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First published online February 15, 2008
Journal of Experimental Biology 211, 790-797 (2008)
Published by The Company of Biologists 2008
doi: 10.1242/jeb.014613
Oxygen profiles in egg masses predicted from a diffusion–reaction model
1 Division of Biological Sciences, University of Montana, Missoula, MT 59812,
USA
2 Department of Biological Sciences, Clemson University, Clemson, SC 29634,
USA
* Author for correspondence (e-mail: art.woods{at}mso.umt.edu)
Accepted 8 January 2008
 |
Summary |
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 TOP Summary INTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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Key words: oxygen, diffusion coefficient, egg mass, metabolism, nudibranch, size, Antarctic, embryo
|
INTRODUCTION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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; Lee and Strathmann,
1998). For masses, unlike adults, O2 delivery problems
are defined by only a few parameters: egg-mass size and shape, embryo
metabolic rate, and O2 diffusion coefficients in egg-mass gel. This
simplicity has attracted attention from those working both on marine systems
(Strathmann and Chaffee, 1984;
Cohen and Strathmann, 1996;
Lee and Strathmann, 1998;
Moran and Woods, 2007;
Woods and Podolsky, 2007) and
on amphibian egg masses (Seymour,
1994; Seymour and Bradford,
1995; Mitchell and Seymour,
2003). For all egg masses, however, predicting transient temporal
and spatial profiles of O2 in masses remains difficult. First,
O2 concentrations at particular points in space and time reflect a
balance of diffusion and reaction; second, the kinetics of O2
consumption in biological systems often are non-linear (i.e.
Michaelis–Menten); and third, estimating O2 coefficients is
challenging, especially when O2 is consumed by metabolic reactions.
Defining time- and space-integrated O2 histories of embryos is
crucial to understanding O2 limitation in nature, and to predicting
how variation affects embryos over development.
Here we develop a model of how egg-mass size and shape, O2
diffusion coefficient, and embryo metabolic rate jointly affect O2
distributions in egg masses. We constructed the model to (1) predict full
radial and temporal profiles of O2 in egg masses, even with
nonlinear (Michaelis–Menten) reaction kinetics, and (2) analyze sources
of error in estimating O2 diffusion coefficients from experimental
data in which external step-changes in O2 concentration were
imposed on intact, living egg masses. Given information about mass size,
embryo O2 consumption rates, number of embryos per unit volume of
egg mass (`embryo density'), and diffusion coefficients, the model predicts
O2 concentrations anywhere in the structure and at any future time
after a change in external O2 concentrations. The model may be
broadly applicable to other biological situations, as the basic set of
mechanisms (oxygen diffusion and consumption) is responsible for establishing
oxygen profiles in other structures, including tumors
(Braun and Beatty, 2007),
engineered tissues (Brown et al.,
2007), insect eggs (Woods et
al., 2005), and vertebrate embryos early in development
(Kranenbarg et al., 2000).
Model development
In the marine literature, a frequently used model calculates maximal egg
mass size (thickness) at which central O2 just goes to zero:
 | (1) |
Although Eqn 1 has been quite
useful (see Lee and Strathmann,
1998; Woods, 1999;
Moran and Woods, 2007) (and
others), it suffers from several shortcomings. First, it predicts neither
transient O2 concentrations arising from changing conditions nor
O2 concentrations in non-central areas. Second, implicit in the
equation is the assumption that embryo metabolic rate is constant for all
O2 levels above zero. Although approximately true when the
half-saturation constant is low, this assumption can lead to serious
underestimates of Rmax when it is not. The new model
developed below provides a more flexible framework for tracking spatial and
temporal changes in O2 and for incorporating different kinds of
reaction kinetics. Similar approaches to analyzing oxygen in egg masses, using
both iterative numerical modeling and finite element analysis, have been
described (Seymour and White,
2006).
Consider an infinite cylinder, in which mass transfer occurs in the radial
direction only. Diffusive transport is described by:
 | (2) |
) (see
Table 1). In radial
coordinates, this equation becomes:
 | (3) |
; Crank, 1975).
An additional transformation, from O2 concentration to partial
pressure, may sometimes be desirable. The transformation can be done by
substituting (into Eqn 3) partial
pressure, P, for concentration, C, and substituting Krogh's
coefficient of diffusion, K, for D, the diffusion
coefficient of O2 (K=Dβ and β is oxygen
solubility). K does a better job of describing how O2
transport changes with temperature, because it takes into account
temperature's effects on both diffusion coefficient (positive) and solubility
(negative).
|
We add metabolic consumption of O2 to the radial equation with
an extra term:
 | (4) |
 | (5) |
).
Another, non-linear possibility is Michaelis–Menten kinetics,
 | (6) |
; Palumbi and Johnson,
1982). For example, embryos of the sea urchin Arbacia
have constant metabolic rates down to approximately 10% of air saturation
(Tang and Gerard, 1932).
However, for embryos in gelatinous masses, gel and other materials around each
embryo will act like large boundary layers for local diffusion, which may make
the reaction kinetics appear more first-order. We explore both types of
kinetics below. We solved the equations numerically using a program written in the R statistical package (v. 2.3.1) with standard 3-point center differencing for second-order terms. The program takes arbitrary initial O2 distributions (radially symmetric) and, for a given cylinder size, O2 diffusion coefficient, reaction kinetics and surface concentration of O2 (which depends on temperature via the parameter β), calculates the radial profile of O2 concentrations at future times. Central O2 concentrations (r=0) are undefined in Eqn 4 and so were obtained by cubic-spline interpolation.
In numerical models tending to steady state, there is the question of when
states are in fact steady. In cylindrical diffusion, the approximate time
is given by:
 | (7) |
.
First-order versus saturating reaction kinetics
Embryo metabolic rates must change as O2 availability declines
from full air saturation (the usual condition under which embryo metabolism is
measured) to low or zero O2 within egg masses. The model allows
analysis of different relationships between oxygen and metabolic rate. In
particular, it shows that for equivalent metabolic rates at air saturation
(here assumed to be air saturation at 11°C in seawater),
Michaelis–Menten kinetics draw O2 down further than linear
kinetics (Fig. 1), because high
reaction rates are sustained at low O2 concentrations. At high
rates of O2 consumption, moreover, Michaelis–Menten kinetics
gives different profiles and, with high Vmax, larger
central areas of anoxia. Overall, though, profile shapes were not particularly
different.
|
 | (8) |
<0.1
metabolism does not affect central O2; when >1,
there is a large effect (Fig.
2); and for 0.1<<1 the effect of metabolism on
central O2 is moderate. A similar dimensionless number can be used
with Michaelis–Menten kinetics, e.g.
Vmaxa2/DKm.
However, because the effect of O2 on metabolism is nonlinear, it is
difficult to develop general rules of thumb.
|
)
estimated O2 diffusion coefficients in egg-mass gel by measuring
central O2 concentrations during an externally imposed step-change
in O2 concentration (from air-saturated to N2-purged)
(see also Seymour, 1994).
Metabolic consumption by embryos was eliminated by microwaving masses briefly
to a high temperature. In such a no-reaction situation, the time course of
central O2 concentrations in cylinders is described by
(Crank, 1975):
 | (9) |
ns are roots of:
 | (10) |
When O2 is consumed, Eqn 9 is invalid: metabolic consumption accelerates the central loss of O2 during the external step-down and retards its rise during the external step back up, and the shapes of the two curves may differ. Here we explore how much they differ and whether the type of reaction kinetics (linear or Michaelis–Menten) affects how O2 is distributed. Because the estimates derived above are computationally intensive, we subsequently explore methods for correcting estimates of D fitted from simpler equations.
Consider first-order (linear) O2 consumption. For
=1 or 4 (moderate and large effects of metabolism, respectively;
see Fig. 2), the equilibrium
central O2 levels in air-saturated water were high and low,
respectively. In both cases, traces of central O2 levels during
external step-down were symmetrical with traces during external step-up
(Fig. 3A). By contrast,
Michaelis–Menten kinetics gave asymmetrical traces
(Fig. 3B): O2 levels
during step-down fell faster than they rose during step-up, and the asymmetry
was magnified at larger Vmax. Thus, information about both
equilibrium O2 levels in air-saturated water, and relative
asymmetry in central O2 traces during step changes, can be used to
evaluate whether metabolism in vivo is better represented by
first-order or Michaelis–Menten kinetics. An example based on real
O2 traces is developed below.
|
We used Eqn 4 to simulate
step-change data under both 1st-order and Michaelis–Menten kinetics.
Simulated data sets were then used to compare simpler methods of estimating
D to results from our fully fitted model. In particular, we fitted
Eqn 9 (which ignores metabolism)
to the traces, with the key assumption that the surface O2
concentration at t=0 was equivalent to the central O2
concentration at t=0. For first-order kinetics, data were simulated
across a range of from 0 to 10, using
D=3x10–6. Subsequent fits of
Eqn 9 to data traces were done
with the nonlinear least-squares function in R. As increased
(increasingly high metabolic rate), estimated D
(Dest) also rose; at =10,
Dest was approximately twice D. To avoid having
to estimate directly, it is also possible to use the depression
of central O2 level as a proxy
(Fig. 4A). In a variety of
simulated conditions, 
80% depression in central O2 levels
doubled estimated D.
|
Because step-down and step-up-traces are (under first order kinetics) mirror images, it is immaterial which trace is used to estimate D. Under Michaelis–Menten kinetics, by contrast, the step-down and step-up-traces have different shapes, and give different estimates of D. How to estimate the true value of D? We simulated central O2 levels after step changes up or down using a range of Vmax and either of two Km values (15 or 45). The degree of divergence between down-versus up-estimates increased with Vmax (Fig. 4B). Usually, however, up estimates were much closer to true simulated values than were down-estimates. In the model context, and likely too in real situations, these results suggest that D should be estimated by stepping from N2-purged water to air-saturated water, rather than vice versa.
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MATERIALS AND METHODS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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) using
ultra-low-melt agarose and fertilized embryos of the Antarctic sea urchin
Sterechinus neumayeri (Meissner). Adult S. neumayeri were
collected on SCUBA from McMurdo Sound and returned to McMurdo Station, where
they were spawned and gametes fertilized following Strathmann
(Strathmann, 1987). Embryos
were reared in stirred cultures of 0.5 µm filtered seawater at
–1.5°C. When they were 4.5 days old (just prior to hatching), 10 ml
of a known concentration of embryos was added to 20 ml of 2% ultra-low-gelling
temperature agarose (Fluka BioChemika, Buchs, Switzerland; gelling temperature
4–6°C) cooled to 3°C with stirring. This solution was drawn into
cylindrical molds, allowed to solidify, gently removed from the mold, and kept
in –1°C running seawater tables until used.
We made masses of three diameters (3.4, 6.7 and 11.8 mm) and three
densities (0, 1.25 or 12.5 embryos µl–1) in a
full-factorial design. Following published methods
(Moran and Woods, 2007), we
measured metabolic rates of 5-day-old embryos (from the same batch used to
make artificial masses) at –1.5 and +1.5°C using a µBOD method
and radial profiles of O2 at –1.5 and +2°C using a
Clark-style O2 microelectrode (Unisense A/S, Aarhus, Denmark). In
addition, O2 diffusion coefficients in embryo-free masses were
estimated using step-change experiments and fitting
Eqn. 9 to traces of central
O2 concentration over time (see previous section; fits done using
the R statistical package
http://www.r-project.org).
All measured parameters were put into the full numerical model
(Eqn 4), assuming that embryos
followed Michaelis–Menten kinetics. Modeled Vmax was
chosen so that it gave measured metabolic rates at air saturation.
km was unknown but assumed to be low (=20 nmol
O2 cm–3).
Estimating D in real egg masses
Artificial egg masses can be constructed without embryos, allowing
straightforward estimation of diffusion coefficients from step-change
experiments. Real egg masses are not so conveniently designed. One solution is
to stop metabolism by killing embryos, e.g. by microwaving masses briefly to
high temperature (Cohen and Strathmann,
1996). However, for cold-temperature Antarctic masses we were
concerned that microwaving, exposure to high temperatures, or other chemical
or physical means of killing embryos might affect gel structure and alter
O2 diffusion coefficients. Our model provides an alternative method
to estimate D without killing embryos. To test the model in natural
systems, we used egg masses of two congeneric nudibranch molluscs,
Tritonia diomedea Bergh 1884 and T. challengeriana Bergh
1884, and of another Antarctic species, Tritoniella belli Bergh 1884.
T. diomedea is common subtidally along the Pacific coast of North
America. We collected adult T. diomedea on SCUBA in Puget Sound (WA,
USA), primarily from Squamish Harbor, returned them to the Friday Harbor
Laboratories, and kept them in running seawater tables (11°C). Many
individuals subsequently laid egg masses in the tables. We collected T.
challengeriana and Tritoniella belli egg masses on SCUBA in
McMurdo Sound, Antarctica, and kept them in running seawater tables (approx.
–1°C) at McMurdo Station. Neither Tritonia challengeriana
nor Tritoniella belli would lay egg masses in the laboratory, so we
used field-collected masses. Positive identification of egg masses to species
was made by observations of adults spawning in the field, and also by later
observations of laboratory spawning events.
For all three species, we measured central O2 concentrations
using calibrated microelectrodes as described previously
(Moran and Woods, 2007). After
the microelectrode was positioned, egg masses were subjected to step changes
in external O2 concentration; the `step-down' was a change from
air-saturated to N2 purged sea water, and the `step-up' was from
N2 purged to air-saturated. Step changes were imposed by
withdrawing chamber water into a syringe and immediately replacing it with
temperature-equilibrated seawater (either N2 purged or
air-saturated, depending on the direction of the change). After a step-down,
to avoid O2 contamination, additional N2 was bubbled
into the egg-mass chamber and the water surface was isolated from the ambient
air. After a step-up, air was bubbled into the chamber. Bubbling also stirred
water around egg masses, thereby reducing boundary layers
(Lee and Strathmann, 1998).
Individual runs typically lasted 1 h (temperate measurements) or 3–4 h
(Antarctic measurements). Drift from O2 consumption by electrodes
was negligible, as (i) equilibrium plateaus in egg-mass O2 levels
were very stable over time; (ii) the stirring effect, measured during
calibration in bubbled versus still water, was 1–2% at most;
and (iii) electrode tips consumed O2 at rates well below rates
measured for individual embryos in masses.
For all three species, paired pieces of individual egg masses (several cm long, cut ends tied with dental floss) were equilibrated in either cold or warm temperatures. For the temperate species, T. diomedea (N=7 paired mass pieces), the two experimental temperatures were 12°C and 22°C. For the Antarctic species, T. challengeriana (N=5 pairs) and Tritoniella belli (N=7 pairs), experimental temperatures were –1.5°C and + 2.0°C. Metabolic rate measurements of these species can be found in the accompanying paper (Woods and Moran, 2008b).
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RESULTS |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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),
namely that larger egg mass diameter, higher embryo densities and higher
temperature all depressed central O2 levels.
|
). Modeled radial profiles of O2 (Fig. 5B) were remarkably similar to measured profiles (Fig. 5A). The main difference was that modeled profiles at +2°C were not quite as low as measured values.
Estimating D in real egg masses
Good fits were obtained by fitting the simple model
(Eqn 9) to central O2
concentrations in real egg masses (Fig.
6). In general, small initial O2 depression gave
symmetrical traces, and up- and down-traces gave similar estimates of
D. By contrast, large initial O2 depression gave both
asymmetrical traces and much higher estimates of D for the down-trace
than for the up-trace (see Fig.
3B). Across all three species at their two experimental
temperatures, the relationship (Fig.
7) between initial O2 depression and divergence in
D was consistent with predictions of the full radial model with
Michaelis–Menten reaction kinetics (see
Fig. 4B). We estimated
D in real egg masses of the three species using just up-traces
(Fig. 8). Estimates of
D were similar across species and temperatures, essentially always
lying between 7–10x10–6 cm2
s–1. However, temperature (warm vs cold within each
species) had no statistically discernable effect on estimated D
values, although the higher temperatures did tend toward higher D. D
in egg masses of the two Antarctic species was very close to independently
established values for D in pure seawater
(9.9x10–6 cm2 s–1)
(Denny, 1993). In the temperate
species, Tritonia diomedea, estimated D values were less
than half their values in seawater at the two (warmer) temperatures.
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DISCUSSION |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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), in that greater mass diameter, higher embryo densities, and
higher temperature all resulted in greater depression of central O2
concentrations.
Estimating D in real egg masses
The logic and mathematics of estimating D in systems without
metabolism are well developed. Most biological systems, however, violate the
no-metabolism assumption to such an extent that simple mathematics are
inadequate. How then should one estimate D? We approached this
problem first by developing a full numerical model describing O2
transport and consumption, with the option to specify different types of
reaction kinetics. We then used the full model to simulate data that could be
analyzed by simpler means. This approach showed that when
Michaelis–Menten kinetics prevail, D is estimated accurately
from the up-trace but not the down-trace.
We applied this conclusion to estimating O2 diffusion
coefficients in egg masses of three nudibranch species, Tritonia
challengeriana, Tritonia diomedea and Tritoniella
belli. Down- and up-traces were highly asymmetrical, indicating that
metabolism was better described by Michaelis–Menten than first-order
kinetics. From up-traces only, D values for all three species were
estimated to be 8x10–6 cm2
s–1. Our analysis shows that potential errors from using
down-traces to estimate D can be large.
Fig. 7 shows that the
divergence in estimates of D between down- and up-traces increases
approx. linearly with degree of initial O2 depression. At the
highest levels of depression (e.g. only 25% of air saturation), estimates of
D from down-traces were 15–20x10–6
cm2 s–1 higher than from up-traces
(9x10–6 cm2 s–1), i.e.
overestimated by 200–300%.
The fitted values of D (Fig.
8) suggest two biological conclusions. First, egg-mass material
itself does not depress O2 diffusion coefficients much below their
values in seawater, particularly in the two Antarctic species. This result
matches Cohen and Strathmann's finding
(Cohen and Strathmann, 1996)
that D in egg masses of the opisthobranch Melanochlamys
diomedea, at 20°C, ranged between 15 and
20x10–6 cm2 s–1
(75–95% of the value of D in seawater at 20°C).
Interestingly, D for M. diomedea was higher than the values
we obtained here for T. diomedea at 22°C
(9x10–6 cm2 s–1). Why
egg masses of the T. diomedea did not have higher D is
unclear.
Second, in none of the three species did temperature have any substantial
effects on D. One could object that a temperature change from
–1.5 to 2°C would be unlikely to have dramatic effects anyway.
However, other measurements of embryo metabolic rates [reported in the
accompanying paper (Woods and Moran,
2008)] showed that, for at least some Antarctic species, such
small temperature shifts stimulated metabolism two- to threefold, which was
equivalent to the degree of metabolic stimulation that we found across a
9°C increase for the temperate embryos of T. diomedea
(Moran and Woods, 2007). Also,
here we measured D in egg masses of T. diomedea across the
larger temperature increment and found no statistically significant effect. It
thus appears that O2 diffusion coefficients in egg-mass gel are
relatively insensitive to temperature.
These data support the idea that changing temperature has larger effects on
metabolic consumption than on diffusive transport of O2
(Woods, 1999). This assumption
stems from known physical effects of temperature on diffusing molecules: all
else being equal, temperature will have only a weakly positive effect on
diffusive transport. In biological systems, however, all else may not be
equal; in particular, increasing temperature could alter physical or chemical
properties of the matrix through which diffusion occurs. For egg masses,
however, our findings here support the assumption that simple, physical models
of temperature's effects on diffusion are adequate.
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APPENDIX |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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), we lay out
equations for these two other idealized shapes. Other solutions are available
for variations on the scenarios below
(Carslaw and Jaeger, 1959;
Crank, 1975).
Plane sheets
For an infinite sheet of thickness 2l, initially at uniform
concentration C0 (in –l<x<l)
and the two surfaces kept at constant concentrations C1,
the solution is:
 | (A1) |
Spheres
For a sphere with initial uniform oxygen concentration
C0 and surface concentration fixed at
C1, the solution is:
 | (A2) |
0:
 | (A3) |
). |
Acknowledgments |
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References |
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TOPSummaryINTRODUCTIONMATERIALS AND METHODSRESULTSDISCUSSIONAPPENDIXReferences |
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gives
larger initial drawdown (see 
