The dual function of appendage movement (food acquisition, ventilation)proved to be the key to explaining the peculiar oxyregulatory repertoire of the planktonic filter feeder Daphnia magna. Short-term hypoxic exposure experiments with normoxia-acclimated animals under varying food concentrations revealed a dichotomous response pattern with a compensatory tachycardia under food-free conditions and a ventilatory compensation prevailing under food-rich conditions. Food-free, normoxic conditions resulted in maximum appendage beating rates (fa) and half-maximum heart rates (fh), which restricted the scope for oxyregulation to the circulatory system. Food-rich conditions (105algal cells ml-1), on the contrary, had a depressing effect on fa whereas fh increased to 83% of the maximum. In this physiological state, D. magna was able to respond to progressive hypoxia with a compensatory increase in ventilation. A conceptual and mathematical model was developed to analyse the efficiency of ventilatory and circulatory adjustments in improving oxygen transport to tissue. Model predictions showed that an increase in perfusion rate was most effective under both food-free and food-rich conditions in reducing the critical ambient oxygen tension (PO2crit) at which oxygen supply to the tissue started to become impeded. By contrast, a hypothetical increase in ventilation rate had almost no effect on PO2crit under food-free conditions, indicating that appendage movement is driven by nutritive rather than respiratory requirements. However, the model predicted a moderate reduction of PO2crit by hyperventilation under food-rich conditions. Since the regulatory scope for an adjustment in fh was found to be limited in D. magna under these conditions, the increase in ventilation rate is the means of choice for a fed animal to cope with short-term, moderate reductions in ambient oxygen availability. Under long-term and more severe hypoxic conditions, however, the increase in the concentration and oxygen affinity of haemoglobin represents the one and only measure for improving the transport of oxygen from environment to cells.

To cope with fluctuations in ambient oxygen tension(PO2amb), oxyregulating water breathers must respond with appropriate adjustments of their ventilatory and circulatory systems. Systemic regulation enables these animals not only to maintain their oxygen consumption rates in the face of declining PO2amb but also to minimize the risk of oxidative stress for the tissues and to reduce the energetic expenditures for pumping medium and blood when PO2amb is high. Whether a ventilatory or a circulatory adjustment is appropriate in a given situation depends on the adjustability of the respective system, the degree of interference with other vital body functions and, perhaps most importantly,the efficiency of a ventilatory or circulatory change.

The planktonic crustacean Daphnia magna (Branchiopoda; Cladocera),a euryoxic species with oxyregulatory capacities(Kobayashi and Hoshi, 1984; Paul et al., 1997), exhibits a remarkable tolerance for environmental hypoxia (i.e. a state of reduced oxygen availability when the PO2amb has fallen below the normoxic values of 20-22 kPa prevailing normally at sea level; Grieshaber et al., 1994). The underlying physiological mechanisms that allow the animal to maintain oxygen uptake under environmental hypoxia range from short-term adjustments at the systemic level (Paul et al.,1997; Pirow et al.,2001) to long-term changes in the concentration and oxygen-binding characteristics of haemoglobin (Hb; Fox et al., 1951; Kobayashi and Hoshi, 1982; Kobayashi et al.,1988; Zeis et al., 2003a,b). Interestingly, the acute systemic responses to progressive, moderate hypoxia seem to deviate from those of other oxyregulating water breathers. Whereas fish (Dejours, 1981; Randall et al., 1997) or decapod crustaceans (McMahon and Wilkens,1975; Taylor,1976; Dejours and Beekenkamp,1977; Herreid,1980; Wheatly and Taylor,1981) typically increase ventilation while keeping cardiac output(= heart rate × stroke volume) more or less constant, the situation seems to be reversed in D. magna. Recent studies(Paul et al., 1997; Pirow et al., 2001) have shown that the heartbeat accelerates without notable changes in stroke volume(compensatory tachycardia) whereas the movements of the thoracic appendages,whose ventilatory function has been demonstrated experimentally(Pirow et al., 1999a), remain almost constant. It appears that hyperventilation is an inappropriate response for D. magna to compensate for a reduction in PO2amb.

This idea is in line with the fact that D. magna is able to increase the concentration of Hb in the haemolymph by more than 10 times when exposed to chronic hypoxia (Kobayashi and Hoshi, 1982). Both responses, the tachycardia and the elevation of Hb concentration, compensate for the reduction in ambient oxygen availability by increasing the oxygen transport capacity of the circulatory system(Pirow et al., 2001; Bäumer et al., 2002). This suggests that the circulatory system rather than the ventilatory system is the limiting and controlling step of the oxygen transport cascade from environment to cell. The reason for the suggested absence of a ventilatory controllability of the oxygen transport cascade could lie in the filter-feeding mode of life. The rhythmical beating of the thoracic appendages has not only a ventilatory function but also serves an important non-respiratory need: food acquisition. The third and fourth limb pairs are equipped with fine-meshed filter combs that enable Daphnia to retain food particles suspended in the ambient medium (Fryer, 1991). Since food particles can be highly diluted in the natural environment, the rate of medium flow required to assure an adequate nutrition could exceed the rate necessary to satisfy the oxygen demand of the animal, as is supposed for other filter feeders such as sponges, lamellibranches and ascidians(Dejours, 1981).

The dual function of appendage movement presumably provides the key to explaining the peculiar oxyregulatory behaviour of D. magna. However,if nutritive requirements rather than respiratory needs drive appendage movement under conditions of limited food availability, what controls this activity under excess food conditions? Several studies have shown that Daphnia spp. exhibits close to maximum appendage beating rates when there is little or no food available, whereas high food concentrations(≥104 unicellular algae ml-1) effect a pronounced deceleration of appendage movement(McMahon and Rigler, 1963; Burns, 1968; Porter et al., 1982). Since the latter effect is inevitably associated with a reduction in ventilatory power, and since the oxygen demand increases as a consequence of the activation of digestive processes(Lampert, 1986; Bohrer and Lampert, 1988)despite lower energetic expenditures for appendage movement(Philippova and Postnov,1988), it is possible that the oxyregulatory responses exhibited under these conditions deviate from those described so far(Paul et al., 1997; Pirow et al., 2001). The aim of the present paper is to analyse the oxyregulatory repertoire of D. magna and to provide a causal mechanistic explanation for its peculiar oxyregulatory behaviour. These issues were tackled by an experimental approach, in which the systemic responses to declining PO2amb were examined under food-free and food-rich conditions, as well as by the use of a conceptual and mathematical model, which made it possible to predict the efficiency of ventilatory and circulatory adjustments in improving oxygen transport to tissue.

### Animals, experimental conditions and statistical analysis

Water fleas (Daphnia magna Straus) were reared under normoxic conditions (80-95% air saturation, 19.5-21.5°C) as described previously(Pirow et al., 2001). Animals examined under food-free (food-deprived group) and food-rich conditions(food-provided group) had body lengths of 2.76±0.17 mm (mean ± s.d.; N=5) and 2.60±0.18 mm (N=11),respectively, with 0-8 parthenogenetic eggs or embryos in the brood chamber. The slight difference in the mean body length of both groups was not statistically significant (unpaired two-tailed t-test: t=-1.69, d.f.=14, P=0.11). The lower number of animals in the food-deprived group was regarded to be sufficient for comparative purposes since similar experiments have already been done before(Paul et al., 1997; Pirow et al., 2001).

In order to measure heart (fh) and appendage beating rate (fa), single animals were tethered by gluing their posterior apical spine to a 1-cm-long synthetic brush-hair with adhesive(histoacryl; B. Braun Melsungen AG, Melsungen, Germany). The animal was positioned lateral-side-down with the opposite side of the brush-hair and one of the large antennae glued onto a cover slip. The cover slip with the tethered animal was transferred into a transparent perfusion chamber(Paul et al., 1997) with the head orientated against the direction of the medium flow. The chamber was sealed and placed onto the stage of an inverted video microscope (Zeiss Axiovert 100; Carl Zeiss, Oberkochen, Germany). While keeping the animal under infrared illumination (>780 nm), the frequency of the periodic movements of the heart and the thoracic appendages were automatically determined as described in detail elsewhere (Pirow et al., 2001).

The experimental chamber was perfused with culture medium (M4; Elendt and Bias, 1990) at a flow rate of 5 ml min-1. During the experiments, the oxygen tension of the medium was lowered gradually from normoxia (21 kPa) to severe hypoxia(<1.5 kPa) with a duration of five minutes for each step. This time interval was sufficient for the animal to attain a new stable level of fh and fa(Paul et al., 1997; Pirow et al., 2001). Different levels in oxygen tension were obtained by using two computer-driven peristaltic pumps (Gilson Minipuls 3; ABIMED, Langenfeld, Germany) that mix normoxic and anoxic media at different ratios(Freitag et al., 1998). Both media were prepared by equilibration with air or with a gas composed of 99.95%N2 and 0.05% CO2. The oxygen tension of the perfusion medium was measured behind the experimental chamber using a polarographic electrode (WTW Oxi 92; Weilheim, Germany). In the experiment with high food concentration, the unicellular green alga Scenedesmus subspicatus was added to both reservoirs at a final concentration of 105 cells ml-1. This concentration had been reported to effect a depression of fa in D. magna(Porter et al., 1982), which was confirmed in a separate experiment(Fig. 1; Table 1). The algal stock solution was prepared by centrifuging the algae at 2000 g (5 min, 4°C) and resuspending the algal pellet in filtered culture medium(cellulose acetate filter; pore size, 0.45 μm). The concentration of algae in the stock solution was determined using a Neubauer counting chamber. The stock solution was then appropriately diluted and kept in complete darkness.

Fig. 1.

Effect of increasing food concentrations on (A) appendage beating rate(fa) and (B) heart rate (fh) of D. magna (2.68±0.22 mm long) at normoxic conditions. Data are given as means ± s.d. (N=4 except for 5.6×104 cells ml-1, where N=2). A repeated-measures ANOVA was performed for all food levels except 5.6×104 cells ml-1. Neither the mean fa (F=49.9, groups d.f.=6, remainder d.f.=18, P<0.001) nor the mean fh (F=28.8,groups d.f.=6, remainder d.f.=18, P<0.001) were the same in animals on all seven food levels. The results of multiple comparisons among pairs of means are shown in Table 1.

Fig. 1.

Effect of increasing food concentrations on (A) appendage beating rate(fa) and (B) heart rate (fh) of D. magna (2.68±0.22 mm long) at normoxic conditions. Data are given as means ± s.d. (N=4 except for 5.6×104 cells ml-1, where N=2). A repeated-measures ANOVA was performed for all food levels except 5.6×104 cells ml-1. Neither the mean fa (F=49.9, groups d.f.=6, remainder d.f.=18, P<0.001) nor the mean fh (F=28.8,groups d.f.=6, remainder d.f.=18, P<0.001) were the same in animals on all seven food levels. The results of multiple comparisons among pairs of means are shown in Table 1.

Table 1.

Results of the Tukey multiple comparison testing among the mean appendage beating rates (fa) and the mean heart rates (fh) of the seven food levels

Food level (cells ml-1)Mean fa (min-1)Food level (cells ml-1)Mean fh (min-1)
1 000 000 97 1 000 178
100 000 129 179
259 3 200 195
32 000 262 10 000 212
1 000 282 32 000 246
3 200 313 100 000 264
10 000 339 1 000 000 269
Food level (cells ml-1)Mean fa (min-1)Food level (cells ml-1)Mean fh (min-1)
1 000 000 97 1 000 178
100 000 129 179
259 3 200 195
32 000 262 10 000 212
1 000 282 32 000 246
3 200 313 100 000 264
10 000 339 1 000 000 269

All seven means of fa and fh,respectively, were arranged in order of increasing magnitude, and the vertical lines beside them represent non-significant sets of means(Sokal and Rohlf, 1995).

All experiments were carried out at 20°C. Animals were allowed to acclimate to the experimental conditions for 50 min before starting the experiment. fh and fa were analysed for the last minute of each step in oxygen tension. Data were expressed as means ± s.d., with N indicating the number of animals examined. For each experiment, in which multiple measurements were made on the same animal under various treatment levels (either food concentration or oxygen partial pressure), differences in mean values(fh and fa) of the different treatments levels were assessed using a repeated-measures analysis of variance(repeated-measures ANOVA; Zar,1999). Statistical differences were considered significant at P<0.05. In the case of a statistical significant difference,multiple comparisons (Tukey test; Zar,1999) among pairs of means using an experimentwise error rate of 0.05 were performed to determine between which means differences exist.

### General description of the conceptual model of oxygen transport

Animals with a body size in the millimetre range are distinguished from their larger counterparts by short transport distances from the body surface to the central body regions. Since diffusive processes are effective for short distances only, millimetre-sized animals can rely to a greater extent on diffusion for providing peripheral tissues (i.e. tissues close to the body surface) with oxygen directly from the ventilated or non-ventilated ambient medium, whereas internal convection is, in principle, only needed to deliver oxygen to the more centrally located tissues, which are too distant from the periphery to be sufficiently supplied by diffusion. Such a pathway deviates somewhat from that of the basic vertebrate model(Taylor and Weibel, 1981; Piiper, 1982; Weibel, 1984; Shelton, 1992), where the transport of oxygen from the environment to the cells is thought to occur along a linear sequence of alternating convection and diffusion steps(ventilatory convection, diffusion across the oxygen-permeable integument,circulatory convection, diffusion in the tissue). The proposed model for the diffusive-convective oxygen transport in Daphnia magna incorporates both a ventilatory-circulatory transport of oxygen to the centrally located tissues as well as a diffusive supply of peripheral tissues directly from the respiratory medium (Pirow,2003).

To keep the mathematical formulation of the oxygen transport cascade as simple as possible, the complex body shape of D.magna(Fig. 2A) is reduced to a cylindrical trunk, which is enveloped by a hollow cylinder representing the carapace (Fig. 2B). The carapace consists of an outer and an inner wall that both enclose a haemolymph space, the carapace lacuna. The cylindrical trunk is further assumed to be composed of a peripheral tissue layer, a haemolymph space (trunk lacuna) and a central tissue cylinder (Fig. 2B). The respiratory medium flows through the space between the carapace and the trunk in a posterior direction while oxygen is released both into the carapace lacuna and the trunk. This design takes into account that the feeding current of D. magna is an important pathway for oxygen(Pirow et al., 1999a) and that the inner wall of the carapace is a significant site of oxygen uptake(Pirow et al., 1999b). Similar to the real situation, the medium flow and haemolymph flow in the carapace lacuna are in concurrent orientation to each other. Leaving the carapace lacuna, the oxygen-rich haemolymph enters the haemolymph space of the trunk and flows in an anterior direction while oxygen is released into both tissue compartments. Reaching the anterior position, oxygen-poor haemolymph then re-enters the carapace lacuna. The circulation of haemolymph takes place in a single circuit that, of course, is a simplification compared with the real situation, where the haemolymph flow branches into subcircuits(Pirow et al., 1999b). The oxygen partial pressure of the inspiratory and expiratory medium is denoted by Pin and Pex (kPa), respectively,whereas that of the haemolymph entering and leaving the trunk is denoted by Pa and Pv, respectively(Fig. 2B).

Fig. 2.

(A) Dorsal view of the microcrustacean Daphnia magna showing the medium flow pattern (white arrows) and the circulatory pattern (black arrows). A dorsal piece of the left carapace valve (chequered area) was removed (for details see Pirow et al.,1999b). (B) Conceptual model for oxygen transport in D. magna based on a cylinder-within-a-tube arrangement. Medium flows through the space between the carapace and the trunk in a posterior direction (open arrows) while oxygen is released both into the carapace lacuna and the peripheral tissue layer of the trunk. This tissue layer is supplied with oxygen from the medium and from a truncal haemolymph space by diffusion(broken arrows). Oxygenated haemolymph leaves the double-walled carapace and then enters the truncal haemolymph space (solid arrows). While flowing in an anterior direction, oxygen diffuses from this haemolymph space both into the coaxial tissue cylinder and the cortical tissue layer (broken arrows). Pin, Pex, inspiratory and expiratory oxygen partial pressures, respectively; Pa and Pv, oxygen partial pressures of the haemolymph entering and leaving the trunk, respectively.

Fig. 2.

(A) Dorsal view of the microcrustacean Daphnia magna showing the medium flow pattern (white arrows) and the circulatory pattern (black arrows). A dorsal piece of the left carapace valve (chequered area) was removed (for details see Pirow et al.,1999b). (B) Conceptual model for oxygen transport in D. magna based on a cylinder-within-a-tube arrangement. Medium flows through the space between the carapace and the trunk in a posterior direction (open arrows) while oxygen is released both into the carapace lacuna and the peripheral tissue layer of the trunk. This tissue layer is supplied with oxygen from the medium and from a truncal haemolymph space by diffusion(broken arrows). Oxygenated haemolymph leaves the double-walled carapace and then enters the truncal haemolymph space (solid arrows). While flowing in an anterior direction, oxygen diffuses from this haemolymph space both into the coaxial tissue cylinder and the cortical tissue layer (broken arrows). Pin, Pex, inspiratory and expiratory oxygen partial pressures, respectively; Pa and Pv, oxygen partial pressures of the haemolymph entering and leaving the trunk, respectively.

### Model assumptions

The following simplifying assumptions are made in the model. (1) Diffusion of oxygen in all compartments (tissue, haemolymph and medium) and across compartment interfaces is only in a radial direction. Axial diffusion is ignored in order to reduce mathematical complexity. (2) Oxygen diffusion across the tissue-medium and medium-haemolymph interfaces is impeded by cuticular barriers of the same permeability. (3) The outer wall of the carapace is assumed to be impermeable to oxygen. (4) Axial convection occurs only in the haemolymph and medium compartments. (5) Convective flows have velocity profiles that are uniform in respect to the radial axis. (6) The mixing of haemolymph leaving the lacunae at the bases of the cylindrical model as well as the re-entrance of the mixed haemolymph into destined lacunae is assumed to occur without a time delay. (7) Haemoglobin as the oxygen carrier in the haemolymph is not considered in order to reduce mathematical complexity. (8) The volume-specific oxygen consumption rate is assumed to be constant throughout the tissue compartments.

### Mathematical formulation and derivation of the numerical solution

Based on assumptions made in the previous section, the following general oxygen transport equation (Groebe and Thews, 1992) accounts for radial diffusion with axial convection and oxygen consumption:
This equation derives from Fick's second law of diffusion(Crank, 1975), extended by two terms for convective oxygen transport and oxygen consumption. P (kPa)is the oxygen partial pressure, t (s) and r (mm) are the time and the radial coordinate, respectively, α (nmol mm-3kPa-1) is the solubility coefficient for oxygen, and a(nmol s-1 mm-3) represents the volume-specific oxygen consumption rate of pure tissue. D (mm2 s-1) is the diffusion coefficient for oxygen whereas h (mm) and v(mm s-1) represent the axial coordinate and the convective velocity, respectively.

The solution of the partial differential equation 1 is approximated by numerical methods, which requires dividing the whole cylindrical body into discrete volume elements. The cylindrical model of height(h0) and radius (r0) is divided in the axial direction in Nax equal intervals of lengthΔ h and in the radial direction in Nrad+1 intervals of length Δr and 0.5Δr, respectively(Fig. 3). This subdivision yields two different kinds of coaxial volume elements: solid and hollow cylinders. As a consequence of this discretization, the radii of the compartment interfaces (e.g. tissue-haemolymph interface) have to be rounded to multiples of Δr. Since the whole cylindrical body is radially symmetrical, it is sufficient to further consider only that region of the median plane that is covered by 0...h0 and 0...r0 (Fig. 3). In this view, each volume element is represented by a discrete grid point. The axial and radial coordinates of the grid points are(j+0.5)Δh and iΔr,respectively, where the indices j and i are integers with j=0, 1,..., Nax-1 and i=0, 1,..., Nrad.

Fig. 3.

Subdivision of the cylindrical model of radius r0 and height h0 for numerical analysis. This subdivision yields coaxial cylindrical and hollow-cylindrical volume elements of height(Δh) with radial extensions being multiples of 0.5Δr. The numerical analysis aims to determine the oxygen partial pressures for the discrete set of grid points (filled circles)representing the volume elements. The axial and radial coordinates of the points are (j+0.5)Δh and iΔr,where the indices j and i are integers with j=0,..., Nax-1 and i=0,..., Nrad. The oxygen partial pressure(Pj,i) of a representative volume element (white rectangle) is affected by diffusive (broken arrows) and convective (solid arrows) exchange processes with the adjacent volume elements. The hatched areas represent those fractions of the respective volume elements that are shifted to the left by convection during the time interval Δt. The white line on the left exemplifies the radial position of a compartment interface that has to be rounded to a multiple of Δr.

Fig. 3.

Subdivision of the cylindrical model of radius r0 and height h0 for numerical analysis. This subdivision yields coaxial cylindrical and hollow-cylindrical volume elements of height(Δh) with radial extensions being multiples of 0.5Δr. The numerical analysis aims to determine the oxygen partial pressures for the discrete set of grid points (filled circles)representing the volume elements. The axial and radial coordinates of the points are (j+0.5)Δh and iΔr,where the indices j and i are integers with j=0,..., Nax-1 and i=0,..., Nrad. The oxygen partial pressure(Pj,i) of a representative volume element (white rectangle) is affected by diffusive (broken arrows) and convective (solid arrows) exchange processes with the adjacent volume elements. The hatched areas represent those fractions of the respective volume elements that are shifted to the left by convection during the time interval Δt. The white line on the left exemplifies the radial position of a compartment interface that has to be rounded to a multiple of Δr.

By using the Taylor's series technique(Faires and Burden, 1993; Crank, 1975) for solving Fick's second law of diffusion, the following explicit finite-difference solution of equation 1 is obtained for all grid points not coinciding with compartment interfaces and excluding the central and outermost grid points at i=0 and i=Nrad, respectively:
$\ {\Delta}P_{\mathrm{j,i}}=\frac{D{\Delta}t}{2i({\Delta}r)^{2}}[(2i+1)P_{\mathrm{j,i}+1}-4iP_{\mathrm{j,i}}+(2i-1)P_{\mathrm{j,i}-1}]+\frac{{\nu}{\Delta}t}{{\Delta}h}(P_{\mathrm{k,i}}-P_{\mathrm{j,i}})-\frac{a{\Delta}t}{{\alpha}}.$
2
This equation makes it possible to calculate the change in oxygen partial pressure (ΔPj,i) at the grid point referenced by j, i during the time interval Δt. Pj,irepresents the oxygen partial pressure at the representative grid point at time point t. The respective oxygen partial pressures of the neighbouring points at the radially inferior and superior positions are Pj,i-1 and Pj,i+1 (Fig. 3). Pk,i, relevant only in the case of axially directed convection, represents the oxygen partial pressure of the neighbouring point at the upstream position. The oxygen partial pressure of the downstream neighbouring point does not enter the convection term, because there is no gradient in the axial direction within the volume elements (axial diffusion is ignored). For grid points at the bases of the cylindrical model(at j=0 and j=Nax-1), Pk,i is appropriately replaced by Pin, Pa or Pv.

From the general solution given in equation 2, three special cases may be easily derived to describe the oxygen transport in the tissue compartments (D=DT, v=0,α=αT), in the haemolymph compartments(D=DH, v=vHT or v=vHC, α=αH, a=0)and in the medium compartment (D=DM, v=vM, α=αM, a=0). In these special cases, the solubility and diffusion coefficients assume the specific values of the respective compartments (tissue - αT, DT; haemolymph - αH, DH; medium - αM, DM). Flow velocities of the haemolymph in the carapace lacuna and in the trunk lacuna are denoted by vHC and vHT,respectively.

After having derived the balance equations for the oxygen transport within the different compartments, it remains to describe the oxygen transfer across compartment interfaces as well as the changes in oxygen partial pressure at the central and outermost grid points. For the central grid points (at i=0) belonging to the tissue compartment, the following approximation is derived from Crank (1975):
$\ {\Delta}P_{\mathrm{j},0}=4\frac{D_{\mathrm{T}}{\Delta}t}{({\Delta}r)^{2}}(P_{\mathrm{j},1}-P_{\mathrm{j},0})-\frac{a{\Delta}t}{{\alpha}_{\mathrm{T}}},$
3
where Pj,0 and Pj,1 are the oxygen partial pressures at the grid points referenced by i=0 and i=1, respectively.
A somewhat more complicated mathematical formulation is required to calculate the changes in oxygen partial pressure at grid points located at compartment interfaces with an additional diffusion barrier (cuticle). In such a case, a grid point referenced by j, i is characterized by two variables,
$$P_{\mathrm{j,i}}^{\mathrm{u}}$$
and
$$P_{\mathrm{j,i}}^{\mathrm{o}}$$
(kPa), which represent oxygen partial pressure at the inner and outer side of the infinitesimal thin diffusion barrier. The following example gives the specific solution for the tissue-medium interface with diffusion barrier:
$\ {\Delta}P_{\mathrm{{\ }j,i}}^{\mathrm{u}}=\frac{{\Delta}t}{V_{\mathrm{{\ }i}}^{\mathrm{u}}{\alpha}_{\mathrm{T}}}\left[\frac{B_{\mathrm{i}-0.5}D_{\mathrm{T}}{\alpha}_{\mathrm{T}}}{{\Delta}r}(P_{\mathrm{j,i}-1}-P_{\mathrm{{\ }j,i}}^{\mathrm{u}})+B_{\mathrm{i}}g(P_{\mathrm{{\ }j,i}}^{\mathrm{o}}-P_{\mathrm{{\ }j,i}}^{\mathrm{u}})-V_{\mathrm{{\ }i}}^{\mathrm{u}}a\right],$
4
$\ {\Delta}P_{\mathrm{{\ }j,i}}^{\mathrm{o}}=\frac{{\Delta}t}{V_{\mathrm{{\ }i}}^{\mathrm{o}}{\alpha}_{\mathrm{M}}}\left[\frac{B_{\mathrm{i}+0.5}D_{\mathrm{M}}{\alpha}_{\mathrm{M}}}{{\Delta}r}(P_{\mathrm{j,i}+1}-P_{\mathrm{{\ }j,i}}^{\mathrm{o}})+B_{\mathrm{i}}g(P_{\mathrm{{\ }j,i}}^{\mathrm{u}}-P_{\mathrm{{\ }j,i}}^{\mathrm{o}})+A_{\mathrm{{\ }i}}^{\mathrm{o}}{\nu}_{\mathrm{M}}{\alpha}_{\mathrm{M}}(P_{\mathrm{j}-1,\mathrm{i}}-P_{\mathrm{{\ }j,i}}^{\mathrm{o}})\right].$
5
The parameters Bi-0.5, Bi, Bi+0.5,
$$A_{\mathrm{i}}^{\mathrm{o}}$$
,
$$V_{\mathrm{{\ }i}}^{\mathrm{u}}$$
and
$$V_{\mathrm{{\ }i}}^{\mathrm{o}}$$
are explained graphically in Fig. 4, whereas g (nmol s-1 mm-2 kPa-1) represents the permeability of the cuticular diffusion barrier. The specific solution for the medium-haemolymph interface with diffusion barrier (i.e. the inner carapace wall) may be obtained by appropriately adapting equations 4 and 5.
Fig. 4.

Oxygen transfer across a compartment interface with additional diffusion barrier (cuticle). The grid point referenced by the indices i and j is located on an infinitesimal thin diffusion barrier separating the tissue from the medium compartment. This grid point is characterized by two variables,

$$P_{\mathrm{{\ }j,i}}^{\mathrm{u}}$$
and
$$P_{\mathrm{{\ }j,i}}^{\mathrm{o}}$$
, which represent the oxygen partial pressure at the inner and the outer side of the circular diffusion barrier, respectively. To calculate the temporal changes in
$$P_{\mathrm{{\ }j,i}}^{\mathrm{u}}$$
and
$$P_{\mathrm{{\ }j,i}}^{\mathrm{o}}$$
(see equations 4, 5), the following geometrical parameters are required: the cylindrical wall areas Bi-0.5, Bi and Bi+0.5, the areas of the hollow-cylindrical bases
$$A_{\mathrm{i}}^{\mathrm{u}}$$
and
$$A_{\mathrm{i}}^{\mathrm{o}}$$
, and the hollow-cylindrical volumes
$$V_{\mathrm{{\ }i}}^{\mathrm{u}}$$
and
$$V_{\mathrm{{\ }i}}^{\mathrm{o}}$$
. Three example equations for calculating these parameters are given. vM,flow velocity in the medium compartment; Pj,i-1, Pj,i+1,
$$P_{\mathrm{{\ }j}-1,\mathrm{i}}^{\mathrm{o}}$$
, oxygen partial pressures of the neighbouring grid points; Δh andΔ r, distance between two grid points in axial and radial direction.

Fig. 4.

Oxygen transfer across a compartment interface with additional diffusion barrier (cuticle). The grid point referenced by the indices i and j is located on an infinitesimal thin diffusion barrier separating the tissue from the medium compartment. This grid point is characterized by two variables,

$$P_{\mathrm{{\ }j,i}}^{\mathrm{u}}$$
and
$$P_{\mathrm{{\ }j,i}}^{\mathrm{o}}$$
, which represent the oxygen partial pressure at the inner and the outer side of the circular diffusion barrier, respectively. To calculate the temporal changes in
$$P_{\mathrm{{\ }j,i}}^{\mathrm{u}}$$
and
$$P_{\mathrm{{\ }j,i}}^{\mathrm{o}}$$
(see equations 4, 5), the following geometrical parameters are required: the cylindrical wall areas Bi-0.5, Bi and Bi+0.5, the areas of the hollow-cylindrical bases
$$A_{\mathrm{i}}^{\mathrm{u}}$$
and
$$A_{\mathrm{i}}^{\mathrm{o}}$$
, and the hollow-cylindrical volumes
$$V_{\mathrm{{\ }i}}^{\mathrm{u}}$$
and
$$V_{\mathrm{{\ }i}}^{\mathrm{o}}$$
. Three example equations for calculating these parameters are given. vM,flow velocity in the medium compartment; Pj,i-1, Pj,i+1,
$$P_{\mathrm{{\ }j}-1,\mathrm{i}}^{\mathrm{o}}$$
, oxygen partial pressures of the neighbouring grid points; Δh andΔ r, distance between two grid points in axial and radial direction.

For all compartment interfaces lacking an additional diffusion barrier,only one variable is required to describe oxygen partial pressure at that location. The following two examples show the specific solutions for the tissue-haemolymph interface (equation 6) and the outermost grid points (at i=Nrad; equation 7):
\begin{eqnarray*}&&\ {\Delta}P_{\mathrm{j,i}}=\frac{{\Delta}t}{V_{\mathrm{{\ }i}}^{\mathrm{u}}{\alpha}_{\mathrm{T}}+V_{\mathrm{{\ }i}}^{\mathrm{o}}{\alpha}_{\mathrm{H}}}\left[\frac{B_{\mathrm{i}-0.5}D_{\mathrm{T}}{\alpha}_{\mathrm{T}}}{{\Delta}r}(P_{\mathrm{j,i}-1}-P_{\mathrm{j,i}})+\frac{B_{\mathrm{i}+0.5}D_{\mathrm{H}}{\alpha}_{\mathrm{H}}}{{\Delta}r}(P_{\mathrm{j,i}+1}-P_{\mathrm{j,i}})\right.\ \\&&\left.\ +A_{\mathrm{{\ }i}}^{\mathrm{o}}{\nu}_{\mathrm{HT}}{\alpha}_{\mathrm{H}}(P_{\mathrm{j}+1,\mathrm{i}}-P_{\mathrm{j,i}})-V_{\mathrm{{\ }i}}^{\mathrm{u}}a\right],\end{eqnarray*}
6
$\ {\Delta}P_{\mathrm{j,N}_{\mathrm{rad}}}=\frac{{\Delta}t}{V_{\mathrm{N}_{\mathrm{rad}}}^{\mathrm{u}}}\left[\frac{B_{\mathrm{N}_{\mathrm{rad}}-0.5}D_{\mathrm{H}}}{{\Delta}r}(P_{\mathrm{j,N}_{\mathrm{rad}}-1}-P_{\mathrm{j,N}_{\mathrm{rad}}})+A_{\mathrm{N}_{\mathrm{rad}}}^{\mathrm{u}}v_{\mathrm{HC}}(P_{\mathrm{j}-1,\mathrm{N}_{\mathrm{rad}}}-P_{\mathrm{j,N}_{\mathrm{rad}}})\right].$
7

The balance equations derived for all grid points were used to calculate(1) the oxygen partial pressure distribution within the model and (2) the total oxygen consumption rate as a function of PO2amb. Solutions were obtained by initially setting the oxygen partial pressure of all grid points to zero and Pin to PO2amb. The numerical calculation was started and continued until quasi steady-state conditions (ΔPi,j<10-6 kPa for all grid points) were reached. For Δt, a value equal to or smaller than 0.005 s was chosen, which proved to be adequate to allow the model system to approach steady-state conditions.

### Selection of parameter values

The selection of reasonable parameter values determining the geometrical extensions and functional properties of the model is a tricky step in the modelling process, especially when parameter values are not precisely known or when the geometrical model deviates in some respects from structural or physical reality. Since model parameters can depend on each other, we defined key parameters from which derived parameters were calculated according to functional relationships (Table 2). Following this approach, the radial extensions of all compartments were derived taking the following assumptions into account. (1)Volume (V) and height (h0) of the cylindrical model are 1.12 mm3 and 2.5 mm, respectively, and refer to a 2.5 mm-long D. magna with no brood in the brood chamber(Kobayashi, 1983). (2) V comprises the tissue and haemolymph compartments. (3) The tissue fraction φ of V is 0.4(Kobayashi, 1983). (4) The flow cross-sectional area (AM) penetrated perpendicularly by the medium flow is 0.4 mm2. (5) The thickness(Δx) of the carapace lacuna is 0.02 mm. (6) The fraction of total tissue (ξ) allocated to the central tissue cylinder is 0.25. (7) The radial distance (Δr) between two grid points is 0.005 mm.

Table 2.

Values and units of parameters used in the model

SymbolValueUnitDescription
Key (primary) parameters
V 1.12 mm3 Body volume (haemolymph and tissue compartments)
h0 2.5 mm Height of the cylindrical model
AM 0.4 mm2 Flow cross-sectional area of the medium flow
Δx 0.02 mm Thickness of the carapace lacuna
Φ 0.4  Tissue fraction of body volume
ζ 0.25  Fraction of total tissue in the central tissue cylinder
a0 0.0058 nmol s-1 mm-3 Volume-specific O2 consumption rate (whole body)
M 0.8333 mm3 s-1 Medium flow rate
H 0.0311 mm3 s-1 Perfusion rate
αM 0.0137 nmol mm-3 kPa-1 Solubility coefficient for O2 in water
αH 0.0123 nmol mm-3 kPa-1 Solubility coefficient for O2 in haemolymph
αT 0.0147 nmol mm-3 kPa-1 Solubility coefficient for O2 in tissue
DM 0.0020 mm2 s-1 Diffusion coefficient for O2 in water
DH 0.0015 mm2 s-1 Diffusion coefficient for O2 in haemolymph
DT 0.0010 mm2 s-1 Diffusion coefficient for O2 in tissue
g 0.0010 nmol s-1 mm-2 kPa-1 Permeability of the cuticular diffusion barrier
Δr 0.005 mm Radial distance between two grid points
Δh 0.1 mm Axial distance between two grid points
Δt ≤0.005 Time interval for numerical calculation
Derived (secondary) parameters
0.120 mm Radius of the central tissue cylinder
0.280 mm Outer radius of the truncal haemolymph space
0.350 mm Outer radius of the peripheral tissue layer
0.500 mm Outer radius of the medium lacuna
r0 0.520 mm Outer radius of the carapace lacuna
a 0.015 nmol s-1 mm-3 Volume-specific O2 consumption rate (pure tissue)
νM 2.09 mm s-1 Flow velocity of the medium in the medium lacuna
νHC 0.13 mm s-1 Flow velocity of the haemolymph in the carapace lacuna
νHT 0.96 mm s-1 Flow velocity of the haemolymph in the trunk lacuna
SymbolValueUnitDescription
Key (primary) parameters
V 1.12 mm3 Body volume (haemolymph and tissue compartments)
h0 2.5 mm Height of the cylindrical model
AM 0.4 mm2 Flow cross-sectional area of the medium flow
Δx 0.02 mm Thickness of the carapace lacuna
Φ 0.4  Tissue fraction of body volume
ζ 0.25  Fraction of total tissue in the central tissue cylinder
a0 0.0058 nmol s-1 mm-3 Volume-specific O2 consumption rate (whole body)
M 0.8333 mm3 s-1 Medium flow rate
H 0.0311 mm3 s-1 Perfusion rate
αM 0.0137 nmol mm-3 kPa-1 Solubility coefficient for O2 in water
αH 0.0123 nmol mm-3 kPa-1 Solubility coefficient for O2 in haemolymph
αT 0.0147 nmol mm-3 kPa-1 Solubility coefficient for O2 in tissue
DM 0.0020 mm2 s-1 Diffusion coefficient for O2 in water
DH 0.0015 mm2 s-1 Diffusion coefficient for O2 in haemolymph
DT 0.0010 mm2 s-1 Diffusion coefficient for O2 in tissue
g 0.0010 nmol s-1 mm-2 kPa-1 Permeability of the cuticular diffusion barrier
Δr 0.005 mm Radial distance between two grid points
Δh 0.1 mm Axial distance between two grid points
Δt ≤0.005 Time interval for numerical calculation
Derived (secondary) parameters
0.120 mm Radius of the central tissue cylinder
0.280 mm Outer radius of the truncal haemolymph space
0.350 mm Outer radius of the peripheral tissue layer
0.500 mm Outer radius of the medium lacuna
r0 0.520 mm Outer radius of the carapace lacuna
a 0.015 nmol s-1 mm-3 Volume-specific O2 consumption rate (pure tissue)
νM 2.09 mm s-1 Flow velocity of the medium in the medium lacuna
νHC 0.13 mm s-1 Flow velocity of the haemolymph in the carapace lacuna
νHT 0.96 mm s-1 Flow velocity of the haemolymph in the trunk lacuna

Data refer to a 2.5 mm-long, fasting D. magna without embryos in the brood chamber at food-free, normoxic conditions and to 20°C.

Whereas the first three assumptions are easy to comprehend, points 4-7 require a brief justification. The value for AM was derived from an experimental study (Pirow et al., 1999a; value not explicitly stated there). The value forΔ x represents a rough estimate since the thickness of the haemolymph space varies across the carapace(Dahm, 1977; Schultz and Kennedy, 1977; Fryer, 1991). The value of 0.25 assigned to ξ is exactly that fraction of a solid cylinder that is encircled by half of the cylinder radius. This choice ensured that there is a sufficiently great sink for oxygen in the central region of the model as would be the case if haemolymph and tissue were more homogeneously distributed within the trunk. The value for Δr was chosen to divide the thin hemolymph compartment of the carapace lacuna into four intervals. The influence of these somewhat arbitrarily chosen parameter values on model behaviour is assessed by a sensitivity analysis in the Results.

The remaining parameters describing the functional properties of the model refer to 20°C and a 2.5 mm-longanimal in the fasting state. The convective flow velocities (vM, vHC, vHT) were derived by dividing the medium flow rate M (mm3s-1) or the perfusion rate H (mm3s-1) by the respective flow cross-sectional area(Rouse, 1978; e.g. vM=M/AM). M was calculated from fa (360 min-1; present study) according to functional relationships (Pirow et al.,1999a), whereas H was obtained from stroke volume (Bäumer et al.,2002) and fh (257 min-1; present study).

The solubility of oxygen in water (αM) was obtained from Gnaiger and Forstner (1983),taking the salinity of 0.2‰ of the culture medium into account. The solubility of oxygen in haemolymph (αH) was assumed to be that of human plasma (58-85 g protein l-1; Christophorides et al., 1969). Values reported for the oxygen solubility in tissue (αT) vary in the range of 0.0097-0.0169 nmol mm-3 kPa-1(Grote, 1967; Grote and Thews, 1962; Mahler et al., 1985; Thews, 1960). For the present model, a value of 0.0147 nmol mm-3 kPa-1 was chosen for the tissue compartment.

Values reported for the diffusion coefficient for oxygen in water(DM) vary in the range of 0.0017-0.0025 mm2s-1 (Bartels, 1971; Gertz and Loeschke, 1954; Goldstick and Fatt, 1970; Grote, 1967; Grote and Thews, 1962; Hayduk and Laudie, 1974; Himmelblau, 1964; St-Denis and Fell, 1971). For the present model, a value of 0.0020 mm2 s-1 was chosen. A variety of data also exist for the diffusion coefficient for oxygen in tissue (DT; corrected to 20°C if necessary), such as 0.0008-0.0020 mm2 s-1 for vertebrate skeletal or heart muscle tissue (Bentley et al.,1993; Ellsworth and Pittman,1984; Grote and Thews,1962; Homer et al.,1984; Mahler et al.,1985), 0.0015 mm2 s-1 for rat lung tissue(Grote, 1967) and 0.0012 mm2 s-1 for rat grey matter(Thews, 1960). For the present model, a value of 0.0010 mm2 s-1 was chosen for the tissue compartment. For haemolymph, we assumed a diffusion coefficient to be 75% of that in water, because similar relationships (70-85%) were reported for both bovine serum (Gertz and Loeschke,1954; Yoshida and Ohshima,1966) and an 8.5% solution of bovine serum albumin(Kreuzer, 1950). The permeability of the cuticular diffusion barrier (g) was calculated by dividing Krogh's diffusion coefficient for oxygen in chitin(1.27×10-6 nmol s-1 cm-1torr-1; Krogh,1919) by the thickness of the cuticle, for which a value of 0.001 mm was assumed (Pirow et al.,1999b).

The volume-specific oxygen consumption rate of pure tissue (a) was derived by dividing the volume-specific oxygen consumption rate of the whole animal (a0) by the tissue fraction of body volume(φ). The value for a0 was obtained by dividing the respiration rate (Y) of 23.6 nmol O2 animal-1h-1 by body volume (V). Y was calculated according to the allometric equation Y=0.3087X0.95(Glazier, 1991) using a dry body mass (X) of 96 μg for a 2.5 mm-long D. magna with no brood in the brood chamber (Kobayashi,1983).

### Circulatory and ventilatory responses to hypoxia

Exposing normoxia-acclimated D. magna to food-free conditions with a gradual decline in ambient oxygen tension(PO2amb) resulted in a statistically significant increase in heart rate (fh) from 257±16 min-1 (N=5) at normoxia (20.9 kPa) to a maximum of 412±33 min-1 at 3.9 kPa, followed by a deceleration of heart beat below 2.3 kPa (Fig. 5A; Table 3). Appendage beating rate (fa) remained constant at a high level of 321-364 min-1 in the range of PO2amb from 20.9 kPa to 2.3 kPa but decreased significantly with a further reduction in PO2amb (Fig. 5B; Table 3).

Fig. 5.

Responses in (A,C) heart rate (fh) and (B,D) appendage beating rate (fa) to decreasing ambient oxygen tensions at food-free conditions (left; N=5) and food-rich conditions (right;105 algal cells ml-1; N=11 except for 0.8 kPa,where N=4). Data are given as means ± s.d. A repeated-measures ANOVA was performed for all data. At food-free conditions,the mean fa (F=25.6, groups d.f.=9, remainder d.f.=36, P<0.001) and the mean fh(F=48.0, groups d.f.=6, remainder d.f.=36, P<0.001) were not the same in animals on all 10 oxygen levels. At food-rich conditions, the mean fa (F=3.1, groups d.f.=9, remainder d.f.=36, P=0.025) and the mean fh (F=15.4, groups d.f.=9, remainder d.f.=36, P<0.001) were not the same in animals on all 10 oxygen levels. The results of multiple comparisons among pairs of means are shown in Tables 3, 4.

Fig. 5.

Responses in (A,C) heart rate (fh) and (B,D) appendage beating rate (fa) to decreasing ambient oxygen tensions at food-free conditions (left; N=5) and food-rich conditions (right;105 algal cells ml-1; N=11 except for 0.8 kPa,where N=4). Data are given as means ± s.d. A repeated-measures ANOVA was performed for all data. At food-free conditions,the mean fa (F=25.6, groups d.f.=9, remainder d.f.=36, P<0.001) and the mean fh(F=48.0, groups d.f.=6, remainder d.f.=36, P<0.001) were not the same in animals on all 10 oxygen levels. At food-rich conditions, the mean fa (F=3.1, groups d.f.=9, remainder d.f.=36, P=0.025) and the mean fh (F=15.4, groups d.f.=9, remainder d.f.=36, P<0.001) were not the same in animals on all 10 oxygen levels. The results of multiple comparisons among pairs of means are shown in Tables 3, 4.

Table 3.

At high food concentration (105 algal cells ml-1),progressive hypoxia induced responses in fh and fa that deviated from those observed under food-free conditions. During the initial normoxic conditions, food-provided animals had a higher fh (344±62, N=11; Fig. 5C) than those without food (257±16, N=5). In response to the progressive reduction in PO2amb, animals of both groups developed a tachycardia that was more pronounced in the food-deprived group. The larger scope for circulatory adjustment in the food-deprived group resulted from a lower initial fh, because the maximum fh at ∼4 kPa was almost the same in both groups(412±33 min-1vs 416±31 min-1). The increase in the mean fh of the food-provided group from 344 min-1 at normoxia (20.9 kPa) to 395-416 min-1at hypoxia (15.3-2.0 kPa) was statistically significant(Table 4).

Table 4.

For PO2amb values higher than 8 kPa, the fa of food-provided animals was always lower, on average by 93 min-1, than that of animals without food. The depressing effect of the high food concentration on the normoxic fawas, however, not as great as expected from the food modulation experiment(cf. Fig. 5D and Fig. 1B). As in the food-deprived group, the reduction of PO2ambhad no effect on the fa of food-provided animals in the range from 20.9 kPa to 6.8 kPa (Fig. 5D; Table 4). However, below 8 kPa, both groups showed diverging changes in mean fa. Whereas the fa of food-provided animals started to rise significantly from 265±48 min-1 at 9.5 kPa to 326±47 min-1 at 3.0 kPa(Table 4), that of the food-deprived group declined non-significantly from 360±50 min-1 at 8.8 kPa to 321±76 min-1 at 2.3 kPa(Table 3).

Comparing the shape of individual response curves for fh and fa with that of the respective mean response curves shown in Fig. 5A-D, there was always a good correspondence between individual and mean curves except for the fa of the food-provided animals. This group showed large interindividual variations in the initial,normoxic fa ranging from 153 min-1 to 356 min-1 and, as a consequence, nonuniform responses to declining PO2amb (Fig. 6A). All animals that had attained an initial, normoxic fa lower than 260 min-1 exhibited a pronounced hypoxia-induced increase in limb beating activity whereas those with a rate higher than 300 min-1 were hardly able to further elevate their fa (Fig. 6B).

Fig. 6.

(A) Individual responses (N=11) in appendage beating rate(fa; solid and broken lines) to decreasing ambient oxygen tensions at high food concentration (105 algal cells ml-1). The profiles were shifted vertically for clarity. Eight of 11 animals showed a hyperventilatory response (solid lines). The dotted lines indicate the two reference points, the initial normoxic level at 21 kPa and the hypoxic level of 3 kPa, which were used to assess the hypoxia-induced changes in fa. (B) Magnitude of hypoxia-induced changes in fa in relation to the initial faprevailing before the start of the hypoxic exposure. The dotted line demarcates positive (open circles) from negative responses (filled circles).

Fig. 6.

(A) Individual responses (N=11) in appendage beating rate(fa; solid and broken lines) to decreasing ambient oxygen tensions at high food concentration (105 algal cells ml-1). The profiles were shifted vertically for clarity. Eight of 11 animals showed a hyperventilatory response (solid lines). The dotted lines indicate the two reference points, the initial normoxic level at 21 kPa and the hypoxic level of 3 kPa, which were used to assess the hypoxia-induced changes in fa. (B) Magnitude of hypoxia-induced changes in fa in relation to the initial faprevailing before the start of the hypoxic exposure. The dotted line demarcates positive (open circles) from negative responses (filled circles).

### Efficiency of systemic adjustments predicted by the model

To estimate the efficiency of ventilatory and circulatory adjustments in improving oxygen transport to tissue, we examined the behaviour of the conceptual model (Fig. 2B) and determined the critical ambient oxygen tension(PO2crit) at which the rate of oxygen consumption decreased to 99% of the maximum(Fig. 7A). The data that were initially entered into the model referred to a fasting 2.5 mm-long D. magna exposed to food-free, normoxic conditions at 20°C(Table 2). For this physiological state, the numerical evaluation yielded a PO2crit of 10.1 kPa. At this PO2crit, the oxygen partial pressure distribution in the medium lacuna and the carapace lacuna revealed an almost complete equilibration of medium and haemolymph at the posterior part of the model (Fig. 8). The depression in oxygen consumption rate resulted from the formation of an anoxic corner in the anterior part of the central tissue cylinder. The convective contribution of the circulatory system to total oxygen supply above the PO2crit was 32% (grey-shaded area in Fig. 7A). A hypothetical doubling of medium flow rate(M) had almost no effect on the PO2crit under these conditions(Fig. 7A). The PO2crit decreased only by 0.3 kPa. By contrast,a doubling of perfusion rate(H) caused a pronounced reduction of PO2crit to 7.9 kPa.

Fig. 7.

Model predictions revealing the efficiency of ventilatory and circulatory adjustments. Solid curves show the predicted dependencies of the oxygen consumption rate (100%24 nmol h-1) upon ambient oxygen tension for the fasting state/food-free conditions (A) and fed state/food-rich conditions (B). Both states differ from each other in the volume-specific oxygen consumption rate (a0), perfusion rate(H) and ventilation rate(M). H and M represent normoxic values. Vertical lines mark the critical ambient oxygen tensions(PO2crit) at which the rates of oxygen consumption decreased to 99% of the maximum. Below PO2crit, the central tissue cylinder experiences an inadequate supply with oxygen. The overproportional decline in oxygen consumption rate in A and B below 4 kPa and 6 kPa (bold arrows),respectively, indicates the incipient impediment of oxygen provision to the peripheral tissue layer. Horizontal arrows indicate the reductions in PO2crit by hypothetically doubling either H or M. The grey shaded areas reflect the amounts of oxygen transported by the circulatory system; the remaining white areas below the solid curves are those amounts diffusing from the respiratory medium directly into the peripheral tissue layer.

Fig. 7.

Model predictions revealing the efficiency of ventilatory and circulatory adjustments. Solid curves show the predicted dependencies of the oxygen consumption rate (100%24 nmol h-1) upon ambient oxygen tension for the fasting state/food-free conditions (A) and fed state/food-rich conditions (B). Both states differ from each other in the volume-specific oxygen consumption rate (a0), perfusion rate(H) and ventilation rate(M). H and M represent normoxic values. Vertical lines mark the critical ambient oxygen tensions(PO2crit) at which the rates of oxygen consumption decreased to 99% of the maximum. Below PO2crit, the central tissue cylinder experiences an inadequate supply with oxygen. The overproportional decline in oxygen consumption rate in A and B below 4 kPa and 6 kPa (bold arrows),respectively, indicates the incipient impediment of oxygen provision to the peripheral tissue layer. Horizontal arrows indicate the reductions in PO2crit by hypothetically doubling either H or M. The grey shaded areas reflect the amounts of oxygen transported by the circulatory system; the remaining white areas below the solid curves are those amounts diffusing from the respiratory medium directly into the peripheral tissue layer.

Fig. 8.

Oxygen partial pressure distribution in the median plane of the radially symmetrical model at the critical ambient oxygen tension(PO2crit) of 10.1 kPa (see Fig. 7A). The upper-case letters along the radial axis mark the different compartments, and the vertical lines indicate the compartment interfaces (A, central tissue cylinder; B, truncal haemolymph space; C, peripheral tissue layer; D, medium lacuna; E, carapace lacuna). Note the formation of the anoxic corner in the central tissue cylinder.

Fig. 8.

Oxygen partial pressure distribution in the median plane of the radially symmetrical model at the critical ambient oxygen tension(PO2crit) of 10.1 kPa (see Fig. 7A). The upper-case letters along the radial axis mark the different compartments, and the vertical lines indicate the compartment interfaces (A, central tissue cylinder; B, truncal haemolymph space; C, peripheral tissue layer; D, medium lacuna; E, carapace lacuna). Note the formation of the anoxic corner in the central tissue cylinder.

To assess the efficiency of systemic adjustments for a fed animal exposed to food-rich, normoxic conditions, new parameter values characterizing this changed physiological state were entered into the model. This state is characterized by an increased metabolic rate(Lampert, 1986; Bohrer and Lampert, 1988), a reduced fa and an elevated fh (cf. Figs 1, 3). The volume-specific,whole-body oxygen consumption rate (a0) was assumed to be 50% higher than that of the fasting state (see Discussion). In addition, M was halved and H was set to 135% of the rate of the fasting state, thus assuming that the observed relative changes in fa and fh translate into the same relative changes in the respective flow rates. For this physiological state,the model yielded a PO2crit of 14.3 kPa(Fig. 7B). A doubling of M reduced the PO2crit by 0.7 kPa, which was more than twice as large as the decrease determined for the fasting state. Nevertheless, the change in H was still most effective, because the doubling of H caused a reduction of PO2crit by 3.0 kPa.

To assess the effect of changes in key parameters other than M and H, a sensitivity analysis was performed using the data from Table 2 (referring to the fasting state and food-free normoxic conditions). The value of each individual key parameter was then either decreased to 50% or increased to 200% of its initial value (all others being equal) and the percentage change in PO2crit was evaluated (Fig. 9). Of all the geometrical parameters tested (h0, AMx, ξ), the sensitivity was highest for h0 (height of the cylindrical model) and for ξ, which defines the allocation of tissue to the two tissue compartments. Variations in the radial and axial grid intervals (Δr, Δh)caused only minor changes (<1%) in PO2crit. The permeability of the cuticular diffusion barrier (g) did not prove to be a limiting factor of the oxygen transport cascade, whereas changes in the diffusing properties of oxygen in tissue(αT,DT) had a great influence on PO2crit. Of all the parameters affecting the convective and diffusive transport of oxygen in the medium and haemolymph compartments, the solubility coefficient for oxygen in haemolymph(αH) had the greatest impact on PO2crit(Fig. 9).

Fig. 9.

Sensitivity analysis showing the effect of individual parameter changes on the critical ambient oxygen tension (PO2crit). The initial state refers to the fasting state(Table 2) with a PO2crit of 10.1 kPa (100%). The value of each parameter (listed by its symbol along the horizontal axis) was decreased to 50% (white) and increased to 200% (grey) of its initial value(Table 2) while keeping all other parameters unchanged.

Fig. 9.

Sensitivity analysis showing the effect of individual parameter changes on the critical ambient oxygen tension (PO2crit). The initial state refers to the fasting state(Table 2) with a PO2crit of 10.1 kPa (100%). The value of each parameter (listed by its symbol along the horizontal axis) was decreased to 50% (white) and increased to 200% (grey) of its initial value(Table 2) while keeping all other parameters unchanged.

The present study revealed that the oxyregulating species Daphnia magna exhibits a greater flexibility in adjusting systemic functions involved in oxygen transport than has previously been assumed(Paul et al., 1997; Pirow et al., 2001). Not only the circulatory system but also the ventilatory system proved to be sensitively tuned to ambient oxygen tension(PO2amb). The extent, however, to which the convective performance of each system could be increased during progressive hypoxia depended on the initial circulatory and ventilatory states, which in turn were largely affected by food availability and the nutritional state of the animal.

Under food-free conditions, the moderate reduction in PO2amb did not influence appendage beating rate(fa), which was used as a measure of ventilatory performance. Heart rate (fh), on the contrary, increased by 61%, indicating a large regulatory scope in the circulatory system under these conditions. This cardio-ventilatory response pattern was in line with the previous results (Paul et al.,1997; Pirow et al.,2001).

A reversed response pattern occurred when the animals had access to food in high concentration. Exposure to food-rich conditions (105 algal cells ml-1) instead of food-free conditions resulted in a lower mean initial fa (265 min-1vs 360 min-1) and a higher mean initial fh (344 min-1vs 257 min-1). In this physiological state, with an individual fa lower than 260 min-1, D. magna was able to respond to a reduction in PO2amb with a compensatory increase in ventilation. The fact that four out of 11 individuals with an fa above 300 min-1 failed to show this hyperventilatory response suggests that other unknown factors counteracted the depressing effect of high food concentrations on fa. The hypoxia-induced tachycardia was less pronounced under food-rich conditions than under food-free conditions because of the higher initial fh. These findings clearly demonstrate that the scope for a short-term improvement of oxygen transport shifted from the circulatory to the ventilatory system. The oxyregulatory repertoire of the millimetre-sized D. magna consequently includes a systemic response pattern that is comparable to that of large-sized, physiologically advanced water breathers such as fish (Dejours, 1981; Randall et al., 1997) or decapod crustaceans (Wheatly and Taylor,1981).

The decelerating effect of high food concentrations on fa has been known for quite some time(McMahon and Rigler, 1963; Burns, 1968; Porter et al., 1982). Above a critical food concentration of 104 algal cells ml-1(Porter et al., 1982), the amount of food retained by the filter combs of the thoracic appendages exceeds the amount that the animal is able to ingest and/or digest(Rigler, 1961). The decrease in fa reduces the imbalance between food supply (the amount of food initially retained) and demand. However, the reduction in fa is obviously not sufficient to remove this imbalance completely, since a higher rate of rejection movements of the postabdominal claw is required to remove superfluous food particles from the filter apparatus (Porter et al.,1982). From the energetic point of view, one would expect a further reduced or intermittent appendage activity, which would make additional rejection movements unnecessary. However, taking the dual function of appendage movement into account, it is likely that the actual fa exhibited under these conditions represents a compromise between the need to reduce energetic expenditures for food collection and the need to satisfy the increased oxygen requirements of the fed animal.

Lampert (1986) as well as Bohrer and Lampert (1988) have shown that the respiratory rate (carbon loss per time interval) of well-fed D. magna is more than twice the rate of starving animals. Taking the change in the respiratory quotient from 0.7 (starved) to 1.15 (well-fed; Lampert and Bohrer, 1984) into account, this elevation in respiratory rate corresponds to a 1.4-fold increase in oxygen consumption rate. The higher oxygen demand arises from the digestion and biochemical processing of food (Bohrer and Lampert, 1988; Philippova and Postnov, 1988) and the tissues involved in this physiological task; i.e. the digestive tract and the fat cells as a storage site of reserve mass are, to a great extent, located in the central part of the trunk rather than in the peripheral body region. This is important to note because centrally located tissues are more dependent on a convective supply of oxygen via the circulatory system than are those in the periphery of the body, which can rely to a greater extent on a direct diffusive provision of oxygen from the ambient medium. Since the open circulatory system of Daphnia spp. lacks any arteries and capillaries, there is no possibility of redirecting a greater share of blood flow to these metabolically activated tissues, and an improvement of local oxygen supply can only be achieved by an increase in total perfusion rate. This could explain why the initial, normoxic fh of food-provided animals was 34% higher than that of starving animals. Besides this explanation, a circulatory compensation for the reduction in external convection cannot be excluded. In the food-modulation experiment(Fig. 1), we observed in two of four animals a complete stop of appendage movement for 1.5 min after the food concentration had been changed from 106 to 0 algal cells ml-1. This behavioural change, which occurred under normoxic conditions, was immediately followed by a sharp increase in fh by 20-24%, and fh remained at this high level until the animals resumed limb beating activity.

The experimental findings of the present study suggest that a circulatory adjustment is the most effective measure of hypoxia adaptation in the planktonic filter feeder D. magna, at least under low food concentrations when fa is at a maximum. The apparent inability to further increase fa under these conditions might result from biomechanical or energetic constraints such as the hydrodynamic resistance of the filter combs and the energetic costs of pumping medium. But even if such constraints would not exist, it is questionable whether an enhancement of ventilatory activity would have any beneficial effect, for example, in enabling the animal to sustain its rate of oxygen uptake at a far lower PO2amb. An experimental test to answer this question is hardly possible. However, since a large amount of physiological information is available for D. magna, this question can be approached theoretically by the use of a conceptual model.

Conceptual models have proven to be a valuable complement to experimental approaches and have made it possible to analyse, for example, the transfer characteristics and transport limitations of the different gas exchange organs of vertebrates (Piiper and Scheid,1975). Contrary to the situation in vertebrates, the millimetre-sized D. magna lacks an arrangement in which the circulatory system links distinct sites for respiratory gas exchange and tissue gas transfer (Taylor and Weibel,1981; Piiper,1982; Weibel,1984; Shelton,1992). The whole integument of D. magna is, in principle,permeable to respiratory gases, and the oxygen is moved from the body surface to the tissues by diffusion and convection as well. We have therefore devised a conceptual model that takes a direct diffusive supply of oxygen to the peripheral tissues via the ambient medium and a convective supply to the centrally located tissues via the circulatory system into account.

The predictions made by this model gave support to the hypothesis raised in the Introduction that the circulatory system is the limiting step of the oxygen transport cascade in D. magna. The model analysis showed that an increase in perfusion rate was most effective both under food-free (fasting state) and food-rich (fed state) conditions in reducing the critical ambient oxygen tension (PO2crit) at which oxygen supply to the tissue started to become impeded. By contrast, an increase in ventilation rate had almost no effect on PO2crit under food-free conditions but a moderate effect under food-rich conditions. Since the regulatory scope for an adjustment in heart rate was found to be limited in D. magna under food-rich conditions, the increase in ventilation rate is the means of choice for a fed animal to cope with short-term, moderate hypoxia. The improvement of oxygen supply in the animal by enhancing ventilatory flow may also include the reduction of fluid and diffusive boundary layers, an aspect that was not considered in the present model. Under chronic and more severe hypoxic conditions, however, the increase in the concentration and oxygen affinity of Hb represents the one and only measure for improving the transport of oxygen from environment to cells.

In the present model, Hb was not considered as the haemolymph oxygen carrier in order to reduce mathematical complexity. This might explain why the critical ambient oxygen concentration of 3.9 mg O2 l-1(PO2crit=7.9 kPa), which was predicted for the fasting state with doubled perfusion rate(Fig. 7A), was higher than the critical oxygen concentrations of 1.3-3.0 mg O2 l-1reported for filtering and respiration rates of Hb-poor D. magna and D. pulex (Kring and O'Brien,1976; Heisey and Porter,1977; Kobayashi and Hoshi,1984). The high sensitivity of the model to changes in the solubility coefficient for oxygen in the haemolymph (αH)indicates that Hb can have a pronounced effect on PO2crit, since it enhances both the convective and diffusive transport of oxygen in the haemolymph. The implementation of Hb-mediated oxygen transport is the next step when extending the model, which will then allow us to provide a causal mechanistic explanation for the expression of specific Hb-related characteristics (concentration, oxygen affinity, cooperativity) in animals of a given anatomical disposition and physiological constitution under different environmental conditions.

• a

volume-specific oxygen consumption rate of pure tissue (nmol s-1 mm-3)

•
• a0

volume-specific oxygen consumption rate of the whole body (nmol s-1 mm-3)

•
• $$A_{\mathrm{{\ }i}}^{\mathrm{u}},{\ }A_{\mathrm{{\ }i}}^{\mathrm{o}}$$

area of hollow-cylindrical bases (mm2)

•
• AM

flow cross-sectional area perpendicular to medium flow(mm2)

•
• Bi-0.5, Bi, Bi+0.5

cylindrical wall areas (mm2)

•
• D

diffusion coefficient for oxygen (mm2s-1)

•
• DH

diffusion coefficient for oxygen in haemolymph (mm2s-1)

•
• DM

diffusion coefficient for oxygen in medium (mm2s-1)

•
• DT

diffusion coefficient for oxygen in tissue (mm2s-1)

•
• fa

appendage beating rate (min-1)

•
• fh

heart rate (min-1)

•
• g

permeability of the cuticular diffusion barrier (nmol s-1mm-2 kPa-1)

•
• h

axial coordinate (mm)

•
• h0

height of the cylindrical model (mm)

•
• i, j, k

indices of grid points

•
• Nax

number of grid points in axial direction

•

number of grid points in radial direction

•
• P

oxygen partial pressure (kPa)

•
• Pa

oxygen partial pressure of the haemolymph entering the trunk lacuna(kPa)

•
• Pex

oxygen partial pressure of the expiratory medium (kPa)

•
• Pi,j

oxygen partial pressure at the grid point referenced by i, j(kPa)

•
• $$P_{\mathrm{{\ }j,i}}^{\mathrm{o}}$$

oxygen partial pressure at the outer side of an infinitesimal thin diffusion barrier (kPa)

•
• $$P_{\mathrm{{\ }j,i}}^{\mathrm{u}}$$

oxygen partial pressure at the inner side of an infinitesimal thin diffusion barrier (kPa)

•
• Pin

oxygen partial pressure of the inspiratory medium (kPa)

•
• PO2amb

ambient oxygen tension (kPa)

•
• PO2crit

critical ambient oxygen tension (kPa)

•
• Pv

oxygen partial pressure of the haemolymph leaving the trunk lacuna(kPa)

•
• H

perfusion rate (mm3 s-1)

•
• r

•
• r0

•
• t

time (s)

•
• v

flow velocity (mm s-1)

•
• V

volume of the cylindrical model (mm3)

•
• vHC

flow velocity of the haemolymph in the carapace lacuna (mm s-1)

•
• vHT

flow velocity of the haemolymph in the trunk lacuna (mm s-1)

•
• $$V_{\mathrm{{\ }i}}^{\mathrm{u}},{\ }V_{\mathrm{{\ }i}}^{\mathrm{o}}$$

volume of hollow cylinders (mm2)

•
• vM

flow velocity of the medium (mm s-1)

•
• M

medium flow rate (mm3 s-1)

•
• X

dry body mass (μg)

•
• Y

respiration rate (nmol animal-1 h-1)

•
• Δh

distance between two grid points in axial direction (mm)

•
• ΔPj,i

change in Pj,i during Δt(kPa)

•
• Δr

distance between two grid points in radial direction (mm)

•
• Δt

time interval (s)

•
• Δx

thickness of the carapace lacuna (mm)

•
• α

solubility coefficient for oxygen (nmol mm-3kPa-1)

•
• αH

solubility coefficient for oxygen in haemolymph (nmol mm-3kPa-1)

•
• αM

solubility coefficient for oxygen in medium (nmol mm-3kPa-1)

•
• αT

solubility coefficient for oxygen in tissue (nmol mm-3kPa-1)

•
• φ

tissue fraction of body volume

•
• ξ

fraction of total tissue allocated to the central tissue cylinder

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