The oxygen gain of diving insects

Respiration Physiology 128 (2001) 229– 233 www.elsevier.com/locate/resphysiol

Short communication

The oxygen gain of diving insects Jose´ Guilherme Chaui-Berlinck a,*, Jose´ Eduardo P.W. Bicudo a, Luiz Henrique Alves Monteiro b a

Departamento de Fisiologia do Instituto de Biocieˆncias da Uni6ersidade de Sa˜o Paulo, 05508 -900, Sa˜o Paulo, SP, Brazil b Po´s-Graduac¸a˜o, Engenharia Ele´trica, Uni6ersidade Presbiteriana Mackenzie, Mackenzie, Canada Accepted 24 July 2001

Abstract The gas gill of diving insects allows gas exchange with the surrounding water, thus extending diving time. Incompressible gas gills can potentially last indefinitely underwater, but compressible gas gills have a definite lifetime. Theoretical models of a dive event have reached opposite conclusions about the oxygen gain (G, the ratio between the duration of the diving event and the time that the initial oxygen content of the bubble would allow the insect to stay underwater). While some authors claim that G has a fixed value independently of the parameters of the dive (e.g. oxygen consumption rate) others claim the contrary. However, these claims are based on numerical solutions of the models. In this study we offer an analytical solution to the problem. The analysis of a model with constant area for gas exchange demonstrates that G cannot have a fixed value, for a fixed gain would imply in a PO2 inside the bubble different from the one occurring as a result of physical constraints of the gas exchange process. © 2001 Elsevier Science B.V. All rights reserved. Keywords: Diving, insects; Gas exchange, diving insects; Gills, gas; Invertebrates, insects; Model, gas gills

1. Introduction In 1915, Richard Ege established the respiratory function (i.e. gas exchange) of air stores carried by diving insects. Also, he clearly showed the importance of nitrogen as an ‘skeleton’ of the air store, thus increasing the diving time (tD). This extension of the diving time beyond the one al* Corresponding author. Tel.: + 55-11-8187519; fax: +5511-8187422. E-mail address: [email protected] (J.G. Chaui-Berlinck).

lowed by the initial content of oxygen in the gas store was then defined as the oxygen gain (Ege, 1915; Rahn and Paganelli, 1968). The concept of oxygen gain (G) applies only to compressible gas stores (i.e. the so-called compressible gas gills or bubbles), since for those insects carrying an incompressible gas store (the plastron), tD is potentially infinite. In 1968, Rahn and Paganelli modelled the gas exchange between the bubble and the surrounding water. They had two models, both based on the equation describing the exchange rate for a gas

0034-5687/01/$ - see front matter © 2001 Elsevier Science B.V. All rights reserved. PII: S 0 0 3 4 - 5 6 8 7 ( 0 1 ) 0 0 2 8 7 - 0

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species between two media (Eq. (1) below). The area for gas exchange was considered under two conditions, as an initial assumption of their models. Therein, in one of those models, the area for gas exchange decreased as the volume of the bubble decreased during the simulated dive. The other model had a constant area for gas exchange during the dive. In their first model, the oxygen partial pressure inside the bubble decreased constantly during the dive, whereas in their second model such a partial pressure attained a stable value, after a transient. Rahn and Paganelli (1968) studied in depth only the constant area model. They concluded that the oxygen gain has a constant value independently of the parameters of a dive, as the oxygen consumption rate, the initial volume of the bubble, the thickness of the boundary layer, etc. In other words, G is a fixed and independent value. In 1994, Chaui-Berlinck and Bicudo analysed a model in which the area decreased as the volume of the bubble decreased during a dive. Two constraints limited the diving time in that study: the degree of hypoxia in the bubble and the volume of the bubble itself (i.e. the bubble had to assure buoyancy to the insect). Contrasting to Rahn and Paganelli (1968) conclusion, ChauiBerlinck and Bicudo (1994) stated that the oxygen gain had no fixed value, G varies as the oxygen consumption, the initial volume of the bubble, and other parameters changed. Both studies modelling the gas exchange process in the bubble (Rahn and Paganelli, 1968; Chaui-Berlinck and Bicudo, 1994) have an intrinsic drawback, they are numerical solutions (numerical integration) of a set of differential equations (see below). In other words, the conclusions drawn from those studies are highly dependent on the range of values of the parameters employed. It becomes difficult to evaluate whether the different conclusions are due to differences in the models or to differences in the range of the parameters only. Thus, is G constant or not? The aim of this study is to present an analytical solution to the fixed area model. Such an analytical solution would allow one to draw a more general conclusion, free from the restric-

tions coming either from numerical solutions or from the differences between the models. 2. Methods

2.1. The model The basic equation employed by Rahn and Paganelli (1968) and by Chaui-Berlinck and Bicudo (1994) is the one describing the rate of gas exchange between two compartments (see Dejours, 1975): dnx DPxixDxA = L dt

(1)

where nx is the number of moles of the gas species x, DPx is the partial pressure difference (Torr) of x between the two media, ix is the capacitance of the medium (mol Torr − 1 m − 3), Dx is the diffusion coefficient of x (m2 sec − 1), L is the thickness (m) of the membrane separating the media (a boundary layer in this case) and A is the area for gas exchange (m2). Notice that this was the term that Rahn and Paganelli (1968) held constant in their analysis. Temperature of the water per bubble per insect system is assumed to be constant during the submersion period and there are four important gas components inside the bubble, inert gases (basically nitrogen), oxygen, carbon dioxide, and water vapour. Water vapour has constant partial pressure in a dive event, corresponding to a 100% relative humidity, and carbon dioxide also maintains a constant partial pressure inside the bubble, due to its high diffusibility to the water (Ege, 1915; Rahn and Paganelli, 1968; ChauiBerlinck and Bicudo, 1994). Therefore, we are left with two major gas species governing the lifetime of the bubble, namely nitrogen (or the inert gases) and oxygen. Considering the fixed area model, all the terms in the right-hand side of Eq. (1) but DPx can be lumped in one constant-value term. Let i stands for oxygen and j for nitrogen (or the inert components). Therefore, the following system of equations describes the total gas exchange between the bubble and the surrounding water, taking into account the oxygen consumption rate by the insect, Q:

J.G. Chaui-Berlinck et al. / Respiration Physiology 128 (2001) 229–233





Ádni ni à dt =I Wi −Pn +n −Q i j Í Ã dnj =J W −P nj j Ä dt ni +nj





(2)

where Wx is the partial pressure of the gas x in the surrounding water and P is the total pressure inside the bubble (given by the sum of the barometric pressure plus the water column above the insect). Notice that the lumped constant terms are I and J. Also, the product P[nx/(nx +ny)] in each equation is the partial pressure of the component (i or j) inside the bubble. Essentially, Eq. (2) is the same of Rahn and Paganelli (1968) study. The volume of the bubble is related to the amount of nitrogen (or inert gases) inside it. Therefore, nj becomes a volume tracer. Let a term h be ni/nj. Therefore, h becomes a hypoxia tracer. We will rewrite system Eq. (2) as:

 





n

Ádh 1 h h à dt = n I Wi −P h+1 −Q−hJ Wj −P h+1 j Í dn j à =J W −P h j (3) h+1 Ä dt



The total diving time is, therefore, the time elapsed from the beginning of the submersion at initial conditions of volume, oxygen consumption rate, depth, etc.) until either h or nj reach some limiting value. The oxygen gain of such a dive event will be the ratio of the time elapsed and the time allowed by the initial oxygen content in the bubble volume (Section 1 and Eqs. (8a), (8b) and (9)).

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solve for dh/dt = 01. At the same time, the maximum value for h is the ratio between oxygen and nitrogen at the surface. Decreasing h, due to Q, even if P is just the barometric pressure with no water column above the bubble, turns dnj/dtB 0. Therefore, dnj/dt B 0 all the time. Notice that this result is simply the diffusion the inert gases into the surrounding water caused by the liquid column above the bubble and/or by the imbalance of partial pressures due to the oxygen consumption. This indicates that once submerged and subjected to any oxygen consumption rate (even Q= 0), a bubble would be always shrinking until it can no longer guarantee buoyancy to the insect (perhaps, before an intolerable hypoxia supervenes). Even though no equilibrium point exists, the dh/dt equation can be equal to zero within a range of Q, in the constant area model. In other words, there exists h(Q), given that Q[0, Qmax], such that the term within brackets in the dh/dt equation equals to zero. This h(Q) value is a stable value, i.e. it does not change during the dive event. Let h* stands for the stable h(Q) value. This h* value is the ratio between oxygen and nitrogen inside the bubble that occurs by virtue of the physics of the gas transfer process. For dh/dt in Eq. (3), one can find the h* value solving the quadratic equation: − JWjh2 + [I(Wi − Wj)+ J(P− Wj)− Q]h + IWj − Q=0

(4)

Let, for the sake of simplicity: a= − JWj b= I(Wi − Wj)+ J(P− Wj)− Q c= IWi − Q

3. Results and discussion

Therefore, given that any existing h* must be a real value greater than zero (i.e., h*R+):

3.1. Equilibrium points and stable 6alues of h

h*=

In searching for an equilibrium point in Eq. (3) (i.e. a value of the pair (h, nj) such that both derivatives in the system became equal to zero simultaneously) one can easily verify that there is no such a point. Notice that a value of h that turns dnj/dt equal to zero cannot simultaneously

Notice that Qmax = IWi, since for Q\ IWi there can exist no h*\0. This clearly illustrates a first mistake in Rahn and Paganelli (1968) numerical

− b− b2 − 4ac 2a

(5)

1 Unless for a very limited and particularly chosen set of parameters, e.g. Q =0, I =0.

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analysis: there is a finite range of the parameters that can guarantee the existence of a stable oxygen partial pressure inside the bubble. We will call such a range as the valid range. Considering that h* would exist (i.e. we are within the valid range) and that dnj/dt B0 for all t, t [t0, tD], one obtains: dh =0 dt dnj B0 dt

ni(t) = ni0 − (Q− K)t

dni = K− Q dt where:



K = I Wi −P

This result can be easily checked inserting back a stable h value in the equation of dni/dt in Eq. (2). Notice that KBQ arises as a condition, otherwise the oxygen content of the bubble would not change or even increase. However, since the nitrogen content is always decreasing (see above), h would not attain a stable value, thus contradicting the former result of the existence of h* within the valid range. Therefore, the oxygen content of the bubble will decrease as a linear function of time:

h 1+h



(6)

(7)

where ni0 is the initial oxygen content of the bubble. There is a lower limit nilim for ni(t), that is, a bubble has to have a minimal volume (although anything prevents us to fix such a limiting value as zero, the conclusions will remain the same).

Fig. 1. Values of h* (Eq. (5), solid line) and h(G) (Eq. (11), dashed line) as functions of Q. The oxygen gain G in Eq. (11) was arbitrary chosen as 7. Notice that these functions have different values (unless in their crossing point) as the oxygen consumption rate Q, a parameter of the dive, changes. This indicates that a fixed oxygen gain is a physical impossibility, since the values of h(G) should be equal to the values of h*, the stable value attained by virtue of the physical constraints of the gas exchange process. In essence, the oxygen gain must change as the parameters of the dive change. Values of capacitances, coefficients of diffusion, area and boundary layer from Rahn and Paganelli (1968) and from Chaui-Berlinck and Bicudo (1994), P = 780 torr. Rahn and Paganelli (1968) give G =8 for a dive event just below the surface.

J.G. Chaui-Berlinck et al. / Respiration Physiology 128 (2001) 229–233

The expected time for a dive, tE, is: tE =

ni0 − nilim Q

(8a)

whereas Eq. (6) gives the real time of a dive event, tD, as: tD =

ni0 −nilim Q −K

4. Conclusions

t Q G= D = (9) tE Q −K Thus, Eq. (9) gives us the oxygen gain for the constant area model presuming the parameters within the valid range.

3.2. Imposing a fixed G 6alue Let us impose a value for the oxygen gain, and let such a value be fixed despite parameters variation. From Eq. (9) we have: G− 1 G

K=Q

(10)

We can the obtain another stable value of h, hG, directly from Eqs. (6) and (10), under the assumption of a fixed G:

 

valid range is not satisfied. In other words, the stable value of the internal PO2 necessary to obtain a fixed oxygen gain despite variations in the parameters of the dive differs from the stable PO2 occurring by virtue of the physical constraints of the gas exchange process, thus leading to a physical impossibility. Fig. 1 illustrates the point.

(8b)

And, the oxygen gain G, as defined, is:

G −1 − IWi G hG = G −1 I(Wi −P) − Q G Q

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(11)

In order for the fixed oxygen gain concept have consistency, hG should be equal to h* in the entire valid range of the parameters. However, a simple comparison between Eqs. (5) and (11) clearly shows that the condition hG h* within the entire

This study offers an analytical solution to the problem of the supposed fixed oxygen gain of diving insects that Rahn and Paganelli (1968) claimed for. Here we demonstrated that the fixed oxygen gain will not occur. A fixed gain value would imply in an oxygen partial pressure inside the air store different from the oxygen partial pressure anticipated by the gas transfer process. In face of such results, we see no further reason for either modelling compressible air stores of diving insects as having constant area during the dive event or to sustain the claim of a fixed oxygen gain during a dive event.

References Chaui-Berlinck, J.G., Bicudo, J.E.P.W., 1994. Factors affecting oxygen gain in diving insects. J. Insect Physiol. 40, 617 – 622. Dejours, P., 1975. Principles of Comparative Respiratory Physiology, Second ed. Elsevier/North-Holland Biomedical Press, Amsterdam. Ege, R., 1915. On the respiratory function of the air stores carried by some aquatic insects (Corixidae, Dytiscidae and Notonecta). Z. Allgem. Physiol. 17, 81 – 124. Rahn, H., Paganelli, C.V., 1968. Gas exchange in gas gills of diving insects. Respir. Physiol. 5, 145 – 164.