## SUMMARY

We developed a novel diffusion–reaction model to describe spatial and
temporal changes in oxygen concentrations in gelatinous egg masses containing
live, respiring embryos. We used the model in two ways. First, we constructed
artificial egg masses of known metabolic density using embryos of the
Antarctic sea urchin *Sterechnius neumayeri*, measured radial oxygen
profiles at two temperatures, and compared our measurements to simulated
radial oxygen profiles generated by the model. We parameterized the model by
measuring the radius of the artificial masses, metabolic densities (=embryo
metabolic rate×embryo density) and oxygen diffusion coefficients at both
ambient (–1.5°C) or slightly warmer (+1.5–2°C)
temperatures. Simulated and measured radial oxygen profiles were similar,
indicating that the model captured the major biological features determining
oxygen distributions. Second, we used the model to analyze sources of error in
step-change experiments for determining oxygen diffusion coefficients
(*D*), and to determine the suitability of simpler, analytical
equations for estimating *D*. Our analysis indicated that embryo
metabolism can lead to large (several-fold) overestimates of *D* if the
analytical equation is fitted to step-down-traces of central oxygen
concentration (i.e. external oxygen concentration stepped from some high value
to zero). However, good estimates of *D* were obtained from
step-up-traces. We used these findings to estimate *D* in egg masses of
three species of nudibranch molluscs: two Antarctic species (*Tritonia
challengeriana* and *Tritoniella belli*; –1.5 and +2°C)
and one temperate Pacific species (*Tritonia diomedea*; 12 and
22°C). *D* for all three species was approximately
8×10^{–6} cm^{2} s^{–1}, and there
was no detectable effect of temperature on estimated *D*. For the
Antarctic species, *D* in egg masses was 70–90% of its value in
seawater of similar temperature.

## INTRODUCTION

A common reproductive strategy among aquatic and marine animals is to embed
embryos in gelatinous masses (Strathmann,
1987; Lee and Strathmann,
1998). For masses, unlike adults, O_{2} delivery problems
are defined by only a few parameters: egg-mass size and shape, embryo
metabolic rate, and O_{2} 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 O_{2} in masses remains difficult. First,
O_{2} concentrations at particular points in space and time reflect a
balance of diffusion and reaction; second, the kinetics of O_{2}
consumption in biological systems often are non-linear (i.e.
Michaelis–Menten); and third, estimating O_{2} coefficients is
challenging, especially when O_{2} is consumed by metabolic reactions.
Defining time- and space-integrated O_{2} histories of embryos is
crucial to understanding O_{2} limitation in nature, and to predicting
how variation affects embryos over development.

Here we develop a model of how egg-mass size and shape, O_{2}
diffusion coefficient, and embryo metabolic rate jointly affect O_{2}
distributions in egg masses. We constructed the model to (1) predict full
radial and temporal profiles of O_{2} in egg masses, even with
nonlinear (Michaelis–Menten) reaction kinetics, and (2) analyze sources
of error in estimating O_{2} diffusion coefficients from experimental
data in which external step-changes in O_{2} concentration were
imposed on intact, living egg masses. Given information about mass size,
embryo O_{2} consumption rates, number of embryos per unit volume of
egg mass (`embryo density'), and diffusion coefficients, the model predicts
O_{2} concentrations anywhere in the structure and at any future time
after a change in external O_{2} 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 O_{2} just goes to zero:
(1)
where *R*_{max} is half the maximum egg-mass thickness,
*F* is a shape factor (6 for a sphere, 4 for an infinite cylinder and 2
for an infinite sheet), *D* is the diffusion coefficient of
O_{2}, *C*_{R} is the O_{2} concentration at
the surface of the egg mass, *M* is the metabolic rate of an embryo,
and *N*/*V* is the number of embryos per unit volume (embryo
density).

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 O_{2} concentrations arising from changing conditions nor
O_{2} concentrations in non-central areas. Second, implicit in the
equation is the assumption that embryo metabolic rate is constant for all
O_{2} levels above zero. Although approximately true when the
half-saturation constant is low, this assumption can lead to serious
underestimates of *R*_{max} when it is not. The new model
developed below provides a more flexible framework for tracking spatial and
temporal changes in O_{2} 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)
where *C* is concentration, *D* is diffusion coefficient,
*t* is time, and *x* is distance
(Crank, 1975) (see
Table 1). In radial
coordinates, this equation becomes:
(3)
Various time-dependent and steady-state solutions to this equation are well
known (Carslaw and Jaeger,
1959; Crank, 1975).
An additional transformation, from O_{2} 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 O_{2} (*K*=*D*β and β is oxygen
solubility). *K* does a better job of describing how O_{2}
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 O_{2} to the radial equation with
an extra term:
(4)
where *f*(*C*) takes on any biologically realistic shape. For
first-order kinetics,
(5)
where *k* is the reaction coefficient. Here, metabolic rate declines
linearly to zero as available O_{2} declines. For first-order
kinetics, a number of analytical solutions are available for different
idealized shapes (Crank, 1975).
Another, non-linear possibility is Michaelis–Menten kinetics,
(6)
which asymptotes at *V*_{max} at high O_{2}
concentrations and falls rapidly to zero at concentrations below
*K*_{m}, the half-saturation (Michaelis) constant. Many
biological systems sustain more-or-less constant metabolic rates down to low
O_{2} concentrations (Yanigasawa,
1975; 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 O_{2}
distributions (radially symmetric) and, for a given cylinder size,
O_{2} diffusion coefficient, reaction kinetics and surface
concentration of O_{2} (which depends on temperature *via* the
parameter β), calculates the radial profile of O_{2}
concentrations at future times. Central O_{2} 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 timeτ
is given by:
(7)
where *a* is cylinder radius. Systems with high metabolic rates arrive
at steady states in times <τ.

#### First-order *versus* saturating reaction kinetics

Embryo metabolic rates must change as O_{2} availability declines
from full air saturation (the usual condition under which embryo metabolism is
measured) to low or zero O_{2} 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 O_{2} down further than linear
kinetics (Fig. 1), because high
reaction rates are sustained at low O_{2} concentrations. At high
rates of O_{2} consumption, moreover, Michaelis–Menten kinetics
gives different profiles and, with high *V*_{max}, larger
central areas of anoxia. Overall, though, profile shapes were not particularly
different.

#### Relative influences of diffusion and reaction: when does reaction matter?

When reaction kinetics are first order, a dimensionless number,
(8)
can be used as a shortcut for determining when metabolism will significantly
depress central O_{2} levels. In general, when <0.1
metabolism does not affect central O_{2}; when >1,
there is a large effect (Fig.
2); and for 0.1<<1 the effect of metabolism on
central O_{2} is moderate. A similar dimensionless number can be used
with Michaelis–Menten kinetics, e.g.
*V*_{max}*a*^{2}/*DK*_{m}.
However, because the effect of O_{2} on metabolism is nonlinear, it is
difficult to develop general rules of thumb.

#### Estimating diffusion coefficients from step-change experiments

Using opisthobranch egg masses, Cohen and Strathmann
(Cohen and Strathmann, 1996)
estimated O_{2} diffusion coefficients in egg-mass gel by measuring
central O_{2} concentrations during an externally imposed step-change
in O_{2} concentration (from air-saturated to N_{2}-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 O_{2} concentrations in cylinders is described by
(Crank, 1975):
(9)
where *J*_{1}(*x*) is the Bessel function of the first
kind of order one, and α_{n}s are roots of:
(10)
For equal but opposite step changes, the time courses are mirror images of one
another. The Appendix gives equations for central O_{2} levels in both
plane sheets and spherical masses.

When O_{2} is consumed, Eqn
9 is invalid: metabolic consumption accelerates the central loss
of O_{2} 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 O_{2} 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) O_{2} consumption. For
=1 or 4 (moderate and large effects of metabolism, respectively;
see Fig. 2), the equilibrium
central O_{2} levels in air-saturated water were high and low,
respectively. In both cases, traces of central O_{2} 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): O_{2} levels
during step-down fell faster than they rose during step-up, and the asymmetry
was magnified at larger *V*_{max}. Thus, information about both
equilibrium O_{2} levels in air-saturated water, and relative
asymmetry in central O_{2} 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
O_{2} traces is developed below.

A further goal was to use step-change data to estimate O_{2}
diffusion coefficients (*D*) for different egg masses under contrasting
environmental conditions. Ideally, *D* would be calculated by fitting
the full model (Eqn 4) to the
data. This approach is feasible only if one has reliable measurements of
embryonic metabolic rate. An easier approach, developed here, is to fit a
simpler equation (e.g. Eqn 9)
that does not incorporate embryonic O_{2} consumption. Estimates of
*D* from such an approach will be biased by *in situ*
O_{2} consumption, but these biases can be mathematically
corrected.

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 O_{2}
concentration at *t*=0 was equivalent to the central O_{2}
concentration at *t*=0. For first-order kinetics, data were simulated
across a range of from 0 to 10, using
*D=*3×10^{–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*
(*D*_{est}) also rose; at =10,
*D*_{est} was approximately twice *D*. To avoid having
to estimate directly, it is also possible to use the depression
of central O_{2} level as a proxy
(Fig. 4A). In a variety of
simulated conditions, ∼80% depression in central O_{2} 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
O_{2} levels after step changes up or down using a range of
*V*_{max} and either of two *K*_{m} values (15
or 45). The degree of divergence between down-*versus* up-estimates
increased with *V*_{max}
(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 N_{2}-purged
water to air-saturated water, rather than *vice versa*.

## MATERIALS AND METHODS

### Construction of artificial egg masses and model parameterization

To test the model, we made artificial egg masses (for details, see
Moran and Woods, 2007) 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 O_{2} at –1.5 and +2°C using a
Clark-style O_{2} microelectrode (Unisense A/S, Aarhus, Denmark). In
addition, O_{2} diffusion coefficients in embryo-free masses were
estimated using step-change experiments and fitting
Eqn. 9 to traces of central
O_{2} 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 *V*_{max} was
chosen so that it gave measured metabolic rates at air saturation.
*k*_{m} was unknown but assumed to be low (=20 nmol
O_{2} 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
O_{2} 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 O_{2} 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 O_{2} concentration; the `step-down' was a change from
air-saturated to N_{2} purged sea water, and the `step-up' was from
N_{2} purged to air-saturated. Step changes were imposed by
withdrawing chamber water into a syringe and immediately replacing it with
temperature-equilibrated seawater (either N_{2} purged or
air-saturated, depending on the direction of the change). After a step-down,
to avoid O_{2} contamination, additional N_{2} 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 O_{2} consumption by electrodes
was negligible, as (i) equilibrium plateaus in egg-mass O_{2} 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 O_{2} 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).

## RESULTS

### Application of the model to artificial egg masses

Measured radial profiles of O_{2}
(Fig. 5A) were largely
symmetrical about egg mass centers, and the various factors had effects
expected from our previous experiments with temperate artificial masses
(Moran and Woods, 2007),
namely that larger egg mass diameter, higher embryo densities and higher
temperature all depressed central O_{2} levels.

Metabolic rates of larval *S. neumayeri* were 8.6 and 10.3 pmol
O_{2} larva^{–1} h^{–1} at –1.5 and
+1.5°C, respectively. This difference corresponds to a Q_{10} of
1.8, and we used this value of Q_{10} to correct the higher metabolic
rate to 2°C for use in the model, the temperature at which radial
O_{2} profiles were measured. Estimates of *D* from medium (6.7
mm) and large (11.8 mm) artificial masses gave values of
8.6×10^{–6} and 10.0×10^{–6}
cm^{2} s^{–1} at –1.5°C and +2°C,
respectively. The known diffusion coefficient of O_{2} in seawater at
0°C is 9.9×10^{–6} cm^{2} s^{–1}
(Denny, 1993).

Modeled radial profiles of O_{2}
(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 O_{2}
concentrations in real egg masses (Fig.
6). In general, small initial O_{2} depression gave
symmetrical traces, and up- and down-traces gave similar estimates of
*D*. By contrast, large initial O_{2} 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 O_{2} 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 7–10×10^{–6} cm^{2}
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
(9.9×10^{–6} cm^{2} 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.

## DISCUSSION

### Application of the model to artificial egg masses

Measured and modeled radial profiles were remarkably similar, suggesting
that the model is reasonable and that our measurements of its parameters were
sufficiently accurate. Radial profiles in Antarctic artificial egg masses were
qualitatively consistent with those found in artificial masses containing
temperate sea urchin embryos that were measured at higher temperatures
(13°C and 22°C) (Moran and Woods,
2007), in that greater mass diameter, higher embryo densities, and
higher temperature all resulted in greater depression of central O_{2}
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 O_{2}
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 O_{2} 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 ∼8×10^{–6} cm^{2}
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 O_{2} depression. At the
highest levels of depression (e.g. only 25% of air saturation), estimates of
*D* from down-traces were 15–20×10^{–6}
cm^{2} s^{–1} higher than from up-traces
(∼9×10^{–6} cm^{2} 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 O_{2} 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
20×10^{–6} cm^{2} 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
(∼9×10^{–6} cm^{2} 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 O_{2} 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 O_{2}
(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.

## APPENDIX

The equations developed in the text apply to cylindrical egg masses –
i.e. structures in which diffusive mass transport is radial. However, egg
masses of particular species may be better approximated as plane sheets (e.g.
dorid nudibranch masses) or spheres (e.g. the globular egg masses of the
opisthobranch *Melanochlamys diomedea*). Here, following Crank
(Crank, 1975), 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 2*l*, initially at uniform
concentration *C*_{0} (in –*l<x*<*l*)
and the two surfaces kept at constant concentrations *C*_{1},
the solution is:
(A1)

### Spheres

For a sphere with initial uniform oxygen concentration
*C*_{0} and surface concentration fixed at
*C*_{1}, the solution is:
(A2)
The central oxygen concentration is obtained by taking the limit as
*r*→0:
(A3)
which is a slightly rearranged version of the equation used by Cohen and
Strathmann to estimate *D* from data on globose egg masses
(Cohen and Strathmann,
1996).

## ACKNOWLEDGEMENTS

We thank the directors and staffs of both the Friday Harbor Laboratories and McMurdo Station for support. SCUBA support was provided by two fearless Dive Safety Officers, Rob Robbins (McMurdo) and Pema Kitaeff (FHL), and their help is greatly appreciated. Others provided invaluable laboratory help and additional SCUBA assistance, including Bruce Miller, Jim Murray, Erika Schreiber and Jon Sprague. Two anonymous reviewers significantly clarified the manuscript. We also thank Dr Creagh Breuner and Dr Peter Marko for extraordinary logistical support while the authors were in Antarctica. This work was supported by the National Science Foundation (ANT-0440577 to H.A.W. and ANT-0551969 to A.L.M.).

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