Consequences of forced convection for the constraints on size and shape in embryos

Abstract

Previously, predictions of the maximum size of biological objects based on oxygen availability have been made for both zero and infinite water velocity around the object. In reality, however, water velocity is always intermediate between zero and infinity. We predicted maximum size and optimal shape of biological objects, pending the velocity of water around them. We assumed oxygen inside the object to be transported by diffusion and outside the object by diffusion and convection. Fick's first law of diffusion describes the inner transport. For the outer transport, we relied on semi-empirical relations between mass transport and flow conditions (Friedlander's equations). To keep mathematical complexity acceptable, we restricted ourselves to the analysis of a sphere and a cylinder in cross flow. If water velocity is low, a spherical shape is most favourable for gas exchange. If water velocity is high, an elongated and flattened shape is more favourable. A size-dependent intermediate velocity exists where shape does not matter (10(-4) m s(-1)for teleost embryos). Teleost embryos are typically exposed to flow velocities equal to or larger than 10(-4) m s(-1), making an elongated shape more favourable than a spherical one. Although teleost eggs are typically spherical, the oxygen-consuming embryos inside are indeed elongated.