A mathematical model for counter-current multiplications in the swim-bladder.

1. A computer model for swim-bladder gas filling has been developed. Phenomenological descriptions of the Root effect (pH-dependent O2 capacity of fish haemoglobin), of the lactic acid production in the gas gland and of the geometry of the rete mirabile are incorporated in the general counter-current equations to give a comprehensive model of gas filling. 2. It is known that pH along the rete is not constant, as supposed in an earlier gas-filling model. It is also known that the Root shift reaction has a different half-time model. It is also known that the Root shift reaction has a different half-time depending on whether the haemoglobin absorbs or releases O2. These particular effects are accounted for in the present model. 3. The model gives gas filling rate and maximum swim-bladder pressure for CO2, O2 and N2. The partial pressure of these gases as well as the concentration of lactic acid and the pH along the rete are also calculated. 4. The model reproduces quite accurately experimental values for gas-filling rate in eel, together with lactic acid, CO2 and O2 concentrations measured at the rete end-points. There is also good correlation between maximum predicted stable swim-bladder pressure and maximum recorded depth for four fishes investigated (r=0-937; P=0-06). 5. The model predicts an enhancement of O2 filling rate and maximum swim-bladder pressure of at least 4 when the reaction rates of the Root shift in eel haemoglobin are 0-2 sec (Root-off) and 10 sec (Root-on), as compared to an instantaneous Root shift. 6. With a swim-bladder pressure of 1 atm and Root-shift reaction rates of equal magnitude, the po2-profile along the rete is nearly linear. When the reaction rates are such as found experimentally in eel haemoglobin, the po2 along the rete is non-linear, with a maximum of approximately 2 atm near the bladder pole of the rete. An experimental verification of this maximum will constitute a crucial test of the model. 7. The calculations show that blood flow through rete can regulate both gas-filling rate and stable swim-bladder pressure. At high pressure, the main factor limiting gas filling is loss of gas through back diffusion along the rete. 8. Maximum po2 in the swim-bladder is highly dependent upon the Root effect. If the Root effect persists up to about 100 atm, as seems to be the case blue hake, maximum po2 is more than 200 atm. When the Root effect is abolished at 10 atm, as is expected in eel, the maximum po2 drops to about 30 atm. 9. The pN2 in the bladder can reach 10-15 atm depending on blood flow, whereas PCO2 will not exceed 1 atm.