1. In Part I the ability of the plastron mechanism of Aphelocheirus to retain its gas film against a pressure deficiency of several atmospheres was demonstrated, and preliminary experiments were carried out in order to establish that the plastron is in fact respiratory in function. In the present paper quantitative evidence is submitted in support of this view. A comparison is made between the efficiency of the cutaneous respiration of the fifth instar nymph and that of the plastron of the adult.

  2. The basic respiratory rate of nymphs and adults was determined by a metal micro-Barcroft respirometer (described in a separate paper, Crisp & Thorpe, 1947) and also by a micro-Winkler method. The basal metabolic rate when the insect is at rest is, for the adult, about 6 cu.mm./hr. O2 uptake and for the nymph about 3 cu.mm./hr. at 20 ° C. This corresponds in each case to 150 c.c./hr./kg. live weight. Insects when active in the apparatus increase their respiratory rate six- or sevenfold, and in a strongly swimming adult the increase is probably much more than this.

  3. In order to obtain a measure of respiratory efficiency which is independent of the momentary rate of metabolism the concept of ‘diffusion resistance’ R is employed, where Δp=fall in tension in atmospheres and q=the quantity flowing in units/sec. R= Δp/q. Another concept, the ‘respiratory index’, is defined as the reciprocal of the minimum oxygen tension difference between the animal and its surroundings required to maintain the basal respiratory rate. These quantities are determined approximately for Aphelocheirus nymphs and adults by measurement of oxygen uptake at low tensions of oxygen in nitrogen. The principles of this method are described.

  4. Similar measurements on adults whose spiracles were blocked with low melting-point wax made possible a comparison between the permeability of the cuticle of adult and nymph. The former was found to be 0·38 × 10−7 c.c. 02/sec./cm. thickness/sq.cm./atm. difference in tension, and the latter was considerably higher, being 1·6 × 10−7 unit. These values are compared with others found in the literature and are of the same order.

  5. The passage of oxygen through the plastron and tracheae of the adult is considered in a number of steps, and the fall in tension calculated at each step. It is shown that the total fall in tension for basal metabolism is about 0·03 atm. on a basis of simple diffusion, making certain necessary simplifying assumptions. This figure agrees with that computed experimentally, at least in order of magnitude, and substantiates the view that the plastron is an efficient respiratory organ over the whole of its surface and operates as a physical gill according to the established principles of diffusion. The chief resistances to diffusion appear to be at the water-air boundary of the plastron, and from the tracheal endings into the tissues.

  6. A comparison of adult and nymphal respiration is given in tabular form, and the necessity for a supplementary mode of respiration in the adult is shown.

  7. The development of the plastron in the fifth instar nymph prior to moulting is described, and gas is shown to appear there during the reabsorption of the moulting fluid without the animal having access to the surface. The gas is present before moulting takes place, and before the plastron becomes connected with the tracheal system. The waterproofing of the hairs is probably not a result of secretion of gas into the plastron, but is a necessary condition before gas can replace the moulting fluid there. Moulting takes place beneath the surface of the water.

  8. A general theory of the operation of the plastron and its relation to an air store is developed. Since the plastron is capable of withstanding external pressure it can be regarded functionally as a special case of the closed tracheal system. Thus not only the plastron but the whole of the tracheal system containing air must be capable of withstanding the excess pressure Δp. In practice some of this excess pressure may be taken up by the elasticity of the external cuticle, some by the elasticity of the tracheal walls themselves, but a single defect, allowing the volume of the air space to decrease continuously, will render the whole system ineffective.

  9. It is shown that the plastron in Aphelocheirus is unlikely to be rendered ineffective by any environmental condition which the animal is likely to meet, though should the animal descend to great depths the strain on the system would be considerable.

  10. The nature of the plastron is shown to be a major factor governing the ecology of the insect. As a result of the development of this method of respiration Aphelocheirus has become independent of contact with the air. This in turn has enabled it to inhabit swift-flowing streams without exposure to the danger of being swept away which visits to the surface would entail. The fact that it does not have to carry an air store has thus enabled it to become a true bottom-living predator heavier than water. It can thus occupy an ecological niche not available to any other comparable predatory insect in the adult stage. On the other hand, of course, its method of respiration necessarily restricts it to well-aerated, that is to turbulent, water.

  11. An appendix ( Appendix II) is supplied in which certain general relations for diffusion in an absorbing medium are deduced, and these are applied to calculations of oxygen diffusion from the tracheal endings.

In a previous paper (Thorpe & Crisp, 1947, later referred to as Part I) the plastronbearing mechanism of Aphelocheirus was described in detail. The ability of this structure under good conditions to retain a layer of gas against a pressure deficiency of several atmospheres was demonstrated, and preliminary experiments were carried out in order to establish that the plastron was in fact respiratory in function. In this paper We submit quantitative evidence in support of this view, and show that efficient respiration is possible through the plastron by ordinary diffusion. A comparison is made between the efficiency of the cutaneous respiration of the fifth instar nymph and that of the plastron of the adult.

The first step in this investigation was to obtain a figure for the basic respiratory rate, and if possible for the oxygen uptake in various states of activity. Owing to the small size of the animal micro-methods were more appropriate, but in the first instance a standard Barcroft manometer was employed. Table 1 gives the results obtained from experiments in which three or four bugs were used at a time.

Experiments were later carried out on individual animals, for which the ordinary Barcroft respirometer was not sufficiently sensitive. For these experiments a new apparatus was devised which operated on the same principle as the Barcroft manometer, but was about 100–200 times as sensitive, and will be described separately (Crisp & Thorpe, 1947). It will be designated a ‘metal Barcroft microrespirometer’, as its most important feature is its construction from a metal block.

Results obtained using this instrument are given in Table 2 and are in general agreement with the figures obtained from the standard Barcroft manometer.

It will be observed from Tables 1 and 2 that the basal metabolic rate (when the insect is at rest) is about 6 cu.mm./hr. for the adult, and for the nymph about 3 cu.mm./hr., at a temperature of 20° C. This corresponds in each case to 150 c.c./hr./kg. live weight, which is rather low in comparison with figures given by Ege (1918) for some other aquatic insects. It will be noted, however, that very considerable variations occur according to the condition of the insect, and the values given above are from selected experiments in which the animal remained quiet for long periods. Moreover, there is definite evidence that the respiratory rate was higher in freshly collected material, probably on account of the slow starvation of the animals under laboratory conditions where the normal food could not be provided in sufficient quantity. Since the animals often died after about 2 months in captivity, it is not unreasonable to attribute the loss of activity and slowing down in metabolic rate to this cause. It is possible, too, that a seasonal fluctuation occurs in the metabolic rate ; though the evidence for this is scanty, the exceptionally high figures recorded from specimens taken early in March suggests that this was possibly the case.

No data on the influence of temperature on respiratory rate was considered necessary, as the maximum temperature likely to be met by the insect in nature is about 20° C. ; lower temperatures reduce metabolism and favour oxygen uptake, so that this figure for the basal rate of metabolism is the maximum that need be considered.

The increase in uptake when the insect is active is clearly illustrated. Adults appear in general to be more active than nymphs, and when struggling violently the respiratory rate may rise six- or sevenfold. Owing to the confined space which the animals occupy during experiments their maximum activity is not reached, and it is probable that a strongly swimming adult would absorb more than six or seven times the basal oxygen uptake.

A small number of observations were carried out on adults under water, using a micro-Winkler technique similar to that described by Fox & Wingfield (1938). In our experiments it was necessary to reduce the scale of the micro-pipette somewhat, as the quantities for estimation were very small. Two or three insects were placed in 8–10 c.c. of water of known oxygen content under a gauze frame, and the surface of the water sealed with Nujol. Samples of 0·5 c.c. were withdrawn by a capillary siphon at intervals and taken up into an Aglar micro-syringe, 0·1 c.c. was first taken up, followed by 0·1 c.c. of manganese sulphate and 0·01 c.c. of Winkler’s reagent. The pipette was then filled up to 0·5 c.c. with the rest of the solution and left standing for 10 min. with occasional shaking. 0·02 c.c. of 50% H2SO4 was finally drawn in and mixed till all the precipitate dissolved, and the resulting solution titrated against N/200 thiosulphate as described below. The order of introducing reagents into the micro-syringe was important in order to obtain a flocculent precipitate of manganous hydroxide.

In titrating the iodine liberated against thiosulphate, two difficulties not mentioned by Fox & Wingfield were encountered. First, we found that a metal syringe needle must be avoided as it rapidly caused reduction of iodine to iodide. Secondly, we found that very careful control observations were necessary in order to obtain a reliable starch-iodine end-point, owing to the failure of starch to acquire a blue colour until a definite excess of iodine had been added. This phenomenon is not important when the total quantity of iodine being estimated is large, but with quantities of the order of 10−5 g. the error is of the order of 50% or more. The following procedure was found to be reliable. A 1% starch solution is diluted 20 times and N/1000 iodine solution run into it until it changes from yellowish brown by transmitted light to a clear or faintly opalescent blue-grey tint. About 0·3 c.c. of this is placed in each of a row of four 0·5 c.c. tubes, and the iodine to be estimated is run into each from the Aglar syringe, 0·005 into the first, 0·01, 0·015 and 0·02 c.c. into the others. These are control tints indicating the colour when the iodine has ‘overshot’ by varying amounts. To the last tube of starch 0·05 c.c. of N/200 thiosulphate is added from a micro-syringe and the iodine solution added to this until a blue tint appears. This is matched successively with each of the control tubes in order to determine the exact quantity of iodine for neutralization. It is important to make up controls just prior to conducting a titration, as they rapidly deteriorate ; they should also be kept at the same temperature as the experimental tube as the intensity of colour appears to be somewhat temperaturedependent.*

These observations gave an average value of 5·5 cu.mm. O2/adult/hr., in agreement with the foregoing. Unfortunately, the Nujol seal was found to be unsatisfactory, differences being obtained according to the oxygen content of the Nujol. The figure above was obtained as a mean between observations using air-saturated and oxygen-free Nujol, but it was afterwards considered advisable in all such experiments to employ mercury only as a seal owing to the solubility of oxygen in organic liquids.

In Part I a number of qualitative experiments are described which indicate that the plastron is efficient as a respiratory organ, but no measure of its relative efficiency compared with cuticular respiration is possible from behaviour records.

Owing to the very flattened shape of Aphelocheirus it is not possible to employ the technique of biological indicators (Thorpe, 1932) to show the relative importance of the various sites of oxygen uptake. It is necessary, therefore, to employ indirect means of assessing the value of the plastron.

A measure of respiratory efficiency which is independent of the momentary rate of metabolism is required, and for this the concept of diffusion resistance is most suitably employed.

If through any closed path of diffusion there exists a fall in tension of the gas of Δp atmospheres, and the quantity flowing is q units/sec., then the diffusion resistance of this path is
formula

Thus, if the path considered is the over-all uptake from the surrounding medium into the tissues, Δp will be the difference in tension between the outer medium and the mean internal tension, while q will be the rate of uptake of oxygen by the tissues. It is sometimes necessary to compare the respiratory efficiency of one organism with another, and for this purpose diffusion resistance alone is not sufficient, because their normal requirements may differ. The most suitable index of oxygen requirement would appear to be the basal metabolic rate qb, and as a measure of respiratory efficiency we shall introduce a quantity, hereafter referred to as the ‘respiratory index’ of I/qbR. The units of this quantity will be (atm.)−1, and it represents simply the reciprocal of smallest tension difference between internal and external media compatible with maintaining basal metabolism.

In general the diffusion of oxygen into the animal may be divided into a number of elementary paths of diffusion of which some may be in series (as, for instance, the different cuticular layers bounding the external surface), and others in parallel (as, for instance, a row of independently opening spiracles). The well-known equations for equivalent resistances may then be employed if it is desired to calculate the over-all resistance from a number of elementary ones :
formula
formula

The most direct method is to measure Δp and q directly in equation (1). The measurement of the internal oxygen tension may be achieved by Krogh’s method (Krogh, 1908), in which a bubble of air is introduced into the animal tissues, and subsequently squeezed out and analysed after time has been allowed for equilibration.

This method was not considered suitable in the case of a very much flattened insect such as Aphelocheirus, since

  • A bubble of size sufficient for analysis would interfere with the normal behaviour.

  • It would be impossible to confine such a bubble to the tissue fluids without contact with the tracheal vessels or cuticle.

  • The introduction of the bubble presents considerable difficulty.

The alternative method employed here is to measure the oxygen uptake of the animal when exposed to successively lower tensions of oxygen. The horizontal axis in Fig. 1 represents oxygen tension, the vertical represents rate of uptake of oxygen (q). Fig. 1 a represents the normal condition when the surroundings contain 20% oxygen; the tissues will not be at uniform tension but will have a distribution as represented by the small hump, at mean tension Δp below the surroundings. The uptake is indicated by the point A. As the external tension is lowered the internal tension will be reduced correspondingly, as is shown in Fig. 1 b. Eventually, however, the oxygen tension in the tissues will begin to reach a limiting threshold (marked P) below which the tissues cannot make use of oxygen and below which their oxygen tension will therefore never fall. As this happens q begins to fall off (Fig. 1 c). When all the tissues are at threshold tension (Fig. 1 d) Δp=p–P and q is limited entirely by the rate of diffusion of oxygen in the system, also determined by p–P, so that in this region the relation between p and q is represented by a straight line (PQ, Fig. 1 e). The relation between q and Δp is also obtainable from this straight line, and from it the diffusion resistance R = Δp/q can be calculated. In all our observations the threshold internal tension is not appreciably different from zero, so that the line PQ appears to cut the origin. This implies that the tissues are capable of abstracting oxygen at very low tensions indeed, in agreement with observations by Maloeuf (1936). However, it is unlikely that tissues deprived severely of their normal oxygen supply will escape uninjured, and this constitutes the chief weakness of the method. Not only are the vital activities and enzyme systems likely to be damaged by prolonged exposure to low oxygen tensions, it is also possible for the animal to go into debt for oxygen and increase its rate of uptake when the external tension is raised. For this reason we have endeavoured to obtain the q–p curves while the oxygen tension is being reduced, but not to maintain the animal at low tensions longer than is necessary. The order in which observations were made is important and is. given in the results.

Fig. 2 A shows the oxygen uptake of adult Aphelocheirus at varying oxygen tensions. These experiments were carried out in a small glass Barcroft apparatus which was not very sensitive and required rather a long period to obtain a satisfactory respiration rate. Consequently exposure to very low tensions was not possible, and the critical region where uptake is proportional to external oxygen tension is scarcely represented. The line PQ from which the diffusion resistance R is calculated gives therefore the maximum value for R. This is seen to be R = 5·2 × 10−3. When expressed in units represented on the graph (cu.mm./hr. and atm.) or expressed in more absolute units (c.c./sec. and atm.) the value is 1·9 × 10−4. The respiratory index can be seen from the graph to be at least 1/0·031 = 32.

Experiments on fifth instar nymphs shown in Fig. 2B were conducted in a micro-respirometer (separately described) and are more satisfactory than the preceding, as it was possible to employ lower oxygen tensions for shorter periods. Moreover, it is possible to observe from these results some of the complications of this method.

Specimen A was maintained at 4% tension for 1 hr. and returned to normal air at 20%. There was initially a rise in uptake shown by point (2) due to the lower internal tension at first, but later the uptake dropped to the point marked (3). Exposure to 2 and 1% oxygen for periods of and 1 hr. respectively did not permanently injure this animal, which on returning to 20% mixture had a much enhanced uptake (point 6).

Specimen B was initially very active with a rather high uptake, and was maintained at 1% for hr. (points 2 and 3). When the oxygen tension was raised, however, recovery was slow and the animal appeared dead (points 4 and 5), until placed in 8% oxygen, when it partially recovered (point 6). In this case it appeared that some damage was caused by prolonged exposure to low oxygen tension, and this was confirmed by the animal failing to recover its initial activity (point 7) and adopting an asphyxia posture for some time after replacing in the aquarium.

Specimen C was the most satisfactory. It was initially very active, and was maintained at the low tensions and 1 % for only and hr. respectively; the observations were made with progressively less oxygen present, and recovery was good.

From these three sets of results the line PQ has been constructed. This gives an over-all diffusion resistance of 3·7 × 104 abs. units, and a respiration index of 33 not measurably different from that obtained for the adult.

In the fifth instar nymph respiration is wholly cuticular, whereas in the adult respiration will take place simultaneously through the cuticle and via the plastron, rosettes and tracheal system. It is therefore desirable to find out what fraction of the uptake in the adult travels by each route, and how the cuticle permeability of the nymph compares with that of the adult.

Attempts made to measure cuticle permeability directly were not successful owing to the great difficulty experienced in completely sealing a small area of cuticle without causing damage. However, it was found possible to obtain information on the cuticle permeability by means of experiments on oxygen uptake in vivo.

If it is assumed that in the limiting region where oxygen uptake is directly proportional to external oxygen tension, the tissues are approximately at zero tension, then the diffusion resistance measured represents the resistance of the whole cuticle. This assumption is a reasonable one, since the internal tissues are likely to be at a fairly uniform tension, but since there may be a small diffusion resistance in addition to that of the cuticle (e.g. the hypodermis) the value obtained for cuticle permeability obtained will be a minimum value. The area of the nymphal cuticle is 75 sq.mm., and the diffusion resistance 3·7 × 104, giving a value for permeability of 3·6 × 10−5 c.c. O2 at N.T.p./sec./atm. pressure difference/sq.cm. area.

In order to obtain the permeability of the adult cuticle in a similar manner, it is necessary to prevent any uptake by the spiracles. This was most conveniently arranged by covering the whole of the ventral surface of the animal with apeizon low melting-point wax. This wax can be melted at a temperature which is not harmful to the bug and sets firmly. The use of soft wax or vaseline as a seal was unsatisfactory owing to the strength of the third thoracic limbs and of the thoracic muscles, which eventually freed the thoracic grooves and allowed tracheal respiration to recommence.

The uptake of sealed adults even in 20% oxygen was always considerably below the basal respiratory rate; the tissues could therefore be assumed to be at zero tension. After each experiment the area of dorsal surface exposed to the atmosphere was measured by camera lucida drawings, and the permeability calculated. The results, shown in Table 3, exhibit considerable variation, as might be expected when different portions of the terga are exposed, but some of the higher values may be due to incomplete sealing and leakage of air into the plastron. It will be seen, however, that the permeability is considerably lower than that of the nymph.

It was not possible to make observations on the permeability of the ventral surface, but its thickness and appearance in section were not in any way different from that of the dorsal, and it is unlikely that its permeability would be significantly different. The total area of the surface of the adult is 1·0 sq.cm., hence at 0·2 atm. oxygen tension the maximum uptake would be, using the mean value for permeability above, 6·1 cu.mm./hr., which is identical with the basal metabolic rate.

It is of interest to compare the values of cuticle permeability to oxygen with those deduced from other investigations. Krogh, using preserved material from Oryctes larvae, gives 1·3 × 10−2 c.c./min./1µ thickness. Fraenkel & Herford (1938), working on the cutaneous respiration of Calliphora larvae, give oxygen consumptions for larvae in which the spiracles are blocked by ligaturing. These figures give a diffusion resistance of 6 × 104. Measurements carried out on larvae of the same species gave the total surface area as 1·0 sq.cm, and the cuticle thickness about 60µ, from which it is possible to calculate the absolute permeability. Table 4 summarizes these results.

It will be seen that the permeability of the nymphal cuticle is of the same order as the soft cuticle of Calliphora erythrocephala larvae, whereas the adult, whose cuticle is harder and more darkly pigmented (a feature usually associated with a greater degree of cross linkage), is of much lower permeability. Krogh’s data, which are taken from isolated cuticle, are of doubtful physiological significance, as the material had been preserved for some time in alcohol.

The lower permeability in the adult would in itself make necessary some auxiliary respiratory mechanism. But there is no obvious advantage to the adult, at least in the form occurring in Britain which is wingless and purely aquatic, in not maintaining a high degree of cutaneous respiration at the same time as evolving a plastron mechanism. On the other hand, we have shown (Part I) the necessity for a rigid structure bounding the animal to take the strain imposed by a lower internal oxygen tension than that existing outside. It is likely that a rigid exocuticle capable of withstanding this is not easily compatible with one of high permeability, and the evolution of the plastron may therefore to some extent involve a reduction in cutaneous respiration. This will help to explain the sudden and complete switchover after the fifth instar larva.

(a) Diffusion resistance of the plastron determined experimentally

The diffusion resistance of the adult cuticle can be calculated from its area and permeability. Taking the latter as 0·85 × 10−5 c.c./sec./sq.cm./atm., and the area as 1·0 sq.cm., the diffusion resistance will be about 12 ×104 in c.c./sec./atm. The diffusion resistance of the adult measured experimentally was found to be not greater than 1·9 × 104 units. Since these two routes are in parallel, we may write
formula
where R is the diffusion resistance of the normal adult. From this relation the plastron resistance is found to be not greater than 2·3 × 104 units.

As a check on this figure, and in order to ascertain where the main diffusion resistance occurs, we have attempted to evaluate the drop in tension at each step in the diffusion path from the external supply in the water to the tissues. Much of the calculation below depends on making reasonable assumptions, but since the over-all figure obtained agrees reasonably well with the experimental data, we consider that the detailed calculations are qualitatively correct and indicate where resistance to diffusion chiefly occurs. The calculation further serves to prove that no physiological process of secretion, or any process involving expenditure of energy is necessary to enable the plastron to function, but that diffusion offers a sufficient mechanism.

(b) Calculation of diffusion of oxygen through the plastron into the tissues

In calculating the diffusion of oxygen into the tissues of Aphelocheirus, via the plastron and tracheal system, it is convenient to divide this path as follows into a number of steps :

  • From the external medium into the plastron.

  • From the plastron to the spiracles.

  • From the spiracles to the tracheal endings.

  • From the tracheal endings into the tissues.

At each step the drop in tension when the respiration is at its basal level (6 cu.mm./hr.) is calculated.

(i) Diffusion into the plastron

The diffusion of gases across a gas-water interface was originally supposed to be subject to a definite physical barrier leading to a drop in tension at the interface. Earlier workers such as Krogh and Ege have measured the rate of gas exchange under ‘ordinary’ conditions and supposed that an invasion coefficient I, specific to each gas, could be defined as q=Δ pI, where q is the quantity of gas passing per second and Δ p is the difference in tension between gas and liquid.

This drop in tension across the interface is now considered to be wholly due to inefficient mixing at the surface layers, and, indeed, Krogh (1919) has later shown that the value of I could be increased indefinitely by more vigorous stirring. It is clear that the apparent drop in tension at the interface is due to the existence of thin diffusion shells of the medium through which the gas must penetrate, prior to reaching the layers of liquid in which convection is sufficiently rapid to mask diffusion. Hence it would be more appropriate to employ a universal constant C, related to the convectional conditions at the surface and the permeability of the medium P, thus I= CP.

It is well known that a fluid medium flowing past a small stationary object forms a ‘boundary layer’ at the surface of the object, in which the viscous forces solely operate. This layer may be treated from the diffusion aspect as an approximately stationary layer whose thickness depends on the dimensions of the object, the velocity of flow and the kinematic viscosity of the medium. Using such an approximation C= I/d, where d is the average boundary layer* thickness. Hence Δp=qd/P. From this relation we have calculated from Ege’s figure for the invasion coefficient of oxygen of 0·029 c.c. at N.T.p./cm.2/min./atm. that under his conditions of experiment the boundary layer was about 12 × 10−4 cm. in thickness. His figure for nitrogen is 0·009, giving d=20× 10−4 cm., which is not in agreement, possibly owing to slightly different experimental conditions. It is of interest to compare these figures with those of Schulman & Teorell (1938), who measured the diffusion boundary layer thickness at a solid-liquid interface and obtained a figure of 30 × 10−4 cm.

It is likely that Ege’s value of I applies to somewhat static conditions, and while recognizing its defects, we have used it to give an approximate order for the tension drop. With good convection this tension drop would obviously be somewhat less than the value obtained.

When respiration takes place across the plastron two diffusion processes occur at right angles to each other:

  • Invasion across the water-air boundary.

  • Diffusion across the plastron.

It is clear that the plastron will not contain oxygen at a uniform tension; there will be a steady fall towards the spiracular orifices. It is important to know how this fall in tension will affect the plastron in its function as a gill, i.e. whether absorption of oxygen takes place all over the plastron or only close to the spiracles.

Analysis of the above-mentioned processes may be made if we treat the line of spiracles down each side of the animal as a groove into which oxygen diffuses in a direction at right angles to the long axis of the body (Fig. 3 a, b).

This treatment, given in  Appendix I, leads to a relation for Δp, the deficiency of plastron tension from surroundings (p in Appendix) in terms of x the distance from the line of the spiracles
formula
where A, B, C and n are constants defined in terms of i0 the invasion coefficient, D the diffusion constant of oxygen in the hair pile, q the rate of respiration, and h, y, x1 the physical dimensions of the plastron.

In determining the value of p the values of i0 and D must be modified to allow for the influence of the hairs. The size of the hairs is comparable with the mean free path of the gas molecules in the plastron (mean free path=9 × 10−5 cm., radius of hair = 10−5 cm.) ; hence a considerable proportion of the collisions will be between the gas molecules and hairs. At any one plane transverse to the direction of flow of the gas, 0·4 of the available space will be blocked by hairs. Hence D will be about 0·10 rather than 0·18 sq.cm./sec. The surface of the water will be reduced by the presence of the tips of the hairs, but, owing to the bulging of the menisci as shown in Fig. 23 of Part I, this effect is less than might be anticipated. The second effect on i0 is likely to be due to the reduction in flow of the water and the resistance to diffusion of stagnant regions between the hair tips. The value of i0, the invasion coefficient, is equivalent to a static water film of thickness 1·2 × 10−3 cm. The size of the hair tips is only 2·0 × 10−5 cm. diameter, and hence their effect on resistance is negligible in comparison with that of the boundary layer. The effect of the hairs on flow will similarly be negligible. It seems likely, therefore, that a reduction of i0 to half its value at a plane air-water surface is an outside limit. In the calculation h is taken as 5 × 10−4cm., x=0·25 cm. and y=0·60, from measurements on Aphelocheirus. This would give the assumed effective area of the plastron as 60 sq.mm.

In Fig. 4 the tension Δp is plotted against distance from the spiracular openings. It will be seen that the shape of the curve is not very sensitive to variations in i0 and D, and that within the likely range of these values the plastron tension is fairly uniform. Naturally there is greater uptake close to the spiracle, but the error in considering the plastron as being at a uniform tension (shown by broken lines in Fig. 4) is very small. Clearly the plastron is efficient over the whole surface of the bug.

Giving i0 a probable value of 4 × 10−4, the mean value of Δp over the plastron will be 0·7% atm., below its surroundings when q=1·67 = 10−6 c.c./sec., while the pressure drop along the plastron, when D = 0·10 c.g.s. unit, is about 0·1% atm. (Fig. 4) giving a total drop of 0·8% atm. to the line of the spiracles.

(ii) Diffusion from the plastron to the spiracles

It will be seen from subsequent data that the spiracles do not necessarily absorb equal supplies of oxygen; it follows that local oxygen tensions round each spiracle will vary. The simplest method of approach is to treat the plastron area close to the spiracular rosette as a diffusion resistance in series with the main tracheal branch leading from the rosette. To calculate this resistance from the outer edge to the centre Of the rosette the relation
formula
may be employed. r1/r0 has an average value of about 40 giving R = 12 × 103 units. This value will be employed below.

(iii) Diffusion through the tracheal system

The complete analysis of the tracheal system would involve infinite labour, but fortunately a greatly simplifying relation, first shown by Krogh (1920) to be true of Tenebrio, is also true of Aphelocheirus. The tables below, based on careful analysis of trunks supplying muscle and gut, show that the total cross-sectional area of a trunk remains unaltered when it is subdivided into finer ramifications. All trunks were assumed to be circular in cross-section, and their diameters were measured by means of a micrometer scale eyepiece. This relation may be expressed
formula
where R is the radius of the main trunk and rs is the radius of a subsidiary trunk. This relation makes possible the estimation of the diffusion resistance of a tracheal ending from a measurement of the diameter of the main trunk and the distance I, from the origin to the mean termination of the tracheoles.

Two further assumptions were made:

  1. That absorption of oxygen occurred entirely at the tracheal endings. This assumption will give a maximum drop in tension, as it ignores any uptake of oxygen via the main tracheal walls and body fluid.

  2. That the absorption of oxygen by tissues was proportional to their tracheal supply. Though not true at any instant when one particular set of organs are in use, it is reasonable to suppose that the tracheae are developed in accordance with the average oxygen economy of the tissues they supply.

In calculating the probable passage of oxygen under diffusion gradients in the tracheal system, Kirchhoff’s laws, applicable to networks of electrical resistances, may be applied to the tracheal tubes treated as diffusion resistances. The diffusion resistance of the main tracheal trunks and the resistance of the branches terminating in the tissues are calculated from the relation
formula
where D is the diffusion coefficient of oxygen in air. The result of this calculation is shown in Figs. 5 and 6. Fig. 5 gives the resistances and tensions through the main divisions of the tracheal system, while Fig. 6 shows the distribution of tensions at the tracheal endings. It will be noted that the tension drop in the tracheal system itself is very small, the mean value being of the order of 0·3 % atm.

(iv) Diffusion into the tissues from tracheal endings

The drop in tension from the tracheal endings into the tissues is not readily calculated owing to the large number of unknown factors. We have studied the tracheal endings and their distribution in muscle and gut in order to determine this pressure drop, but the values obtained cannot be considered to indicate more than an approximate order of magnitude. To treat the problem one of us (D. J. C.) has developed a number of equations for diffusion of oxygen in a medium which is absorbing it simultaneously (see  Appendix II).

Four representative tissues were studied on the assumption that these were typical for the whole animal. These were (1) hindgut, (2) crop, (3) dorsoventral thoracic muscle, (4) leg muscle (third thoracic).

The value of the absorption of oxygen (m) was computed to be in the order of I·0−1·5 × 10−4 c.c. O2/sec./c.c. of living tissue; the permeability of the tissues P is taken from Krogh’s (1919) data as 2·3 × 10−7 c.c./sec./cm.2/atm./cm.

The results of these calculations are shown in Table 6. The table indicates that the maximum drop in tension from the tracheae and to the tissues is in the order of 3%. This figure represents the tension drop at points most remote from the tracheoles, so that the average value will be rather less than this. Also it should be remembered that some diffusion occurs through the walls of the larger tracheae and via the body fluids which are kept in motion by the heart. This may somewhat short-circuit the paths of diffusion. As a very rough estimate the average fall in tension from the tracheal endings to the tissues is 1–3 % atm.

The drop in tension at each step may now be summarized in Table 7. From this over-all drop in tension, the diffusion resistance is found to be about 2 × 104. This is in sufficient agreement with the observed value (2·3 × 104) to justify the broad lines of the calculation and to reject any theory which involves selective absorption. It is of interest to note that the tracheal and plastron resistances are very small compared with the resistance on entering the plastron and the resistance from the tracheoles into the tissues.

A comparison of the data for the adult and the nymph given in Table 8 makes clear the necessity for a supplementary mode of respiration after the fifth instar. The main factors are :

  • Decrease in permeability of the adult cuticle (see p. 279 above).

  • Increase in activity in the adult.

  • Increase in uptake/surface area in the adult.

It is also of interest to compare this table with the results of behaviour experiments in which nymphs, normal adults and adults treated with wetting agents, were exposed to water at low oxygen tension (Part I). The oxygen tolerance for adults and nymphs is almost identical. Down to a tension of 3–4% both show an initial shock response to severe oxygen lack, but after about 1 hr. become acclimatized and regain some activity though still not normal, normal behaviour requiring about 5–8% saturation. At 2% saturation both fail to regain activity unless replaced in higher saturation and would eventually die. This result agrees well with the observed critical oxygen concentration of 3 % saturation for basal metabolism.

Adults in which the plastron is removed do not appear normal unless kept in air-saturated water, and even there tend to be lethargic. On the other hand, their response to low oxygen tension is not as severe as would be expected. At 8 % saturation they behave subnormally, while at tensions from 5–3 % they show progressive deterioration in behaviour but recover slowly in air-saturated water. Their behaviour is thus markedly different from the normal adult, but as the spiracles themselves are not fully closed and some degree of oxygen debt can undoubtedly be tolerated, these experiments do not emphasize the importance of the plastron as clearly as measurements on oxygen uptake. It is also important to note that any respiratory structure based on diffusion is reversible, and a highly efficient plastron will rapidly reduce the tissues to a low oxygen tension, if this exists in the surrounding medium. Hence the closure of the plastron may well desensitize the animal to conditions of oxygen want, so that its effect is not as pronounced as would be expected in a short interval of time (see calculation in Part III, p. 319).

In all the nymphal stages the plastron is lacking and the spiracular system is closed—as can easily be proved by immersion of the whole animal in a stained oil under reduced pressure. The cuticle of the earlier nymphal instars is, of course, very much thinner than that of the adult; but as the moults pass it thickens rapidly and by the time the fifth instar is fully developed is in most places nearly equal to that of the adult. The structure of the fifth instar nymphal cuticle differs in many respects from that of the adult, as will be seen from Figs. 9 and 10 and from Pl. 6d in Part I.

In the nymph the hypodermis is overlaid by a transparent endocuticle which shows itself in unstained frozen sections as composed of about twenty fairly well-defined horizontal lamellae. This in its turn is covered by an exoçuticle of a moderate degree of pigmentation in which pore canals or something similar to them can be clearly seen. Overlying this again comes a relatively thick epicuticle which in many places and with all but the highest magnifications appears structureless. Under good conditions, however, some sections show in places an appearance as if the pore canals of the exocuticle are continued through the epicuticle, giving rise on its surface to exceedingly minute spines, sometimes curved and sometimes straight (see Fig. 10). These are the only structures in the nymph which in any way resemble the plastron hairs on the adult. The epicuticle varies a good deal in thickness from place to place and one frequently finds spots where it can hardly be seen at all, in which case the exocuticle with its pore canals looks like a very stiff hair pile not containing air but fully wetted. This, however, appears to be an artefact. Although the exocuticle often shows brilliantly in section with a dark background illumination, this is not due to any contained air, since neither strong butyl alcohol nor cetyl pyridinium chloride will alter the appearance of either sections or the whole insect, even after an hour’s treatment. At any time during the period of digestion and reabsorption of the nymphal endocuticle the layers of adult endocuticle can be seen in process of formation beneath it. The exocuticle first appears as a structureless layer in which close-packed transverse laminae are soon seen. These become more conspicuous externally, this process seeming to constitute the early differentiation of the epicuticle (cf. Wigglesworth, 1933). The separation of these laminae by splitting gives rise to the hair-pile system of the future plastron, but the hairs are not at this stage obviously bent over at the tips. This may, however, in part be due to difficulties in microscopic observation of such minute and delicate structures. The fact that the hair pile of the adult is well formed before much of the old endocuticle is absorbed, seems to show quite clearly that the waterproofing of the plastron is a relatively late stage in adult cuticle formation. When all but about one layer of the old endocuticle has been reabsorbed, the moulting fluid decreases in quantity and a pale sheen appears first on the ventral part of the meso- and metathorax and in the region of the intersegmental membrane of the abdomen, suggesting that with the commencement of the process of absorption of the moulting fluid a gas is coming from solution and invading the plastron. There is no suggestion at this early stage of any continuity between the gas layer developing in the plastron and the gaseous contents of the tracheal system which still appears closed. The sheen begins to show through the transparent nymphal cuticle at a time varying from 48 hr. to a few days before the moult is actually due. When the insect is nearing moult the sheen spreads more rapidly and is particularly conspicuous in the region of the rosettes and of the organs of pressure sense where it may show brilliantly under the old skin. Nymphs do not come to the surface to moult but in the aquarium cling to plants or stones some distance below the surface. The skin breaks by means of two splits at right angles forming an inverted ⊥ at about the level of the second abdominal segment. Cast skins showing tracheal linings are found attached to the points on the abdomen beneath each rosette and also to the sites of the thoracic spiracles, and the adult comes out into the water with the plastron surface perfectly dry and complete. That not even contact with gas bubbles in the water is necessaiy to start the process of aeration of the plastron is shown by the fact that we have had perfectly good specimens with sheen complete produced during experiments in which access to the surface or to gas bubbles was prevented.

The process of waterproofing of the hair pile raises an interesting physicochemical problem. Although qualitatively this phenomenon is not different from many examples of the filling of closed tracheal systems with gas, the surface forces involved are very much greater owing to the large surface area of the hairs and the small size of the gas space between them (see Part I). Normally at a solid-water interface, any mobile polar groups become orientated towards the water and nonpolar groups away from it, since in this manner the free energy of the system is minimal. If the surface is dried reorientation may occur and the non-polar groups become external and the surface consequently less wettable. It might be suggested that gas secretion into the plastron would expel the moulting fluid from the surface and thus render it hydrophobic by the above mechanism. This view seems untenable since :

  • Secretion of gas would have to take place against an excess pressure of at least 4 atm. (Part I). This does not seem likely.

  • Even if gas were secreted the surface is still saturated in aqueous vapour and an adsorbed layer of water would be present, hence waterproofing would not occur.

It is necessary therefore to assume that some structural modification of the external molecular layer of the epicuticle occurs whereby hydrocarbon groups are directed outwards even though water is present at the interface. The production of an unwettable layer would automatically drive out the water from the plastron, since the surface forces would then be reversed. Two possibilities might be conceived. Either this hydrophobe layer is formed during the development of the epicuticular hairs and a secretion of gas into the moulting fluid induces the appearance of gas; or alternatively, a chemical change in the epicuticular surface, making it hydrophobic, causes the moulting fluid to retreat from the plastron.

The maintenance of a difference in pressure between the plastron space and the surrounding medium based on the mechanical principles outlined in Part I, gives the plastron a functional resemblance closer to the closed tracheal gill than to the air store, though the latter may, to a limited extent, behave as. a physical gill.

For this reason, the invasion of nitrogen does not play the important role that Ege (1918) has emphasized in connexion with the air store; but the necessary conditions for the tracheal gill that the walls of the tracheal air sacs, etc., should be capable of withstanding such pressure as the diffusion gradient may impose, as shown by Koch (1936), are equally relevant to the plastron.

We may develop the ideas of these authors more rigidly as follows :

Let us assume with Krogh (1910) that the passage of a gas g across a liquid-gas interface demands an ‘invasion coefficient’ ig, so that the amount passing
formula
where A = area of the surface, and t1 and t2 the tensions of the gas on either side.

The carbon dioxide tension is everywhere maintained at a very low level on account of its solubility in the surrounding medium and the buffering action of the body fluids. The water-vapour tension is almost constant throughout the system and may also be neglected in the treatment outlined below.

Let us first consider an air store of volume V, area of surface A. Let t0 and tn be the tension of oxygen and nitrogen in the outside medium and p0 and pn the partial pressures in the air store. The invasion coefficients of oxygen and nitrogen are i0 and in and the metabolic consumption of oxygen q. The net gain of oxygen by the air store will be
formula
The net gain of nitrogen similarly will be
formula
We shall now assume that the pressure exerted on the walls of the air store is constant, equal to P, and equals the sum of the gas tensions inside, since the air store, by definition, cannot withstand a pressure difference on its walls. Thus
formula
Combining (19), (20) and (21),
formula

From equation (22) it can be seen that the greater the pressure P on the system, the more rapid the decrease in volume. This equation will be employed in a later publication in which the behaviour of air stores in aquatic insects will be considered.

Equations (19) and (20) may now be used for conditions in the plastron. If the gas exchange has reached a steady state, since the volume of the plastron does not alter,
formula
Hence
formula
and
formula
The total pressure within the plastron p is therefore
formula
If the external pressure on the outer surface is P, and the depth of immersion of the animal is h, ρ being the density of water and g the acceleration of gravity, the difference of pressure which must be maintained by the plastron Δp will be
formula

This formula is an elaboration of that given by Krogh (1910) and Ege (1918).

Normally under saturated conditions, P=t0+tn; but a tension difference is still necessary to maintain respiration on account of the pressure and solubility increase with depth (kρg) and the diffusion resistance q/Ai0.

The very small effect of increase in pressure on the chemical potential of dissolved oxygen has been ignored, since it would not affect these conclusions at depths normally considered.

In order to show the magnitude of the pressure difference that must be sustained, Table 9 has been inserted. The calculations apply to the bug Aphelocheirus, in which the plastron has an area of 100 sq.mm., and a basal metabolic rate of 6 cu.mm./hr. The ‘active’ metabolic rate is assumed to be 10 times this value.

The efficiency of the plastron in maintaining an effectively closed tracheal system and preventing the invasion of water must be clearly distinguished from its efficiency as a respiratory organ. The maintenance of the pressure difference Δp for a respiration rate q is a necessary condition for respiration, but it is also necessary that the oxygen diffusion gradient through the animal is large enough to maintain this rate q. In equations (23)(27) we have been concerned solely with pressure equilibration and we have assumed that this was so. Calculation of p0, the tension of oxygen in the plastron, shows that when the oxygen tension t0 outside the animal is low, p0 may require to be negative to maintain respiration. In practice, of course, p0 must have a positive value depending on the internal diffusion resistance, so that the conditions marked with an asterisk (*) in Table 9 are hypothetical, oxygen want being the limiting factor to the animal’s existence.

It should also be noted that the above treatment assumes a uniform tension of oxygen throughout the air store or plastron. This has been shown to be approximated to by Aphelocheirus (see § 6 (b) (i)), and in a larger air store the lack of uniformity would be quite insignificant.

Several interesting and significant points can immediately be seen from this table. Unsaturation of the surrounding medium whether in oxygen or nitrogen has the same effect in increasing the pressure which the plastron must withstand if it is still to function as a gill. This effect of unsaturation is most marked, of course, when the surrounding fluid is gas-free; the plastron must then withstand the full pressure of the atmosphere and the hydrostatic head, minus, of course, the small pressure of aqueous vapour which has been omitted from the treatment for simplicity.

Should the insect descend to great depths, the strain on the system will be considerable, a limitation already pointed out by Koch (1936) in the case of a closed tracheal system, of which the plastron is really a special case. In this connexion it should be explained that not merely the plastron, but the whole of the tracheal system including air (i.e. tracheae, tracheoles, air sacs and tracheal endings), must be capable of withstanding the excess pressure Δp. In practice, of course, some of this pressure may be taken up by the elasticity of the external cuticle, some by the elasticity of the tracheal walls themselves ; but a single defect allowing the volume of the air space to decrease continuously will render the whole system ineffective. However, a small change in volume allowing cuticular elements and tracheal endings to take up the strain would be followed by a limited solution of oxygen and nitrogen until equilibrium gas exchange was re-established.

The above discussion gives a fairly complete picture of the manner in which the plastron operates. We must now consider how changes in the environment may upset the efficient working of the organ and what degree of latitude is allowed.

The main dangers may be divided into two classes :

  • (a)

    Breakdown in the waterproofing structure with consequent asphyxia.

  • (b)

    Breakdown in respiration in spite of plastron remaining uninjured.

  • (a)

    The plastron in Aphelocheirus is wetted by surface-active materials which reduce the contact angle of the water-hydrocarbon surface to about 6o °, corresponding to a surface tension of about 25 dynes/cm. This is much lower than any value likely to be met in nature. Since the habitat is normally a swiftly flowing stream, where the surface tension will be close to its maximum value, this danger is quite negligible. Even in static water a surface tension of less than 40 dynes/cm. is very unlikely. The only substance producing a sufficient lowering of contact angle would be relatively strong solutions of organic alcohols, acids, etc., or soaps; none of these occur in natural waters.

The second manner in which wetting might occur is on account of increase in depth, causing a rise in the pressure Δp, which must be sustained by the plastron. The breakdown pressure is, however, in the order of 4 atm., corresponding to a depth of 120 ft., which is quite outside any likely habitat. Even if the water is impure (γ=40 dynes/cm., θ= 90 °) a pressure of i atm. is allowed corresponding to 30 ft. in depth. These conditions would only occur in deep ponds or lakes, and it is doubtful if stagnant water with this amount of impurity would contain sufficient oxygen for Aphelocheirus to survive.

Another danger affecting insects which descend into deep pools or lakes is the effect of hydrostatic pressure on the tracheole endings. Normally the hydrostatic pressure is balanced by osmotic pressure of the tissues (Wigglesworth, 1930). Increase in external pressure would force minisci farther away from the tracheole endings, and this would presumably lower the respiratory efficiency. Though we have no evidence on this specific point it has been demonstrated experimentally that the living animal can withstand sudden excess pressure of atm. for at least 2 min. Although the adults exhibited an attitude of asphyxia when the pressure was increased to atm., nymphs were still-active at the higher pressure, and both recovered rapidly and behaved entirely normally after a few hours, no permanent harm having been done. The behaviour of the adults under these very artificial conditions suggests a shock effect. Hence the animals can withstand considerable depths, but they would be clearly put at a disadvantage if suddenly swept to a depth of 50-60 ft.

The effect of lack of oxygen in the surrounding medium will not greatly affect the breakdown pressure (see Table 9), since the maximum increase in Δp (in oxygen-free water) is only 0·2 atm., provided nitrogen is present in saturation. It is very unlikely that nitrogen would be deficient.

Old individuals are sometimes found in which continual abrasion of the belly has worn off the hair pile leaving dark patches on the mid-ventral line. Such areas will not be employed to collect oxygen, but since the whole dorsal and ventral plastron surface has been shown to be efficient as a physical gill, the loss in efficiency will be proportional to the area of these patches which is usually small compared with the whole plastron surface.

(b) The second danger is one which would be encountered under conditions of low oxygen tension. Two independent types of experiment give about 3 % saturation as a lower limit for survival of nymphs, while for normal activity the figure is probably closer to 10-15% saturation. It is unlikely that the oxygen content of rapidly flowing streams would often fall as low as 10%. Butcher, Pentelow & Woodley (1937, p. 1427) have demonstrated that in the Rivers Lark and Itchen the oxygen concentration falls regularly at night, though rarely as low as 8% saturation (0·08 atm. tension oxygen), except perhaps in very sluggish regions or deep holes. On the other hand, conditions in lakes are less favourable. Birge & Juday’s figures for dissolved oxygen in Lake Mendota (1911) show that below 10 m. the tension was less than 3 % atm. (< 1 c.c./l.). Present records of Aphelocheirus from lakes do not suggest that it wanders far from the shore. In ponds and still waters with much vegetation the situation would of course be very different, for Butcher et al. (1930) have shown that in addition to the effect of decaying organic matter in the mud, living plants can at night very rapidly depress the oxygen content. This factor alone seems fully sufficient to account for the absence of Aphelocheirus from such an environment.

The plastron of the adult therefore appears to be a satisfactory organ for its normal environment, and the dangers to which it is exposed are not greater than those which obtain for insects carrying tracheal gills. Since the plastron requires one diffusion gradient less than the tracheal gill (through the outer cuticle) it is relatively more efficient, provided a safe mechanism has been evolved, as in Aphelo-cheirus, to avoid waterlogging.

But while the plastron system of the adult Aphelocheirus is thus comparable in efficiency and adaptability to the tracheal gill systems of many aquatic larvae it is really with the respiratory adaptations of other adult aquatic insects that we must compare it if we are to get a true measure of its importance in the life and evolution of the group. When we make this comparison we see at once that the nature of the plastron system is, indeed, a major factor in the ecology of the animal. As a result of the development of this method of respiration Aphelocheirus has become independent of visits to the surface. This in turn has enabled it to inhabit swift-flowing streams without danger of being swept away during visits to the surface and, since it does not have to carry an air store, also to become a true bottom-living predator heavier than water. It can thus occupy an ecological niche not available to any other predatory adult insect. On the other hand, of course, its method of respiration necessarily restricts it to well-aerated, that is to turbulent, water.

We are greatly indebted to Prof. D. Keilin, F.R.S., for his kindness in allowing us to use the mercury-sealed gasometer in his department. Without this facility the experiments on the effect of gas mixtures with low oxygen content could not have been carried out.

APPENDIX I Analysis of diffusion processes in the plastron

Let the diffusion constant of oxygen within the hair pile be D, the invasion coefficient be i0 through the air-water surface, and let x be the co-ordinate measured from the position of the assumed groove at right angles to it. The problem is a two-dimensional one if it is further assumed that conditions are essentially similar along the y-axis.

Let h be the thickness of the plastron, and p be the tension difference between the plastron and the outer medium at a point x.

Consider the element of plastron dx, dy, ABCD (Fig. 3 b and c).

Across AD oxygen will enter at a rate
formula
Across AB and CD diffusion will occur according to the Fick equation; at AB
formula
Accumulation within ABDC will therefore be
formula
At x=x1 (outer boundary) (dp/dx)x1= 0 and at x = 0, (dp/dx)0 is maximal and accounts for the whole oxygen intake. If the total intake is q, and y=total groove lengths,
formula
The solution in terms of these boundaries is
formula

APPENDIX II Diffusion of oxygen into respiring tissues

Let the absorption of oxygen be m units per c.c. of tissue and the permeability constant of the medium be P.

  1. Rectilinear flow. From a plane at AA’ at tension p1 to any parallel plane at distance t
    formula
    Solution for x=0, p =p1,x = t, dp/dx = 0 will be
    formula
    Hence pressure drop to the plane at distance x=t is
    formula
  2. Radial flow. (a) From an inner cylindrical surface of radius r to an outer of radius r1,
    formula
Solution for p =p0, r=r0,
formula
Hence pressure drop to a radius r1 is
formula
(b) From an outer cylindrical surface of radius r, to an inner of radius r0,
formula
Solution for r=r1,p=p1,
formula
formula
And in the limiting case r0=o,
formula

In both fore- and hindgut and Malpighian tubules the tracheation appears to be limited to a surface network around the gut, from which diffusion must take place through the gut epithelium. In order to find the tension fall in this tissue the diffusion is considered in three stages (Fig. 7).

This treatment, although not rigid, is sufficiently approximate for the present purpose, and gives a value of Δp which is maximal.

  1. Diffusion with absorption through a cylinder of radius r1=1/2, where l is the mean distance between the tracheoles r0 is the radius of a tracheole.

  2. Diffusion without absorption through the annulus bounded by r0 and r1 of oxygen required by the region AA’–BB’.

    For diffusion through the whole annulus:
    formula
    Since diffusion is restricted to half the annulus we shall assume twice the fall in tension,
    formula
    and q = 2m (t–r1)r1 where t is the thickness of the gut. Therefore
    formula
  3. Diffusion from AA’ to BB’ assuming AA’ to be at uniform tension. From equation (7) above,
    formula
Hence total pressure drop
formula

The striated musculature is arranged in bundles on the outside of which there exists a close network of tracheoles, and a few can be seen to pass inside.

It is very difficult to determine the distribution at all exactly, but we have taken two extreme possibilities.

  • The tracheal supply is confined to an envelope from which diffusion occurs to the centre of the muscle bundle (Fig. 8). A treatment similar to that applied to the gut based on equations (10), (14) and (16) gives
    formula
  • The tracheae interpenetrate the whole bundle, their density being equal to that observed at the surface. This value may be found from equation (10).

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*

These complications of method will perhaps have been rendered unnecessary by the recent development of sodium starch glycollate as an indicator for iodometric analysis. (See Peat, S., Bourne, E. J. & Thrower, R. D. Nature, 159, 810–11, 14 June 1947.)

A path of diffusion in which no diffusing material enters or leaves the system other than at the origin and termination.

*

This will be strictly a diffusion boundary layer, and will not necessarily have identical form with the hydrodynamical boundary layer.