AMER. ZOOL., 39:244-252 (1999)
Egg-Mass Size and Cell Size: Effects of
Temperature on
Oxygen Distribution1
2 3
H. ARTHUR WOODS -
Department of Zoology University of Washington Seattle, Washington 98195
SYNOPSIS. TWO processes strongly influence the distribution of oxygen within egg
masses and cells: the supply of oxygen by diffusion and the consumption of oxygen
by embryos and mitochondria. These processes are differentially sensitive to temperature. The diffusion coefficient of oxygen depends only weakly on temperature,
having a Ql0 of approximately 1.4. In contrast, the consumption of oxygen depends
strongly on temperature, having a Ql0 between 1.5 and 4.0. Thus, at higher temperatures, the ratio of oxygen supply to demand decreases. I show, by extending
a model of oxygen distribution within metabolizing spheres, that maximal egg-mass
sizes and cell sizes are predicted to be smaller at higher temperatures. For egg
masses, definitive data are not yet available. For ectothermic cells, this prediction
appears to be supported; cells from a variety of ectothermic organisms, unicellular
and multicellular, are smaller when the cells are produced at warmer temperatures. Establishing a specific connection between this pattern and oxygen distributions requires demonstration of (1) oxygen concentration gradients within metabolizing spheres and (2) central oxygen concentrations low enough to affect function. Egg masses from a variety of taxa show steep oxygen concentration gradients
and often are severely hypoxic or anoxic in central locations. Severe hypoxia appears capable of retarding development or killing embryos. Similar kinds of data
for ectothermic cells have not yet been collected, but the literature on oxygen
gradients within mammalian cells suggests that intracellular gradients may be important.
hypoxia or anoxia near their centers, and
therefore that spheres produced at higher
temperatures should be relatively smaller
than those produced at lower temperatures
(von Bertalanffy, 1960; Bradford, 1990; Atkinson and Sibly, 1997a, b).
This approach to size is rooted firmly in
biophysics. Another common approach is to
consider temperature and size in the context
of the life history (Stearns, 1992)—for example, by focusing on temperature's effects
on growth rate, asymptotic size, and size at
maturity (e.g., Berrigan and Charnov, 1994;
Ernsting, 1995; Perrin, 1995; Svenster,
1995; Atkinson and Sibly, 1996). In some
cases these two approaches have been applied to the same problem. Amphibian egg
size, for example, appears to vary inversely
with
temperature both inter- and intra-spe1
From the symposium Aquatic organisms, terrestricifically
(Berven, 1982; Bradford, 1990).
al eggs: early development at the water's edge presented at the annual meeting of the Society for Inte- Bradford (1990) has suggested that this regrative and Comparative Biology, 3-7 January 1998, lationship may be related to gas exat Boston, Massachusetts.
change—that eggs at high temperature are
2
E-mail: [email protected]
constrained to be small by relatively high
1
Present Address: Department of Biology, Arizona
metabolic
rates of embryos. Berven (1982),
State University, Tempe, AZ 85287-1501.
INTRODUCTION
Despite vast differences in scale, gelatinous egg masses and ectothermic cells
share two simple and fundamental properties. With isometrically increasing size, volume increases more rapidly than does surface area. And with increasing temperature,
oxygen consumption increases more rapidly
than does oxygen supply (von Bertalanffy,
1960). Together, these observations indicate
that size and temperature should strongly
affect the distribution of oxygen within metabolizing spheres (egg masses and cells).
In this paper, I will suggest that relatively
large spheres living at relatively high temperatures are likely to experience severe
244
TEMPERATURE, EGG-MASS SIZE, AND CELL SIZE
in contrast, has discussed tradeoffs between
egg size and egg number, adult growth, and
future reproduction, and has suggested that
low temperatures, by prolonging the embryonic period and the time to feeding, may
select for large eggs. In general, biophysical
and life-history approaches are complementary; biophysical theory determines what is
possible, and life-history theory analyses
why traits occupy particular positions within these bounds.
The central idea of the biophysical hypothesis presented here, that oxygen consumption rises more rapidly with temperature than does oxygen supply, is a specific
instance of a general principle known for
much of the century, viz., rates of reaction
are more sensitive to temperature than are
physical processes such as diffusion and
electrical conductivity. Snyder (1908, 1911)
applied this principle in his studies of factors controlling various biological processes. He suggested, for example, that the relatively high <2io of smooth muscle shortening time (2.0) reflected an underlying
chemical reaction and that the relatively
low Qw of smooth muscle relaxation time
(1.4) reflected an underlying physical process, such as diffusion. These ideas were
discussed extensively by Belehradek (1935;
includes references on much of the early
work) and recast specifically in the context
of animal size by von Bertalanffy (1960),
who argued that animals at higher temperatures grow more rapidly toward a smaller
final size because of the different <2,os of
catabolism and anabolism. The connection
between temperature and size via temperature's effects on reaction and diffusion has
been discussed more recently by Bradford
(1990), van der Have and de Jong (1996),
and Atkinson and Sibly (1997a, b).
Below, I extend a simple mathematical
model of oxygen diffusing into a metabolizing sphere (Harvey, 1928; Krogh, 1941;
Boag, 1969; Dejours, 1975; Lee and Strathmann, 1998). The model calculates the critical radius at which the oxygen concentration at the center of the sphere just drops to
zero. By making the three parameters (the
solubility, diffusion coefficient, and consumption rate of oxygen) temperature sensitive in a biologically realistic way, I show
245
that the critical radius is predicted to decrease with temperature. I connect the model result to egg-mass and cell size by suggesting that (a) both structures are likely to
exhibit significant radial concentration gradients of oxygen under some conditions,
and (b) under these conditions, the oxygen
concentration in the center probably falls
low enough to disrupt function.
A MODEL OF SPHERE SIZE
Consider a metabolizing sphere, representing a roughly globular gelatinous egg
mass or cell (other shapes are discussed below). Because the contents of the sphere
consist mostly of non-convective materials—mucopolysaccharides or aqueous cytoplasm—oxygen arrives at the center of
the sphere principally by diffusion (Jones,
1986; Strathmann and Strathmann, 1995).
In these spheres, incoming oxygen is consumed by oxidative respiration at sites distributed throughout the non-convecting material—embryos in egg masses and mitochondria in cells. The distribution of oxygen within the sphere will be determined by
the relative magnitudes of oxygen supply
and consumption. Clearly, oxygen concentrations will be relatively high (approaching
ambient) near the surface of the sphere and
will be lower toward the interior.
For such a metabolizing sphere, it is possible to calculate the radius at which the
oxygen concentration at the center just
drops to zero. This radius defines a size limit; a sphere with a larger radius must experience anoxia in some finite region of its
center, with certain disruption of egg-mass
or cellular functions that depend on aerobic
respiration. Here I calculate the critical radius C?max) using an equation developed by
Harvey (1928) and Gerard (1931) for analysis of oxygen distributions within cells and
recently applied to egg masses by Lee and
Strathmann (1998):
(1)
where Co is the concentration of oxygen at
the surface of the sphere, D is the diffusion
coefficient of oxygen, and M is the rate of
consumption of oxygen per unit volume of
246
H. ARTHUR WOODS
the sphere. From Eq. 1, it is clear that a
higher surface oxygen concentration or oxygen diffusion coefficient would permit a
larger maximum radius and, in contrast,
that a higher metabolic rate per volume
would force a smaller maximum radius.
The equation is applicable to both egg masses and cells because the equation's parameters differ with scale. Most importantly,
the per-volume rate of oxygen consumption
by an egg mass (embryos separated by nonrespiring gel) is vastly lower than the pervolume rate of oxygen consumption by a
cell.
I extend the model above by considering
how Rmax depends on temperature. The
three parameters in the equation above, Co,
D, and M, are sensitive to temperature, each
in a different way. (1) In the absence of
boundary layers (a simplifying assumption), the surface concentration of oxygen,
Co depends primarily on oxygen solubility
in water. Solubility is well known to decrease with temperature (Fig. 1A; Carpenter
1966), having a Ql0 of roughly 0.85 between 10 and 20°C. For an egg mass in air,
the boundary layer will be quite small, and
the surface concentration of oxygen will depend on the solubility of oxygen in the outermost fluid layer. (2) The diffusion coefficient of oxygen in water rises slowly with
temperature (Fig. IB; Seymour, 1994); Qw
approximates 1.4 between 10 and 20°C.
The absolute value of D within egg-mass
gel and aqueous cytoplasm appears to be
lower than it is in water (Burggren, 1985;
Jones, 1986; Cohen and Strathmann, 1996).
However, unless egg-mass gel or cytoplasm
undergoes some structural change with increasing temperature, the temperature-dependence of D in these substances is likely
to be similar to the temperature-dependence
of D in water. (3) The oxygen consumption
per unit volume (AT) rises rapidly with temperature, having a Qw between 1.5 and 4
(Fig. 1C). A wide variety of ectotherms follow this pattern, and it is likely that embryos in egg masses and individual cells within
ectotherms do so as well. At stressfully
high temperatures, oxygen consumption
rates may fall with temperature; in this paper, however, I will restrict attention to the
10
20
30
40
Temperature (°C)
FIG. 1. Effect of temperature on (A) the solubility of
oxygen in seawater, (B) the diffusion coefficient of oxygen in frog egg jelly, and (C) the rate of consumption
of oxygen by resting goldfish. Solubility data are redrawn from Table 3 in Carpenter (1966). Diffusion
data are redrawn from Table 1 in Seymour (1994); values were obtained by multiplying diffusivities in water
by 0.76. Metabolic data are redrawn from Fry and Hart
(1948). Lines are least-squares fits of second- or thirdorder polynomials.
lower, non-stressful part of the operating
range.
The critical observation is that oxygen
supply (determined jointly by Co and D) rises slowly with temperature, but that oxygen
consumption (M) rises rapidly with temperature (Fig. 2). Thus, at higher temperatures,
the oxygen gradient within a metabolizing
sphere will be steeper, and the radius at
which the oxygen concentration at the center falls to zero (/?max) will be smaller. This
result can be obtained analytically by rewriting the parameters in Eq. 1 to make
them temperature-sensitive:
247
TEMPERATURE, EGG-MASS SIZE, AND CELL SIZE
QMIO
0.8
1.5
of 0.6
i
2.5
3.5
0.4
oi 0.2
40
10
20
30
Temperature (°C)
FIG. 2. Effect of temperature on solubility of oxygen
in seawater, diffusion coefficient of oxygen in frog egg
jelly, and oxygen consumption rate of resting goldfish.
Data are the same as in Figure 1, except they are rescaled by their values at 10°C.
C = C r e f G c l 0 <™"°;
10
15
20
25
Temperature (°C)
30
FIG. 3. Model (Eq. 3) predictions of relative Rmax (the
radius at which the oxygen concentration just drops to
zero at the center of the sphere) as a function of temperature. Predictions are scaled to their values at 10°C.
Each line represents predictions based on a different
Qw for oxygen consumption (QMIO)- T n e GioS of the
solubility of oxygen and the diffusion coefficient of
oxygen are set to 0.85 and 1.4, respectively. The single
values for the Qlos a r e simplifying assumptions; it can
be seen from Figure 1 that the (2ios vary with temperature.
M = MTefQM^T-T"')no
(2a-c)
where Tref is the reference temperature, Cref,
Dref, and MKi are the values of Co, D, and derestimate the true Rmax. Third, oxygen
M, respectively, at Tref, and QCIO, QDl0, and transport occurs homogeneously throughout
QMl0 are the g l 0 s of surface oxygen con- the sphere. In reality, transport of oxygen
centration, diffusion coefficient, and con- probably occurs at different rates in cytosumption rate, respectively. Substituting plasm and cellular lipids (Sidell, 1998). See
these equations into Eq. 1 and rearranging Tai and Chang (1974) and Dutta and Popel
(1995) for fuller theoretical treatments of
gives
transport within heterogeneous material.
(7--7V f)/10
Fourth, oxygen consumption is distributed
(3) homogeneously throughout the sphere. In
Q
fact, oxygen consumption is distributed unevenly, occurring in embryos within egg
where z = V6Cref DrJMKt, Because gM10 > masses and mitochondria within cells; how2cio £?DIO in most circumstances, Rmax must ever, local oxygen gradients around these
decrease with temperature (Fig. 3). This loci of consumption probably do not stronganalysis supports the verbal suggestions of ly influence global gradients from surface
von Bertalanffy (1960), Bradford (1990), to center (Jones, 1986; Seymour and Roberts, 1991). More significant violations will
and Atkinson and Sibly {1991 a, b).
Application of this model to egg masses occur when embryos or mitochondria are
and cells requires several simplifying as- clustered (Jones, 1986). Fifth, the rate of
sumptions. Although none of these assump- oxygen consumption is independent of amtions is likely to be absolutely true, minor bient oxygen concentrations above zero. Inviolations will not change the qualitative dividuals of many taxa respire at a constant
predictions of the model. First, boundary rate under a range of oxygen pressures—
layers around the sphere are small (see Lee for example, fertilized eggs of Arbacia and
and Strathmann [1998] for analysis and dis- larvae of Limulus polyphemus respire at escussion of this assumption). Second, the sentially constant rates above oxygen presmass-specific rate of oxygen consumption sures of 40 torr (Tang and Gerard, 1932;
is independent of mass. If mass-specific Palumbi and Johnson, 1982). Similarly, mimetabolic rate falls with increasing mass tochondria in mammalian cardiac myocytes
{e.g., Hemmingsen, 1960), Eq. 1 will un- operate normally when sarcoplasmic myoc
248
H. ARTHUR WOODS
globin saturation is above 30% and are not
severely inhibited until myoglobin is practically deoxygenated (Wittenberg and Wittenberg, 1989). However, because oxygen
consumption falls with concentration below
some low level, the actual radius at which
the concentration falls to zero (RmaK) will be
larger than that predicted by Eq. 3.
APPLICATION OF THE MODEL TO EGG
MASSES AND CELLS
The model predicts, as a function of temperature, the radius (/?max) at which the central oxygen concentration falls to zero.
Showing that this prediction is, in fact, relevant to egg-mass size and cell size depends on answering three empirical questions in the affirmative. First, are egg masses and cells smaller when produced at
higher temperatures? Second, are there radial concentration gradients of oxygen
within egg masses and cells? And third, do
oxygen concentrations at the centers of
these spheres fall low enough to affect performance?
Extension of the model to other shapes
The model developed thus far refers explicitly to spheres, that is gelatinous egg
masses or cells that are at least roughly
globular. However, many egg masses and
cells do not resemble spheres. Opisthobranch gastropods, for example, lay egg
masses in a spectacular diversity of shapes,
many cylindrical or thin and ribbon-like Are egg masses and cells smaller when
(Hurst, 1967). Likewise, cells—especially produced at higher temperatures?
those in metazoa—often occur in masses or
Definitive data on whether egg masses
epithelia rather than as isolated spheres.
Fortunately, equations similar to Eq. 1 exist are smaller when produced at higher temfor describing oxygen distributions in other peratures are not yet available. Two kinds
idealized shapes such as sheets and cylin- of data would be useful: comparisons of
ders (Boag, 1969; Lee and Strathmann, egg-mass sizes produced by populations or
1998). Furthermore, many frog embryos species inhabiting different thermal habitats
develop in an inner chamber surrounded by (e.g., different latitudes) and by individuals
an outer shell of jelly (Seymour and Brad- in experimentally altered thermal habitats.
ford, 1987), and in some frog egg masses A possible example of the former has been
(e.g., those of Chiromantis xerampelina; suggested by Moore (1949). In studying
Seymour and Loveridge, 1994), larvae geographic variation in populations of Rana
swim following hatching in a chamber pipiens, he found that egg masses from the
within an outer shell of foam; in both cases, New York area were globular and relatively
oxygen distribution is probably better de- large whereas those from Coahuila, Mexico
scribed by a model in which oxygen dif- consisted of many small masses, each confuses through an outer shell and is con- taining a few eggs.
In contrast, a number of studies suggest
sumed in an inner convection chamber
that
cells of ectothermic organisms reared
(Seymour and Bradford, 1987; Seymour
and Roberts, 1991; Seymour and Love- in relatively warm environments are smaller
ridge, 1994). All of these equations may be than cells of conspecifics reared in cool enmade temperature-sensitive as I have done vironments. This response is well-known in
with Eq. 1, leading to size-vs-temperature Drosophila melanogaster (Alpatov, 1930;
predictions qualitatively similar to those ob- Robertson, 1959; Partridge et al., 1994)—
for example, flies reared at 25°C had wing
tained above (unpublished).
Therefore, the qualitative prediction ob- cells that were about 79% the area of those
tained from Eq. 3—that i?max is smaller at in flies reared at 16.5°C (Partridge et al.,
higher temperatures—does not depend 1994). Nematodes reached a smaller adult
strongly on geometry. As a rule of thumb, size when reared at high temperatures (Van
this prediction will obtain whenever oxygen Voorhies, 1996); and because nematode cell
diffusion is an important component of the number is likely to be invariant across temsupply pathway and whenever oxygen con- peratures, the effect is probably due to
sumption rises more rapidly with tempera- changes in cell size (Van Voorhies, 1997).
Red blood cells from individual cichlids acture than does oxygen supply.
TEMPERATURE, EGG-MASS SIZE, AND CELL SIZE
climated to 20°C were about 78% the size
(measured as area) of red blood cells from
individuals acclimated to 10°C (Van Voorhies, 1996). Similarly, planarians reared at
22°C had cells that were about 80% the size
of those in animals reared at 7°C (Romero
and Baguna, 1991). Unicellular organisms
apparently also show this pattern: in a survey, Atkinson (1994) reported that one of
one bacteria and six of seven protists
showed reduced size at high temperatures.
249
Kennedy and Jones, 1986; Takahashi and
Doi, 1995). Although the magnitudes of
these gradients are controversial (Jones;
1986; Wittenberg and Wittenberg, 1989),
they appear to be at least 1—2 torr under
some conditions. Similar data on ectothermic cells will be necessary before deciding
whether intracellular oxygen gradients in
ectotherms are substantial.
Do central oxygen concentrations fall low
enough to affect performance ?
Are there radial concentration gradients
Low central oxygen concentrations in
of oxygen within egg masses and cells ?
egg masses appear quite capable of retardEgg masses of both marine invertebrates ing development or killing larvae. Larval
(Booth, 1995; Cohen and Strathmann, gastropods near the egg-mass center gen1996) and frogs (Seymour and Roberts, erally develop less rapidly and hatch later
1991) show strong oxygen gradients from than larvae near the surface (Chaffee and
surface to center. For example, even early Strathmann, 1984; Booth, 1995). Asynchroduring embryogenesis oxygen concentra- nous development has also been observed
tions at the centers (3—5 mm from the sur- in egg masses of Rana sylvatica at high
face) of the globular, gelatinous egg masses temperatures—an effect which was reof Melanochlamys diomedea (an opistho- moved by bubbling the rearing water with
branch gastropod) were less than 70% of oxygen (Moore, 1940). Dead larvae and exthe oxygen concentration at the egg-mass treme hypoxia were coincident in the censurface (Cohen and Strathmann, 1996). In ters of egg masses of L. tasmaniensis (Seylater stages (gastrula-trochophore and veli- mour and Roberts, 1991).
With data presently available, deciding
ger) central oxygen concentrations fell to
zero. Booth (1995) described similarly whether cells of ectotherms regularly exsteep oxygen gradients in the large, sau- perience central oxygen shortage is not possage-shaped, gelatinous egg masses of Po- sible. However, oxygen concentrations at
linices sordidus. Likewise, a globular, non- the centers of mammalian cells appear to
foamy egg mass of the frog Limnodynastes fall low enough to disrupt function. Vantasmaniensis (embryos at stage 19) exhib- derkooi et al. (1991) compiled data on a
ited a steep oxygen gradient from the sur- variety of tissues from five mammalian speface (c. 21 kPa) to the center 1.5 cm away cies and found that many of them experi(0 kPa) (Seymour and Roberts, 1991). ence oxygen pressures less than 10 torr;
These measurements support the conclu- pressures adjacent to cell membranes may
sions of earlier work on frog egg masses be considerably lower (Jones, 1986; Witten(Savage, 1935; Moore, 1940; Barth, 1946; berg and Wittenberg, 1989). Even shallow
but see Burggren, 1985). Thus, the occur- gradients (1-2 torr), therefore, could signifrence of steep oxygen gradients within ge- icantly affect cellular function—especially
latinous egg masses appears to be wide- as the Kms of most oxygen-using enzymes
are higher than the physiological oxygen
spread.
Whether significant radial oxygen gradi- concentrations to which they are exposed
ents exist in ectothermic cells is unknown. (Vanderkooi et al., 1991). Again, similar
Some evaluation is possible by drawing on data on ectothermic cells have not yet been
the literature concerning gradients within collected.
However, a recent study of temperature
mammalian cells, in which measurements
made with a variety of optical techniques and egg size does not support the idea of
have demonstrated intracellular oxygen gra- central oxygen shortage at high temperadients (Tamura et al., 1978; Katz et ai, tures. Ernsting and Isaaks (1997) collected
1984; Wittenberg and Wittenberg, 1985; eggs from female beetles (Notiophilus bi-
250
H. ARTHUR WOODS
guttatus) housed at 10 or 20°C, finding that
eggs produced at 20°C were roughly 17%
smaller by dry weight (which implies, assuming equivalent densities between treatment groups, that radii of warm-produced
eggs were roughly 6% smaller). At a common developmental temperature of 20°C,
however, eggs produced at the two temperatures developed and survived at equivalent
rates. This result indicates that the oxygen
supply was adequate to both small and large
eggs and therefore cannot account for the
size pattern.
DISCUSSION AND CONCLUSIONS
The studies reviewed above suggest that
the biophysical hypothesis developed here
is plausible but that its generality remains
unclear. For egg masses, internal oxygen
gradients and the negative effects of severe
hypoxia on embryos appear to be well-established, but the existence of the pattern
(small size at high temperature) has not
been shown. For ectothermic cells, in contrast, the inverse relationship between size
and temperature is well-known, but the existence of intracellular oxygen gradients
and the disruptive effects of low oxygen
have not been demonstrated. Comparison of
metabolizing spheres at two vastly different
scales is useful, as it points clearly to areas
for further study.
In closing, I consider two additional issues outside the scope of the model. First,
besides size reduction, several other responses to high temperature may relieve
oxygen stress. For example, the respiratory
density could be lowered by embedding
fewer embryos within the same volume of
egg-mass gel (Strathmann and Chaffee,
1984; Lee and Strathmann, 1998) or by
maintaining fewer mitochondria within the
same volume of cytoplasm. The latter strategy may be reflected in several species of
fish, in which individuals acclimated to
high temperature have fewer mitochondria
in muscle cells than fish acclimated to low
temperature (Johnston and Maitland, 1980;
Sidell, 1983). Another potential response to
high temperature is shape change. For a
fixed volume of material and respiratory
density, the greatest possible ratio of surface area to volume will maximize oxygen
availability. Spring and summer breeding
frogs may use this strategy: egg masses of
three spring-breeding species (Rana sylvatica, R. palustris, and R. pipiens), which are
laid in cool water (10-15°C), are roughly
spherical or oval, whereas egg masses of
two summer-breeding species (/?. clamitans
and R. catesbeiana), which are laid in
warmer water (c. 25°C), are thin films containing widely dispersed eggs (Moore,
1940). It is not known whether, depending
on temperature, individual adults produce
egg masses of different shapes or whether
cells assume different conformations. Finally, some egg masses avoid diffusive constraints by generating heat-driven convection currents through interstitial channels
(Pinder and Friet, 1994; Seymour, 1995).
For example, late stage embryos of R. sylvatica have oxygen requirements that are
apparently met by the convection of large
volumes (>15 1/d) of pond water through
the spherical (>10 cm diameter) egg mass
(Seymour, 1995). Convective currents such
as these probably are unavailable to cells.
Second, although the analysis developed
here suggests that metabolizing spheres
should be small at high temperatures, this
reasoning does not address the inverse
question of why spheres should be large at
cold temperatures. For cells, it is not clear
that being large in the cold is adaptive. For
egg masses, the answer may be related to
protection. Masses protect embryos from a
variety of biotic and abiotic threats, including predation, infection, extreme salinity,
heat, desiccation, and ultraviolet radiation
(Todd, 1981; Pechenik, 1986; Biermann et
al., 1992; Grant, 1995; Woods and DeSilets, 1997). At cooler temperatures, therefore, larger egg masses may better protect
embryos without simultaneously causing
oxygen shortage.
ACKNOWLEDGMENTS
I thank K. Martin and R. Strathmann for
inviting me to participate in the egg-mass
symposium. D. Atkinson, D. Berrigan, C.
Breuner, T. Daniel, J. Hoekstra, R. Huey, J.
Kingsolver, G. Odell, R. Strathmann, members of the mathematical biology group in
the Department of Zoology at the University of Washington, and an anonymous re-
TEMPERATURE, EGG-MASS SIZE, AND CELL SIZE
viewer provided valuable discussion and/or
critique. This work was supported by an
NSF Predoctoral Fellowship and an NSF
Mathematical Biology Training Grant (BIR
92-56532).
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Corresponding Editor: Paul Verrell