentration gradient across the COC decreases and oxygen concentrations in the oocyte increase because the cumulus mass consumes less oxygen. Therefore we have adopted the curve for digitonin-treated hepatocytes to represent the O2 dependence of the cumulus as the ‘worst-case’ scenario in the sense that oxygen concentrations are not overestimated.
Thus, we use the curve for digitonin-treated hepatocytes in Jones (1984) to give the O2 dependence for both the cumulus and oocyte (i.e. the values of and in the consumption-rate terms of the model). With P50 
2µmol L –1, the half-maximal consumption rate for the cumulus (ß/2) is given by the Arrhenius law thus:
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Dividing through by ß and rearranging gives:
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Thus we set = = 1.39 x 10–6 mol L–1 for both cow and mouse COCs.
As discussed later, because oxygen diffuses readily through cellular material and consumption terms for both the cumulus mass and the oocyte are specifically included in our equations, the appropriate diffusion coefficient is that of oxygen in physiological saline, i.e. D = 2.5 x 10–9 m2 s–1, Jones (1984). The transfer coefficient h is a measure of the impedance to oxygen transport provided by the boundary between the follicular fluid and the cumulus mass; the higher the value, the lower the impedance. Because our model assumes the cumulus mass to be a homogeneous mixture of intra- and extracellular materials, there is no physical membrane boundary separating the cumulus mass from the follicular fluid and we expect little, if any, resistance to oxygen transport into the cumulus mass. Numerical experiments over a range of values of h revealed that there was little impedance for h > 1, effectively equivalent to setting the O2 concentration at the boundary equal to the concentration in the follicular fluid (i.e. C(b) = Cf ) and we have chosen to use a value of h = 10 m/s. A summary of the parameter values used in our equations is given in Table 2.
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where solubility is taken to be 1.2 x 10–6 mol/l per mmHg.
Oxygen concentration: solutions
The governing equations were solved using MATLAB for each of two sets of parameters (for cow and mouse), and varying the oxygen concentration Cf in the surrounding fluid. Figure 1 shows the oxygen concentration gradient across a bovine COC when the oxygen concentration in the follicular fluid is at 10%. The shape of the curve is typical of other values also. Figure 2 shows the change in oxygen concentration with radius in the bovine cumulus mass only for a number of values of Cf, and Table 3 gives the oxygen concentration at the oocyte–cumulus boundary, for various values of Cf. As can be seen, there is a relatively small concentration gradient across the cumulus radius indicating that, in preovulatory antral follicles, a relatively small amount of the oxygen entering the COC is actually consumed by the cumulus mass, with the majority diffusing through to the oocyte.
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and Table 3 give similar results for the murine COC. Even though the relative proportion of oxygen consumed per unit volume of cumulus mass is higher in a murine COC than a bovine, there is still very little (maximally approximately 0.5% O2) consumed by the cumulus mass, because the cumulus volume is smaller than that of the oocyte.
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Discussion |
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Our model also assumes spherical symmetry, but differs from previous work in that we have incorporated a term that allows for nonlinear variation of the oxygen consumption rate with oxygen concentration in a manner that fits experimental data. Previous studies (Gosden & Byatt-Smith 1986, Byatt-Smith et al. 1991) consider the cases of constant rate of oxygen consumption and consumption rate proportional to concentration, which fail to give accurate solutions at low and high oxygen concentrations respectively. Gielen & Kranenbarg (2002) combined these two cases when considering oxygen diffusion in small organisms; O2 consumption was taken to be piecewise linear-constant at high O2 concentrations, and proportional to concentration below some threshold O2 concentration. This is more accurate, but requires that the boundary between regions of high and low O2 concentration be determined (e.g. by solving a complex transcendental equation). By introducing a nonlinear term to represent the rate of oxygen consumption as a function of O2 concentration, we eliminate the need to determine regions of high and low concentration and have constructed a model that can be better fitted to experimental data and hence should be more accurate over the entire range of possible oxygen concentrations.
Our model also differs from these previous similar studies in considering two regions (oocyte and cumulus) sharing a common boundary, both of which consume oxygen at different nonlinear rates. Multi-compartment models can be found in the literature to represent nutrient diffusion in other biological systems (for example Hoofd et al. 1990, Bruggeman et al. 2001, Kumar et al. 2004) but have not been applied to cumulus–oocyte complex systems.
Our results for both cow and mouse show a small concentration gradient from the follicular fluid across the cumulus mass to the oocyte, and hence demonstrate that only a small fraction of the oxygen entering the COC system is consumed by the cumulus mass during diffusion across the cumulus layer. Thus the level of oxygen at the surface of the oocyte is approximately 5–15% lower than that in the follicular fluid, depending on O2 concentration and metabolic activity of cumulus cells between species. However, this result is sensitive to the value of the diffusion coefficient D for oxygen across the oocyte–cumulus complex.
There has been much discussion in the literature on the appropriate value of the oxygen diffusion coefficient for a medium consisting largely of intracellular matter/fluid (Hills 1970, Tai & Chang 1974, Jones & Kennedy 1982, Jones 1986, McCabe et al. 2003). Values ranging from that for pure water at 37 ° C (Dw = 2.5 x 10–9 m2/s) to 3 orders of magnitude smaller have been used. For example, Gosden & Byatt-Smith (1986) used a value of 7 x 10–12 m2/s, but later (Byatt-Smith et al. 1991) considered Dw as well as values 5–10 times smaller. Our review of the literature leads us to conclude that values of the diffusion coefficient differing significantly from Dw are due to assumptions made in deriving the models used to determine the diffusion coefficient from experimental data, as well as error in experimental measurement. For example, the common neglect of oxygen consumption by some parts of a biological system will result in a low apparent diffusion coefficient. We note that Jones (1986), commenting on a diffusion coefficient ~Dw /70, states that that "no physical or chemical basis is known to account for such a low value in cells". Jones (1986) also quotes experimental studies that indicate a value ~10–9 m2 s–1 for the diffusion coefficient of oxygen in tissue, which corresponds to Dw. Recently, McCabe et al.(2003) discussed the experimental evidence for enhanced oxygen diffusion in respiring tissue, such as in a COC, and quote a diffusivity of D = 3 x 10–9 m2 s–1 for liver cells. This is not significantly different (only 1.2 times) to the value in water used here. An even more recent study of oxygen levels in a very different system, the mammalian ocular lens containing closely packed fibre cells (McNulty et al. 2004), finds a diffusion coefficient "indistinguishable from that of oxygen in water" using a spherically symmetric diffusion model including consumption. Since our model includes specific terms for all oxygen consumption by both cumulus mass and oocyte, we conclude that the value of the diffusion coefficient appropriate to our model is that for oxygen diffusion in pure water at 37 ° C, which is also the value quoted for oxygen diffusion in physiological saline.
The significance of our results should have impact in several areas of oocyte biology and maturation. First, as already mentioned, in studying O2 diffusion across a preantral follicle, Gosden & Byatt-Smith (1986), with reference to Jones and Kennedy (1982), used a diffusion coefficient of 7 x 10–12 m2/s, almost 3 orders of magnitude smaller than that of O2 in water. This results in a considerably steeper concentration gradient across the oocyte–granulosa cell complex, and hence a much lower oxygen concentration at the oocyte–granulosa cell boundary, than is obtained using the diffusion coefficient for O2 in water (data not shown). The small diffusion coefficient was, in fact, obtained by comparing experimental data for rat hepatocytes with predictions from a model that assumes no oxygen consumption in a rat hepatocyte excepting for that by an isolated mitochondrion at the centre of the cell (see Jones & Kennedy, 1982). This will result in a significant underestimation of the diffusion coefficient and, consequently, low predictions of oxygen concentrations from models that use it. Hence, the results from Gosden & Byatt-Smith (1986), which suggest that the developing oocyte in the pre-antral follicle is certainly anoxic, should be reviewed.
Secondly, there is no consensus as to the optimal O2 concentration for in vitro maturation of oocytes. The early embryo appears to be sensitive to either atmospheric or hypoxic O2 levels in several mammalian species, resulting in decreased development (e.g. Thompson et al. 1990). Intermediate concentrations of O2, which appear to approximate in vivo concentrations, are beneficial for development (Harvey et al. 2002). Therefore, it seems likely that O2 levels for in vitro maturation should follow a similar trend, in that levels approximating in vitro concentrations would be more likely to be beneficial for oocyte maturation, rather than the majority current practise of the use of atmospheric levels. Recently, Hashimoto et al.(2000) demonstrated that for in vitro maturation of bovine oocytes in the presence of 20mM glucose, 5% O2 during maturation yielded more blastocyst-stage embryos than 20% O2. This also correlated with a reduction in the intra-oocyte level of reactive oxygen species. The relatively high glucose level was required to overcome deficiencies in meiotic maturation, which are glucose concentration-dependent during in vitro maturation (Sutton-McDowell et al. 2005). Such changes in substrate concentration may alter oxygen consumption rates by the COC. However, we believe that such changes would lead to only small deviations in oxygen uptake, as many more molecules of substrates such as pyruvate and glucose are oxidised for every additional molecule of oxygen consumed.
One critical assumption made for our mathematical model is that the COC is close to spherical. This is clearly not the case, where a "stalk" of granulosa cells fixes the cumulus mass to the membrana granulosa. Around this region, one would expect the O2 concentration at the surface of the oocyte to be less than indicated by the values given in this paper, allowing the establishment of a polarised O2 concentration around the oocyte. Whether or not such a gradient has physiological relevance is unknown, although the concept of localised intracellular ATP production and polarity in mitochondrial distribution is clearly discussed in the literature (e.g. for review, see Van Blerkom 2004).
In conclusion, we have developed a mathematical model to describe the perfusion of oxygen across the COC. This model reveals that follicle lumen concentrations of oxygen most likely reflect what is seen by the oocyte itself. Although it is known that the cumulus-free oocyte is dependent on oxidative phosphorylation for developmental competency (Van Blerkom et al. 1995), dependency of the oocyte on its own mitochondrial activity during maturation, especially when significant cell-communication occurs via gap junctions with the cumulus layer, is unknown (Van Blerkom 2004). However, even if oxygen consumption by the oocyte is less than the values utilised here, the gradient of O2 would be even less than our current model predicts. Overall, we believe our model makes physiological sense and therefore may be useful in determining the sensitivity of maturing COCs to different oxygen concentrations. Furthermore, our model can also be applied to other questions relating to diffusion of molecules across spherical structures occurring within reproduction, such as follicles and early embryos.
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Footnotes |
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References |
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TopAbstractIntroductionMaterials and MethodsResultsDiscussionAcknowledgementsReferences |
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