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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

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

H. Arthur Woods^1,* and Amy L. Moran²

¹ Division of Biological Sciences, University of Montana, Missoula, MT 59812, USA
² Department of Biological Sciences, Clemson University, Clemson, SC 29634, USA

View larger version (22K): [in this window] [in a new window]   Fig. 1. Model predictions of steady-state O2 profiles in cylindrical egg masses under two different kinds of reaction kinetics: (A) 1st-order and (B) Michaelis–Menten.

View larger version (4K): [in this window] [in a new window]   Fig. 2. Predicting the steady-state drawdown of O2 in the middle of a cylindrical egg mass under first-order reaction kinetics. A dimensionless number, , incorporating information on the reaction coefficient, egg mass size and O2 diffusion coefficient (=ka2/D), predicts how far central O2 levels will fall below surface concentration. At small , the relationship is transport-dominated; at large the relationship is metabolism-dominated.

View larger version (21K): [in this window] [in a new window]   Fig. 3. Model comparisons of 1st-order (A) versus Michaelis-Menten (B) reaction kinetics on time courses of central O2 concentrations during step-change experiments (i.e. external concentration changed from 280 to 0 nmol cm–3 and then vice versa). Under first-order reaction kinetics, larger gives larger initial drawdown (see Fig. 2), and the step-down and step-up-traces are always symmetrical. Analogously, under Michaelis–Menten kinetics, larger Vmax gives greater initial drawdown. However, here the step-down and step-up-traces are asymmetrical, with the asymmetry exaggerated by larger initial drawdown.

View larger version (12K): [in this window] [in a new window]   Fig. 4. Estimating D by fitting simple equations to complicated biological situations. When O2 is consumed by metabolic processes, estimating D in biological structures is non-trivial, as metabolism draws down interior O2 levels and alters the time course of change after external step changes. Here we simulated step change traces under both first-order and Michaelis–Menten reaction kinetics and then fitted a simple equation (Eqn. 9) to the simulated traces. Under first-order kinetics (A), estimated D was good when initial drawdown (or ) was small. At higher k, giving greater initial drawdown, fitted D increasingly overestimated the known D used in the simulations (3x10–6). Under Michaelis–Menten kinetics (B), estimated D was again good when initial drawdown was small. At higher Vmax, giving greater initial drawdown, the down- and up-traces were highly asymmetrical. In this case, fitted D from the up-trace always gave values closer to the known, simulated value.

View larger version (28K): [in this window] [in a new window]   Fig. 5. Measured (A) and modeled (B) radial profiles of O2 concentration in artificial egg masses. Masses were constructed from very low melting point agarose and 5-day-old fertilized embryos of the Antarctic sea urchin Sterechinus neumayeri. The model was parameterized using separately measured values of embryo metabolic rate, O2 diffusion coefficient, cylinder size, and embryo density. In the model panels (B), horizontal black lines represent expected O2 levels (air saturated throughout).

View larger version (17K): [in this window] [in a new window]   Fig. 6. Representative fits of Eqn 2 to central O2 concentrations in an egg mass (A) with little initial O2 depression (Tritonia challengeriana egg mass at –1.5°C; fits shown in green) or (B) with substantial initial O2 depression (Tritonia diomedea egg mass at 22°C; fits shown in orange). x- and y-axes are scaled differently in the two panels. When initial drawdown is slight, estimates of D from down- and up-traces are similar, whereas when initial drawdown is large they are divergent (see also Fig. 7).

View larger version (10K): [in this window] [in a new window]   Fig. 7. Divergence in estimated D from down- and up-traces as a function of initial O2 depression. As initial depression increased, traces became progressively more asymmetrical, leading to larger differences in estimated D. Regression line (y=3x10–5x–4x10–6) was fitted to the pooled data set.

View larger version (12K): [in this window] [in a new window]   Fig. 8. Estimated O2 diffusion coefficient (D) in egg masses of the three species in this study. Paired pieces of egg masses of Tritonia challengeriana and Tritoniella belli were subjected to either –1.5 or +2.0°C, and egg masses of Tritonia diomedea were subjected to 12.2 or 22.3°C. The longer broken line represents the O2 diffusion coefficient in seawater at 0°C, and the two shorter lines are for the two warmer temperatures. Values are means ± s.e.m. (N=5–7).