Oxygen profiles in egg masses predicted from a diffusion–reaction model

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, 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 O₂consumption 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, O₂diffusion coefficient, and embryo metabolic rate jointly affect O₂distributions 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).

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 R_max 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).

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 O₂distributions (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 O₂concentrations at future times. Central O₂ concentrations(r=0) are undefined in Eqn 4 and so were obtained by cubic-spline interpolation.

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 V_max, larger central areas of anoxia. Overall, though, profile shapes were not particularly different.

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

A further goal was to use step-change data to estimate O₂diffusion 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 O₂ consumption. Estimates of D from such an approach will be biased by in situO₂ 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 O₂concentration at t=0 was equivalent to the central O₂concentration at t=0. For first-order kinetics, data were simulated across a range of

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 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 N₂-purged water to air-saturated water, rather than vice versa.

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 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 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 O₂ cm^–3).

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 challengeriananor 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 N<sub>2</sub> 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).

Measured radial profiles of O₂(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 O₂ levels.

Metabolic rates of larval S. neumayeri were 8.6 and 10.3 pmol O₂ larva^–1 h^–1 at –1.5 and+1.5°C, respectively. This difference corresponds to a Q₁₀ of 1.8, and we used this value of Q₁₀ to correct the higher metabolic rate to 2°C for use in the model, the temperature at which radial O₂ 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^–6cm² s^–1 at –1.5°C and +2°C,respectively. The known diffusion coefficient of O₂ in seawater at 0°C is 9.9×10^–6 cm² s^–1(Denny, 1993).

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.

Good fits were obtained by fitting the simple model(Eqn 9) to central O₂concentrations 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 7–10×10^–6 cm²s^–1. However, temperature (warm vs cold within each species) had no statistically discernable effect on estimated Dvalues, although the higher temperatures did tend toward higher D. Din egg masses of the two Antarctic species was very close to independently established values for D in pure seawater(9.9×10^–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.

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 O₂concentrations.

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 O₂transport 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 ∼8×10^–6 cm²s^–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 15–20×10^–6cm² s^–1 higher than from up-traces(∼9×10^–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 20×10^–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(∼9×10^–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.

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).

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.).