diffusion in insect wjng muscle, the most ...intricate system of unidirectional valves in the...

J. Exp. Biol. (1964), 41, 229-2S6 2 2 9With 12 text-figures

Printed in Great Britain

DIFFUSION IN INSECT WJNG MUSCLE,THE MOST ACTIVE TISSUE KNOWN

BY TORKEL WEIS-FOGH

Zoophysiological Laboratory B, Juliane Maries Vej 36,Copenhagen University, Denmark

(Received 12 July 1963)

INTRODUCTION

During flight, insect wing muscle is a strictly aerobic tissue. Per unit protein itconsumes fuel and oxygen at rates paralleled only in freely suspended bacteria but,in contrast to the microbiological systems, this activity takes place in a very concen-centrated and highly organized tissue so that the steady-state rates per unit volumerepresent an absolute record in biology (see Weis-Fogh, 1961, and Table 1). Fromthe point of view of diffusion, many wing muscles are large and may exceed 10 mmin length and 2 mm. in width. Moreover, in some Diptera the fibres are of giantdimensions. In the very active Rutilia, for instance, the cross-section is 1800 fi longand more than 500 fi wide. The transport of fuel and oxygen from the surroundings tothe sites of consumption and the reverse transport of catbon dioxide therefore repre-sent a challenge to the biologist both in relation to transport in the liquid phase and inthe intricate system of air tubes, i.e. in the tracheal system.

Since Krogh's studies on tracheal respiration (1920 a, b) it has been quite clear thatdiffusion in the air tubes plays a major role in the transport of the respiratory gaseseven over quite long distances, often supplemented by ventilation of the larger trunksor air sacs. Recently, however, Krogh's deductions have been heavily criticized partlyon the basis of very detailed measurements of diffusion pathways in silk-worms(Nunome, 1944, 1951) and partly on the basis of a number of simplifications inKrogh's treatment (Buck, 1962). I have found most of these arguments unfounded,as will be shown, but it is true that the number of quantitative concrete examplespublished is still so limited that there is a danger in relying even upon establishedqualitative principles. I hope to remedy this situation by analysing the most extremecase in existence.

Fortunately, some wing muscles are constructed according to simple geometricalprinciples so that it has been possible to reduce the diffusion problems to one-dimen-sional, steady-state equations (Appendix).

MATERIAL AND METHODS

The fresh pterothorax, from dragonnies (Aeshna spp., caught in flight) and frommature desert locusts (Schistocerca gregaria Forskal) was injected with cobalt naph-thenate, according to Wigglesworth (1950). Instead of converting the salt to the blackcobalt sulphide by means of H2S it was found more satisfactory to immerse the pre-paration in 1 % ammonium suphide in water, titrated down to pH 8 by means of HC1.

230 TORKEL WEIS-FOGH

It was then fixed in 4% formaldehyde for 2 days, washed for 1 day, soaked in 15 %gelatine at 370 C. for 1 day, transferred to 25 % gelatine for 1 day, embedded andfixed in 4% formaldehyde for 2 days. The preparation was washed briefly beforebeing sectioned on the freezing microtome. The section thickness was adjusted sothat it corresponded approximately to the distance between the origins of the secondarytracheae from the primary trunk. The sections were transferred to glycerol-water 1:1and after evaporation of most of the water, they were mounted in glycerol-gelatinebetween two cover slips placed on a slide. In this way, distortion and shrinkage wasso small that it could be neglected.

THE PROBLEMS

(a) Metabolic rate

When insects do not fly the metabolic rate m usually varies between o-oi and 0-04 ml.O2 per gram body weight per minute so that we may assume that resting wing musclesconsume about 0-02 ml. O2/g. muscle/min. During steady-state flight the rates are50-400 times greater, as it is seen from Table 1. It is apparent that there is no generalrelationship between size and rate. In the calculated examples, the diffusion equationswere therefore solved for m = 1, 2, 4 and 8 ml. Oj/g./min. (and for 0-02, in brackets).The corresponding figures for the consumption of glucose are given in Table 6. Thesmallest rates are probably representative for insects which do not perform sustainedflight but the muscles of most good flyers, whether small or large, require 1*5-3 ml.O2/g./min. Some of the large Hymenoptera and Diptera approach the exceptionalrate of 8.

Table 1. Oxygen uptake of insect wing muscle during steady-state flight

(For some of the insects, the necessary exchange of air between muscle and surroundings percontraction cycle ( = per wing stroke) has been calculated as a percentage of the musclevolume, assuming that the muscle absorbs 1/4 of the oxygen present in atmospheric air(1/4x21% = S%).)

(ml. O,/g. (100 x ml.muscle/min.) air/g./stroke)

Locust, Schiitocerca, (Weis-Fogh, 1952)Dragonfly, Aethna, (Weis-Fogh, 19646)Butterflies and moths, (Zebe, 1954)Aphid, Aphis (Cockbain, 1961)Fruit fly, Drotophila, (Chadwick & Gilmour, 1940)Wasp, Vespa erabro, (Weis-Fogh, 1964ft)Blowfly, Lucilia, (Davis & Fraenkel, 1940)Honeybee, ApU mellifera, (Hocking, 1953)

(b) Supplies and supply routes

During a single performance the wing muscles may consume more fuel than thatcorresponding to the total dry matter of the active fibres so that the stores of glycogenand fat in the muscle may be ignored in this context. The fuel is therefore transportedto the muscles by means of the blood which bathes them and consists of carbohydrates,fats or their break-down products (Weis-Fogh, 1952). It is most likely that trehalosein the blood is split to glucose on its passage through the fibre membrane and it isunlikely that the fatty acids are broken down as far as to acetate before they diffuse in,

1 -4-2-8i-8

1-4-3-51-4-1-82-0-2-32-6-3-3

5-67-3

2-6-S-22

—

—

0-3-0-40-9-1-1

—

i-o

Diffusion in insect wing muscle, the most active tissue known 231

because this would mean a great waste of energy. We may therefore choose glucose asa model substance for the diffusion of fuels.

A typical wing muscle receives its supply of air from a longitudinal, primary tracheaor air sac which is directly connected with a spiracle (Weis-Fogh, 1964a). From theprimary supply, a system of secondary tracheae branch off transversely between thefibres and give off tertiary tracheae at regular intervals (see dragonfly muscle) whicheventually split up into tracheoles some of which may become ' internal' by identingthe fibre membrane (Edwards, Ruska & de Harven, 1958; Vogell, Bishai, Busher,Klingenberg, Pette & Zehe, 1959; Smith, 1961a). A wing muscle is a prismaticstructure of parallel fibres and its total volume consisting of fibres, blood and air tubestherefore decreases with shortening by a fraction corresponding to the relative shorten-ing. This must lead to muscular pumping of blood and air in and out. In the secondcolumn of Table 1, the amount of fresh air is calculated which is needed per strokeif 25 % of its original oxygen content is used up. In the different insects this amount isof the same magnitude as the change in volume per stroke or even larger, and only anintricate system of unidirectional valves in the primary tracheae could ensure anadequate supply by muscular pumping. Such valves are not present, however, and inlarge insects morphological inspection and direct experimental evidence demonstratethat the primary supply routes are ventilated strongly and in such a way that the com-position of the air is almost the same as in the major air sacs, in locusts about 5 % CO2

and 15 % Oa (Weis-Fogh, 1964 a, b). The pumping therefore causes blood movementsand, as will be indicated, some small ventilation of the secondary tracheae which, how-ever, is not essential but leads to a greater exchange of blood between the interior andthe exterior of the muscle.

The problems are then confined mainly to (a) the exchange of fuel between bloodand tissue and (b) to the exchange by diffusion of O2 and CO2 between the primarytubes and the sites of combustion. The transport across fibre membranes and tracheolarwalls will not be dealt with.

(c) On diffusion

We are dealing only with diffusion in the stationary or steady state where there is nochange in concentration at any point in the system. In a mixture of substances the nettransport J of a given substance passing perpendicularly through a unit reference areaper unit time is given by Fick's first law.

where c is the concentration of the substance, x the co-ordinate normal to thereference surface, and D the diffusion coefficient for the substance in the given system.In the c.G.s. units it has the dimensions of cm.2 sec."1. In most biological systems D isnot known but only the permeability constant P which is the flow of a substance throughunit area per unit time when the average concentration gradient is unity and when theflow is measured in the steady state under certain conditions and with the assumptionthat D remains independent of c (cf. Barrer, 1941; Jost, 1952).

Since the respiratory gases diffuse both in the gaseous and in the liquid phase it isconvenient to express the concentration gradient in similar terms throughout, bymeans of the partial pressure gradient dpjdx. According to Henry's law, the volume of


gas at N.T.P. dissolved per unit volume of liquid is V = apj'760, where a. is the absorp-tion coefficient. The volume of gas at N.T.P. transported according to equation (1) is

The diffusion coefficient of CO2 is about 15 % lower than for O2 in most systems butsince the absorption coefficient for C0a in pure water is 28 times greater than that ofO2> the permeability constant P for COa has been found to be about 36 times greaterthan for oxygen in animal tissues which usually absorb more CO2 than water (Krogh,1919). For the same difference in partial pressure Ap, CO2 is then transported about36 times quicker in tissues, which we shall call tissue diffusion, than Ot but 15 % slowerin the tracheal system, which we shall call air-tube diffusion.

The permeability constant P for 02 in air at 200 C. and atmospheric pressure is11 ml. min."1 cm.~2 atm."1 cm."1, while it is 3-4 x io"6 in water and 1-^x io"6 in musclefrom the abdominal wall of the frog (Krogh, 1919). Like insect wing muscle, frog muscleis poorly vascularized and contains little myoglobin so that the facilitating effect onO2-diffusion of high concentrations of haemoglobin and similar oxygen carriers shouldbe absent (cf. Scholander, i960; Hemmingsen & Scholander, i960). Moreover, astrictly aerobic muscle can hardly afford to decrease the oxygen tension to the low valuesnecessary for this facilitated transport to be operative. For a given Ap tissue diffusionthen requires io6 times more area than air-tube diffusion in order to transport a givenamount of oxygen. It is therefore convenient to introduce the hole fraction a as thesummed cross-sectional area of all air tubes in a given tissue surface divided by thetotal area of that surface. In insect wing muscle, a usually ranges between io"1 andio"3 (see later) which means that the permeability for O2 is io8 to io6 times greaterthan in non-tracheated muscle and for CO2, 50-5000 times greater.

The permeability constant for glucose in muscle is not known but if we adopt thediffusion coefficient D in water (0-57 x io"6 cm.2 sec."1; see Jost, 1952), we shallcertainly not overestimate the exchange but rather underestimate the necessary con-centration difference by a factor of two or more.

DRAGONFLY WING MUSCLE

The major wing muscles of Odonata run parallel to each other in the dorso-ventraldirection and each muscle is a prismatic structure made up from parallel fibres which,in turn, are collected in small longitudinal bundles or lobes. Each lobe contains afew hundred fibres and remains of equal thickness from end to end. The lobes arearranged as thick lamellae radiating out either from the axis of the cylinder-likemuscle (Fig. 3) or from the central plane of symmetry in muscles with an elongatedcross section (Figs. 1,2). The primary air supply to a muscle consists of one or twoshort tracheae coming directly from a spiracle. They enter the ventral end of themuscle belly and run centrally to the dorsal end where they leave the muscle andwiden into a dorsal air sac (Miller, 1962; Weis-Fogh, 1964a). In a muscle with anapproximately circular cross-section there is only one central trachea (Fig. 3) but inthe larger flattened muscles the primary supply consists of two or three longitudinalair tubes connected by many wide and closely spaced transverse tracheae, like thesteps of a ladder (Clark, 1940). Both in the central type and in the ladder type, the

Diffusion in insect toing muscle, the most active tissue known 233

important feature is that the secondary air tubes radiate out from the primary tubesin an exceedingly regular fashion. Thus, for every 20-25 fi along the axis each lobereceives one secondary trachea which enters the central edge of the lobe and runsalmost perpendicular to the fibres towards the peripheral edge, branching only in thetransverse plane to form a structure resembling the fruit grower's 'espalier tree'.Moreover, at different levels the ' espalier trees' are almost identical in shape and theybranch between the same fibres so that it is possible to reconstruct the tracheal supplyin detail from rdktively few transverse sections.

0-5 1-0

Fig. 1 Fig. 3

Fig. 1. Transverse section of the dorso-ventral wing muscles in the left half of the metathoraxof a large dragonfly Aeshna cyanea. The injected tracheae are drawn in black. Nomenclatureaccording to Neville (i960); (a) first basalar muscle, (6) second basalar, <c) tergosternal, (d)anterior coxoalar, (e) posterior coxoalar, if) first subalar, and (g) second subalar muscle. Lobe Ain Figs. 2 and 4 is indicated.

Fig, 2. Part of the second basalar wing muscle in Fig. 1. The blood-filled spaces are in blackand the injected trachea* are faintly dotted.

From, Figs. 1-3, it is seen that in some muscles the distance from the primarytrachea to the periphery may approach 1 mm. On the other hand, the blood (black inFigs. 2 and 3) bathes the lobes completely and no point is more than 75 ft from a freeblood pool. As to volumes, 75 % of the cross-section in Fig, 2 corresponds to fibres and68% in Fig. 3, The blood occupies 12 and 18 %, the secondary air tubes about 8 and10% and the primary tubes 5 and 4 %, respectively. More than half the volume of theclefts through which the trunks of the ' espalier tree' run consists of air tubes and therest is blood. The blood then constitutes 15-20% of the total muscle volume.

We shall now analyse the transport of gas and fuel in Aeshna wing muscle step bystep, choosing as our main examples the almost rectangular lobe A in Figs. 1 and 2


and the two large radial lobes B and C in Fig. 3. For each step we shall try to generalizethe results obtained so as to facilitate the comparison with other muscles. Duringflight the oxygen uptake is i-8 ml. Og/g./min.

Fig. 3. Transverse section of the tergosternal muscle from the left mesothorax of Aesknacyanea, showing lobes B and C. The blood-filled spaces are in black and the injected tracheaeare faintly dotted. The circles indicate the estimated equivalent diameter of the ventilatedprimary supply tubes for the two lobes in question. Further distally, the transport of re-spiratory gases takes place only by diffusion.

(a) Diffusion in primary tracheae

Like many wing muscles in locusts and other large insects, the muscle in Fig. 3belongs to the centro-radial type (Weis-Fogh, 1964a). In the simplest case it iscylindrical and the radius r of the primary tube is about one-fifth to one-tenth of themuscle radius R, corresponding to 1-4% of the muscle volume. This amount of air isadequate only for one or two contractions (Table 1) so that a very effective exchange ofair is essential. Table 2 shows that the renewal of air during activity cannot take placeby diffusion since Ap for oxygen would exceed 0-05 atm. if the muscles are more thanabout 1 mm. long, and we have not even considered diffusion between the respiringmuscle and the spiracle. In Schistocerca and Aeshna most muscles are 7 mm. or longer.But it is also seen that the resting metabolism can be kept up by diffusion alone. Thesedeductions are consistent with the observation (a) that the active thorax is stronglyventilated by a specific thoracic pumping mechanism coupled with the wing move-ments (Weis-Fogh, 1964a, b) and (b) that the muscles of an isolated resting thoraxremain alive and reactive for days.


In small insects with no ventilation during flight, the centro-radial type would beadequate only in. a thorax less than 1 mm. high and only at moderate metabolic rates.In the latero-linear case, however, where the primary supply is an air sac which spreadsout over the surface of the muscle and has a cross-sectional area similar to that of themuscle, diffusion is adequate up to the size of a locust, as is probably the case with thecontroller-depressor muscles of Schistocerca (Weis-Fogh, 1964a).

Table 2. The maximum distance L (in /i) for the diffusion of oxygen in the longitudinaldirection of the axially placed primary tube of a centro-radial type muscle, accordingto Fig. 12 D and equation (7) when Ap = 0-05 atm.

ActiveRest , * >

Metabolic rate m (ml. Ot/g./min.)... 002 1 2 4 8

T = R/io (7>4°o)(14.800)

1,0502,100

7401,480

520

1,04037O74O

In the latero-linear type of muscle whose primary supply may have a cross-sectional area similar tothat of the muscle, these distances should be multiplied by 5 to 10, cf. Drosopfula on p. 248.

The necessity for ample ventilation of the primary tracheae in the centro-radial typeof muscle will be apparent also from the analysis of gas diffusion in the subsequentbranches of the tracheal tree.

Fig. 4. Transverse section of lobe A in Figs. 1 and 2. The thickness is 29 ft. The primary tubeis indicated at the extreme left, the tapering secondary tube runs horizontally, and the tertiarytubes branch off in the vertical direction, ending in tracheoles. Note the close packing of thefibres. The broken lines indicate the number of tertiary branches in the section in addition tothe top branch.

(b) Air-tube diffusion in the small lobe A

The secondary, and the tertiary supply was studied in transverse serial sections,each 26 fi thick. In about 50 sections (1300 ft) there was no variation as to the lay-outseen in Fig. 4. The block diagram in Fig. 5 represents 7 sections. The muscle inquestion has a ladder-type primary supply of which two of the 'horizontal steps' areindicated. A longitudinal row of holes shows the origin of the secondary tubes whichswell up at their bases to become small collapsible sacs before the tracheae enter the

2 3 6 TORKEL WEIS-FOGH

lobe. These long tapering tubes follow a longitudinal cleft which occupies 3 % of thevolume. One secondary tube is given off for every 21 /i on the average. The diameterswere measured in 17 sections at eight relative distances £ from the entrance ( x 800linear magnification, eye-piece 9crew micrometer). Fig. 6 shows that the diameter ofthe secondary tubes decreases linearly from the entrance (£ = o) to the tip (£ = i-o).However, at £ = 0-84, detailed measurements of the tertiary system under oil immer-sion (x 1600) showed that each secondary tube ends and continues as three tertiarytubes each of which soon branch into two, giving rise to an increase in equivalent

Fig. s. Reconstruction of the tracheal supply to a lobe of a wing muscle in a dragonfly (lobeA in Figs, i, 2 and 4), as explained in the text.

diameter (crosses in Fig. 6). Apart from these terminal tubes, the tertiary tubesbranch off normal to the secondary tube and at quite regular intervals correspondingto the ' diameter' of the polygonal muscle fibres, i.e. at about every 20 /*. The course ofthe tertiary tubes among the fibres also follows the same pattern section after sectionbut it was only possible to measure the diameter of each tube at its origin. However,it is possible roughly to estimate in which way the summed cross-sectional areasbelonging to each tertiary tube vary as one approaches the periphery. There are about25 tertiary tubes per secondary trachea and each of them supplies a column 21 /i talland belonging to 5-6 fibres (140/25). Inspection of fresh muscle in glycerol showedthat each fibre is surrounded by a dense system of transverse tracheoles 4-5 /i apart,resembling the fingers of a hand 'gripping' round its fibre segment (see Fig. 5). Thismeans 20-30 terminal tracheoles per tube. Since the average diameter of the tertiarytubes was about 1 ft at the origin and the diameter of the smallest tracheoles visibleabout 0-2 /*, the ratio between the area at the origin of a tertiary tube and at the summedareas at the tips of the 'tree* is about ia/o-22 x 25 = 1. It is therefore reasonable toassume that the summed diffusion area in the tubes branching off from the secondarysupply remains almost constant for the major part of their course, but we shall cal-culate Ap also for the unlikely case that the area decreases considerably towards theperiphery.


Diffusion in the gas phase may now be divided into two steps. For both we choosem = 2 ml. Oa/g./min.

Diffusion in secondary tubes. The length L of the lobe is 407 x io~*cm. and thehole fraction a = 1-5 x io~2, calculated as the summed cross-section area of thesecondary tubes at their entrance into the lobe divided by the corresponding area ofthe lobe perpendicular to the tubes. Since the cross-section of the lobe is approxi-mately rectangular, the necessary difference in partial pressure A/> was calculated foroxygen according to equation (6). The upper curve in Fig. 6 shows that from theentrance at £ => o to where the tertiary tubes begin at £ = 0-84, diffusion is adequateif A/> ^ 0-035 a t m- Ventilation due to muscular pumping of the sac-like parts betweenthe lobe and the primary tubes will increase the effective diameter of the centraltubes but neither this nor some ventilation of the proximal part of the secondarytubes will have much effect, Ap being so small.

to

Q

004

002

6

4

2

i - i

-

" .

. o

-

-

-

1 1

1 1 1 1 1

Secondary tube

1 1 1 1 1

1

o

•

Tertiary

tubes

1

X

X

0-2 0-4 0-6

Relative distance

0-8 10 = £

Fig. 6. Diameter of the secondary tube in lobe A (lower curve) and the necessary difference inpartial pressure Ap of oxygen (upper curve) as functions of the relative distance £ from theprimary tube.

Judged merely from their appearance, tracheal systems with linearly decreasingdiameter like that in Fig. 6 are probably quite common, but so are systems with con-stant diffusion area (Krogh, 1920a, for Cossus larva; Thorpe & Crisp, 1947, forApheiocheirus; Locke, 1958, for Rhodnius). In Table 3, I have therefore calculatedthe maximum depth of active wing muscle which can be supplied by transversetracheae penetrating the muscle surface. When a is unusually large, as in locust wingmuscles (0-07), the distance may exceed 1 mm. but for the more common value of


IO~2, 0-5 mm. is about the limit. The small value of io~3 is relevant for the next stepand also for diffusion in dipteran muscle fibres.

Diffusion in terminal tubes. The depth of tissue to be served at either side of themedian cleft is 65 fi but, because of branching, we shall use 75 /i. The hole fractionof the tertiary tubes at the cleft is a = i-o x io~3i According to equations (4), (5) and(6) Ap is 0-005 atm. in case of constant diffusion area, which is the most likely, o-oiatm. in case of linearly decreasing area and 0-023 a t m- m the quite unrealistic case oflinearly decreasing diameter. This step is therefore particularly favourable for diffusion.

As far as the terminal part of the lobe is concerned, these low figures for Ap werechecked by considering the end of the lobe as a half cylinder with tubes radiating outfrom the axis towards the periphery (Fig. 12 J and equations (16) and (17)). The holearea a was 3-5 x io"3 and the resulting Ap was 0-003 a t m- f°r a constant diffusion areaand 0-007 for a linearly decreasing area.

Table 3. Maximum distance {in fi)for the diffusion of oxygen in air tubes of wing musclessupplied according to the latero-Unear type and when Ap = 0-05 atm.

(In (A) the diffusion area of the air tubes is constant, Fig. 12 B and equation (4), and in (B)the diffusion area decreases linearly with the distance from the surface of the muscle, equation(5)-)

ActiveRest ,

Metabolic rate m (ml. O,/g./min.)...Hole fraction a at muscle surface

a = s x io"1 ABABABAB

O'O2

(I5,8oo)(11,800)

(7,4OO)(5,300)

(2,4OO)

(l,7OO)

(740)(530)

I

2,2401,670

1,050

74O

33O

235

"575

2

1,5801,180

74°525

235170

7555

4

1,120

830

525370

170120

5035

8

790

59O

37O260

120

85

3525

In other words, it is only necessary to establish a difference of less than 0-05 atm.in the partial pressure between the primary supply and the terminal end of thetracheoles in order that diffusion alone may account both for the inwards transport ofoxygen and for the outwards transport of carbon dioxide.

(c) Air-tube diffusion in lobes B and C

These lobes (Fig. 3) can be considered sections of a centro-radial muscle, theprimary tracheae of which are wider than the morphological tube because of ventilationof the proximal sac-like part of the secondary tubes, as indicated by the circles drawn.The secondary tracheae were measured and treated in the same way as those of lobe A,but the injection was not as perfect so that the diffusion areas in Fig. 7 representminimum values. The diffusion pathways are considerably longer, 890 and 735 firespectively, and in both cases the diffusion area varies in a characteristic way with thedistance from the adopted centre of the ventilated tube. The summed area remainsalmost constant until the first major branching but then it declines linearly, as shownby the straight lines. Consequently, Ap for the secondary system was calculated in twosteps, as indicated in Fig. 7, the peripheral 75 fi of tissue being supplied by the tertiary


and tracheolar tubes (the third step). According to equation (n) , the first step withconstant area requires that A/> is only 0-004 atm. in lobe B (a = 6-6 x io~2) and 0-014in lobe C (a = 2-8 x io~J). This means that ventilation by muscular pumping of thesepartially exposed, wide tracheae should have little significance for the transport ofoxygen but may be of some consequence for the transport of carbon dioxide by meansof blood. The second step, however, is critical since Ap is 0-114 atm. in lobe B and0-074 m l°be C (equation (136)), a being 8-8 x io"3 and 3-9 x io"3 respectively. Inother words, the drop in partial pressure necessary is 0-12 atm. in B and 0-09 atm. inC. If we adopt the results from lobe A, the total drop in tension in the air tubes fromthe primary supply to the fibre surface would then amount to as much as 0-13 and010 atm. respectively. There is no obvious possibility for any improvement becausethe main diffusion resistance is located in the peripheral thin tubes.

150

100

SO

50

Primary1 Tertiary

400

Distance r from centre (p)

600 800

Fig. 7. Summed cross-sectional area of the tracheal tubes in lobes B and C of Fig. 3, as afunction of the distance from the equivalent axes of the primary tube. The thin continuouslines show the variation of the area adopted for calculating Ap.

Taking account of tissue diffusion, the total difference in partial pressure from theprimary tube to the sites of combustion must be greater by about 0-05 atm. whichmeans that these muscles cannot become larger unless the tracheal system is re-organized and, also, that the central trachea must be ventilated very efficiently. InSchistocerca the air in the primary trunks contains about 15% O2 during flight butthis is close to the possible limit in dragonflies since 13 % may be needed to ensure anadequate exchange in certain lobes. We must therefore expect that the dragonfly

16 Exp. BioL 41, 2


thorax is ventilated more strongly than the locust thorax, as is also borne out bymeasurements (Weis-Fogh, 1964 ft). Moreover, the carbon dioxide tension in theinterior of dragonfly muscle fibres must be high during flight so that transport byblood may be more important than in locusts. This is considered a more probableexplanation than cooling for the synchronized movements of the thoracic spiracles aftercessation of flight (Miller, 1962) until the tension has decreased to a low value again.

Table 4. Maximum radius R (in fi) for the diffusion of oxygen in the radiating air tubesof a cylindrical, tracheated muscle of the centro-radial type (Fig. 12H, I)

(The radius of the primary trachea is r0 = Rjio and the possible drop in oxygen tensionin the radiating secondary tracheae is Ap = 0-05 atm. The diffusion area decreases linearly withthe distance from the axis (equation (13a)) but if it is considered constant as in equation (11),all distances should be increased by about 55 %).

ActiveRest , • ,

Metabolic rate m (ml. OJg./min.)... 0-02 1 2 4 8

Hole fraction a of wall of primary tracheaa = io"1 (6300) 890 630 440 310a = 5 x 10"1 (4400) 630 440 310 220a = io~* (2000) 280 200 140 100

Since the centro-radial type of muscle is so common among large insects, Table 4was computed and it is seen that even at medium rates with a large number of holesin the central tube, such a muscle may hardly exceed 1 mm. in diameter unless, as indragonflies, the primary system is exceedingly well ventilated with the consequent riskof an impaired water balance. If a of the central tube is as large as o-i, the maximumpossible ratio / between tube and muscle diameter can be estimated from equations(12) and (14). In order that the central tube should not occupy too large a volume, weshall choose / = 0*2 as the upper limit. The maximum diameter for such muscles isthen 2-4 mm. when the consumption is 2 ml. Og/g./min. but decreases to 1-2 mm.at the higher rates. The dimensions of insect wing muscles are consistent with thisestimate except for those of the giant belostomid bug Lethocerus uhleri. According toMoller (1921), the tracheal supply is similar to that of dragonfly muscles but thediameter sometimes exceeds 5 mm. However, the secondary radiating tracheae donot taper. On the contrary, they expand into terminal air sacs when they reach theperiphery, covering the surface like intestinal villi. In such exceptional cases thesecondary tracheae are undoubtedly ventilated.

The above examples stress that the diffusion area of tracheal systems often varies ina complicated way with distance, as was also found by Nunome (1944), so that itmust be analysed in every case considered.

(d) Tissue diffusion

It is characteristic of dragonfly wing muscle that the fibres of a lobe adhere closelyto one another so that the diffusion of respiratory gases and of fuels must be treatedseparately.

Dissolved gases in dragonfly muscle. The tracheoles do not indent the fibre membraneas in many other insects but are confined to an envelope on the surface (Smith, 1961 ft).Although the diffusion of oxygen to the interior may be calculated rather exactly from


equation (18) in Thorpe & Crisp (1947), the dense meshwork of tracheoles makes itreasonable to use the more simple expression of equation (96), i.e. to consider thefibres as being completely surrounded by a layer of air. In this way we obtain themaximum possible radius R for a fibre seen in Table 5. Since the average cross-sectional area of a dragonfly fibre was 350 /J2, the maximum diffusion distance isabout 10 /i, i.e. close to the possible limit when m = 2 ml. Oj/g./min. and Ap = 0-05arm. We saw from the previous examples that Ap cannot be significantly greater thanthis value and, without internal tracheoles, the fibres of most insect wing musclesmust be less than 20 fi in diameter. It is possible, however, that the arrangement ofthe dragonfly fibre into alternating radiating lamellae of ribbon-like fibrils andflattened sarcosomes (Smith, 1961 b) is an adaptation to this situation since surfacesof equal oxygen tension become crenelated rather than cylindrical with valleys at thenon-consuming fibrils and crests where the consuming sarcosomes are located. Sincethe necessary tension usually depends on the second power of the distance to beserved, this tends to facilitate exchange by diffusion and such measures could beexpected to have evolved in a system pressed to the limit (see also cockroaches in theDiscussion).

Table 5. Maximum distance (in n)for the diffusion of oxygen in the dissolved state froma source into active and resting muscle tissue {figures in brackets) when Ap = 0-05 atm.

((A) distance L for rectilinear diffusion from a flat surface, Fig. 12 A and equation (3). (£)radius -R for diffusion from the surface of a cylinder to the axis, Fig. 13 F and equation (96).(C) R — r0 for radial diffusion from an air tube of diameter 1 /* (r0 = 0-5 /*) into the tissuecylinder surrounding it, Fig. 12 E and equation (8).)

Activemetabolic rate m Rest , * \

(ml. Oj/g./min.)... 0-02 1 2 4 8

A: L(ji) (84) 11-8 8-4 5'9 42B: R(ji) (118) 167 n-8 84 59C: R-ro(]i) (41) 7-4 54 4-6 3-6

General comments on tissue diffusion of gas. Table 5 also illustrates some generalproperties of tracheal systems. In his recent review, Buck (1962) claims that if simplediffusion were to account for the transport the 'mean distance from tracheole wall tocytoplasmic sites of Oa consumption cannot exceed o-z/i' under any conceivablecondition even when Ap = 0-2 atm. He uses this and other arguments to show thatthere must be a considerable diffusion out from the larger tracheal trunks and also tocast doubt upon the principles of simple diffusion as the main transport mechanismfor gas in tracheated tissues.

However, he does not disclose his method of calculation and Table 5 clearlydemonstrates that his conclusions are untenable even in the most active tissue known.In example A, oxygen diffuses in freely from the flat surface and it is seen that whenAp = 0-05 atm. a sheath of active muscle will be served which is 4-12 fi thick. Thisillustrates how far out from a large tube direct diffusion is sufficient (still far in excessof 0-2 fi) and it is obvious that large air tubes are completely inadequate because theycannot come close enough to the sites of consumption unless they occupy the majorpart of the muscle volume. In contrast, example C shows that under similar con-ditions a tube only 1 fi in diameter can serve a cylinder, which is 7-15 ft in diameter,

16-3


of which 2-6 % of the volume is air. In other words, if, as we have seen, exchange ofair within the tube system is sufficient to permit a difference in oxygen tension betweenthe tracheolea and the sites of combustion of maximally 0-05 atm. the distance betweenthe tracheoles need not be less than about 6—8 fi in a block of the most active muscletissue. In many muscles it is about 3 fi, as discussed later, so that the safety factor is2-3. It is also clear that during evolution flying insects must either have decreased thefibre diameter considerably (example B), which is not the case, or they must have'invented* indenting tracheoles, which is the case. The wall thickness of tracheolesamounts to only 100-300 A and has therefore not been taken into account (cf. Smith,1961a).

Since the permeability constant of carbon dioxide in animal tissue is about 36 timesgreater than for oxygen, all distances in Table 5 should be increased by a factor of36^ = 6. There is therefore no problem with respect to tissue diffusion. As todiffusion in the tracheoles and in the tertiary tubes, tissue diffusion clearly representsa shunt but it will not be very effective because of the relatively large hole fractioninvariably found in wing muscle. The main problem concerns exchange in thesecondary system which becomes the major factor determining the carbon dioxidetension in the tissue, as already discussed.

Table 6. Diffusion of glucose from the surface of a cylindrical muscle fibre to the axis forvarious values of its radius R

(The figures show the minimum difference in concentration At between surface and axisin mg. glucose per ioo g. muscle (mg. %) and are based on diffusion in pure water, accordingto equation (96). The corresponding figures for a flat sheet of tissue supplied from the oneside and whose thickness L equals it are twice as big, according to equation (3).)

ActiveMetabolic rate m

in mL Ot/g./rrun....in mg. glu./g./sec

R = 500/*

R = 200/1

R = 100/*

R = 50/*R = 20/*

R = 10/*

Rest0-02

4-S x io-«

(49)(08)

(02)

(005)(0008)(0-002)

1

2-25 x io~*

MS40

1 0

2-s0-4

O-I

a4-5 x 10-'

495

792 0

SO

o-80-2

49 x 10-1

990

1584 0

9 91-6

0-4

818 x i o - '

1980

316

792 0

3-2

0 8

Diffusion and fuel concentration in various muscles. In dragonflies there is no effectiveway for fuel to enter the muscle lobes like those in Fig. 4 other than by diffusion fromthe surface because all fibres adhere to one another with almost no extracellular space.In lobe A the average distance to the central fibres is 65 /i and, according to equation(3), the concentration difference Ac of glucose would amount to at least 33 mg./ioo g.tissue (33 mg. %), provided that the bulk diffusion coefficient in muscle equals thatin pure water. If diffusion of glucose in muscle is hampered to the same extent as thatof oxygen in isotropic gelatine solutions and tissues (Krogh, 1919) and if we disregardadditional barriers represented by plasma membranes, Ac would be 33/0-42 = 80mg. %. If simple diffusion prevails throughout, the concentration of the bloodimmediately surrounding the lobe cannot be less than about 100 mg. % to account forthe supply. Many Diptera and Hymenoptera have large fibre diameters (cf. Pringle,1957). In the honey bee Apis mellifera, for instance, R = 100 fi and Table 6 shows

Diffusion in insect wing muscle, the most active tissue knozon 243

that c must be about 79/0-41 === 200 mg. %. In the very active tachinid fly Rutiliapotinia, the distance from the membrane to the central parts of the flat fibre (cf.equation (3)) is about 260 ji (Tiegs, 1955), giving the exceptionally high value ofAc = 535/0-41 === 1300 mg. % for a metabolic rate of only 4 ml. Og/g./min. InDiptera with such giant fibres, but not in Hymenoptera, recent investigations indicatethat the plasma membrane is invaginated into deep narrow longitudinal clefts (Auber,1961; D. S. Smith, personal communication) so that some exchange takes place bymuscular pumping (see below). But in most flying insects, there is no doubt that thedifference in fuel concentration inside the fibres must be very high since mixing of thesarcoplasm of these close-packed parallel systems is unlikely in view of the smalldegree of shortening (0-5-1 %).

In this context, the exact nature of the fuel is of little significance. It could beargued that, although Ac between plasma membrane and fibre axis is large, the bloodconcentration need not be particularly high since, for example, glucose may becomephosphorylated in passing the membrane and thus being prevented from diffusingback. This could hardly be so in Aeskna muscle because the fuel must diffuse both inand out of the peripheral fibres in order to reach the central ones, indicating that itpermeates the membranes freely. Consequently, the fuel concentration in the bloodmust be considerable. There is, however, a more compelling reason why flying insectsshould contain large amounts of trehalose and other fuels dissolved in the haemolymph.

Muscular pumping and fuel concentration. In many insects the wing muscles are solarge that diffusion is quite inadequate for transport from the surface to the interior,as is seen from Table 6. The only other mechanism for exchange is muscular pumpingbut, as an example will show and because of the inefficiency of such a system, theconcentration of fuel in the blood must be increased much above the level necessaryfor diffusion into active fibres. The bee muscle, for instance, consumes 18 mg.glucose per 100 g. muscle per second and if 20% of the volume is blood this means90 mg. consumed per second per 100 ml. blood entrapped in the muscle. The muscleshortens about 200 times per second but only by 0-5 % of its length, giving rise to apumping of maximally 100 ml. per 100 g. muscle per second. Under ideal conditions,the blood surrounding the muscle must therefore contain at least 90 mg. % moreglucose than the amount necessary for simple diffusion into the fibres. However,muscular pumping moves air as well as blood and its tidal character will make it avery ineffective mechanism when, as here, the relative stroke volume is considerablysmaller (40 times) than the relative amounts of air and blood entrapped. We can there-fore conclude that the maximum concentration in Apis blood must be many hundredmg. %, in accordance with measurements of blood sugar (2% blood sugar or more,Beutler, 1937).

In locusts and dragonflies, the volume change is also about 100 ml. per second per100 g. muscle but since the rate of consumption corresponds to 20-30 mg. glucoseper 100 ml. blood and the pumping volume per second may approach 25% of theentrapped air and blood, the situation is less critical. However, one must expect atleast 100 mg. % of glucose or of another fuel to represent a minimum requirement forflight. If these concentrations are compared with the figures for resting muscle inTable 6, it is justified to suggest that the very high concentration of trehalose in insectblood (cf. Howden & Kilby, 1961) can be considered a direct and essential adaptation


to flight. It also follows from all that has been said that the main function of muscularpumping is to move blood and not air in contrast to what I thought earlier (Weis-Fogh, 1956). The limited solubility of lipids presents a special problem which isdiscussed on p. 247.

DISCUSSION AND SOME COMPARATIVE RESULTS

In connexion with the different steps in the transport processes in dragonfly musclewe have already discussed several problems of general interest and illustrated them bymeans of Tables 2-6, but a more comprehensive and comparative survey remains.

Visible end of tracheoles 160

120

80

-40

o

ia

10 15 20 25

Distance from splrade (arbitrary units)

Fig. 8. The variation of the summed cross-sectional area of the tracheae in a segment of a"ilk worm as a function of the distance from the spiracle (open circles), according to Nunome(1944). The filled circles indicate the diameter of the equivalent air tube.

(a) Diffusion in insect larvae

The surprising result of Krogh's analysis (1920 a) was that Ap in the small Tenebriolarvae and in the large caterpillars of Cosstts and Lasiocampa need not exceed 0-02 aim.in order to account for the exchange of respiratory gases from the spiracles to thetracheolar' end trees', i.e. to the beginning of the tracheoles. This has been questionedby Nunome (1944) on the basis of a careful analysis of silk worms. He found that thediffusion area does not remain constant as assumed by Krogh but varies with thedistance from the spiracle (Fig. 8) in a way reminiscent of the muscle in Fig. 7. How-ever, instead of calculating Ap as done in the present study he has chosen to use aconstant area throughout and applied the smallest area in the entire pathway, thesummed cross-sectional area of the visible ends of the tracheoles, indicated by a crossin Fig. 8. In other words, he has substituted a wide tapering diffusion path for a thintube of uniform diameter. This is bound to give values of A/> which are at least oneorder of magnitude too high and since he found Ap = 0-17 he has, in fact, establishedKrogh's result on the basis of a much more detailed analysis.


(b) Diffusion in locust wing muscles

Gas. The tracheal supply of locust muscle is exceedingly rich but much less regularthan in dragonflies so that exact estimates are difficult to make. As an example, weshall consider the large dorsal longitudinal wing muscle. In the mesothorax it belongsto the latero-linear type and, according to fig. 8 in Weis-Fogh (1964a), the holefraction a at the surface of the muscle is as large as 0-07. The mesothoracic muscle issupplied in a similar way although the primary tubes consist mainly of large paralleltracheae on the mesial surface. In transverse sections (Fig. 9) it is seen that the fibresare much more loosely packed than in the dragonfly muscles so that more than halftheir surface area is in direct contact with the haemolymph (Fig. 10A). All majorclefts and most of the minor ones are packed with branching tracheae (not drawn).The summed area of the secondary tubes therefore seems to remain more or lessconstant with the distance from the mesial surface of the muscle. Even in the case of alinearly decreasing area, however, Table 3 shows that for a = 0-05 diffusion in thesecondary, system is sufficient.

Fig. 9. Cross-section of the dorsal-longitudinal wing muscle in the metathroax of the locustSchistocerca gregaria (a) being the mesial and (b) the lateral side. Only the major tracheae areshown. Note the abundant system of crevices filled with blood and tracheae.

It is different with the fine air tubes because of the relatively large fibre diameter ofabout 50 /i (Fig. 10 B). According to Table 5, there is no possibility for an adequatesupply of oxygen unless the tracheoles indent the surface and enter the interior, as infact they do (Tiegs, 1955; Vogell et al. 1959). In order to get an approximate estimateof the summed cross-sectional area of the entering tracheoles divided by the surfacearea of a fibre, i.e. of the hole fraction a, the number of entering tracheoles were countedin 25 fi thick sections. They are drawn as dotted lines in Fig. 10B. Inside the fibresthe density of tracheoles is higher than indicated because each tube bends and runsparallel to the fibre axis for a considerable length. Table 7 demonstrates that even thesimple radial arrangement is sufficient for an adequate supply of oxygen up to fibrediameters of 200 /i when, as here, the hole fraction amounts to about io~* (in the


actual case 1-5 x io~*). It was assumed that the entering tracheoles are o-Zfi wideand that they taper gradually towards the interior. There is therefore no problemwith respect to the diffusion of gases. In Chortoicetes terminifera which is also a goodflyer, Tiegs (1955) found the tracheoles arranged as in Schistocerca and, according tohis photographs figs. 125-126, with approximately the same density.

1001I I I I I I0 SO//

Fig. 10. Cross-sections of fibres of the dorsal-longitudinal wing muscle of the locust Schisto-cerca gregaria. (A) shows the loose packing of a group of fibres and (B) shows the number oftracheoles which indent a fibre section 25 /* tall.

Table 7. Calculated radius R (in fi) of a cylindrical muscle fibre supplied with oxygen bydiffusion in internal tracheoles which radiate from surface towards axis with linearlydecreasing cross-sectional area (Fig. 12G and equation (10)

(Ap = 0-05 atm. Compare R in Table 5, case B.)

Active

Metabolic rate m (ml. O,/g./min)..

Hole fraction a at fibre surface

RestO-O2

(10400)(3300)(1050)

1480470150

1050320105

74023075

5201655°

Fuels. As has already been discussed, the transport of fuel to the interior of musclesas large as those in Fig. 9 represents a special problem, but the combined effect of ahigh trehalose concentration (500-2000 mg. %;'Howden & Kilby, I96i)and of mus-cular pumping undoubtedly accounts for an adequate supply during the first periodof flight. As is indicated for cockroaches (Polacek & Kubista, i960) locusts completelyuse up trehalose during prolonged flight (students' practicals in this laboratory) andthe performance then depends on the combustion of fat (Krogh & Weis-Fogh, 1951;Weis-Fogh, 1952). The loose packing and the relatively small diameter of the fibrescharacteristic of locust wing muscle are probably an adaptation to this type of meta-

Diffusion in insect wing muscle., the most active tissue known 247

bolism, since fatty acids consist of large molecules of limited solubility in haemolymphand fibre plasma. If the fibres are about 50 /i in diameter and if the rather large mole-cular weight of fatty acids compared with that of glucose is outweighed, as far as dif-fusion speed is concerned, by the higher caloric value, Table 6 shows that the con-centration difference between the plasma membrane and the axis need amount only to1 mg. % in locusts (diameter 50 fi), 5-20 mg. % in most butterflies (George &Bhakthan, i960; 80-180 fi) and 5-10 mg. % in some beetles (Darwin & Pringle, 1959;80-140 /i). The solubility of stearic acid in water corresponds to about 30 mg. % at250 C. and increases with temperature to 100 mg. % at 370 C. (Handbook of Physicsand Chemistry, 42nd ed.). There is therefore no difficulty in providing the individualfibres with sufficient fuel and since Tietz (1962) found that the haemolymph ofresting locusts (Locusta) contains 800 mg. % esterified fatty acids (mainly in combina-tion with protein) and 2-6 mg. % free fatty acid, diffusion and muscular pumpingtogether may account for the transport of even long-chain fatty acids. Moreover, inwing muscles of Lepidoptera and Coleoptera there are many fat-storing cells locatedbetween the fibres. They possess a considerable capability for fat synthesis (Zebe,1958) and also a high lipase activity (George & Bhakthan, i960).

0 0-5 mm.1 I I I I I

Anastomosis 1

Head Abdomen

Leg1.

Leg 3-Anastomosis 2.

Leg 2.

B

Fig. 11. The major thoracic air sacs and tracheae of Drosopfrila melanogtuter drawn from theoriginal injection preparations of Wigglesworth (1950). (A) a young adult prior to the ex-pansion of the air sacs and (B) an adult after expansion. In contrast to locusts there are twolarge anastomoses between the right and the left side of the thorax, as indicated.

(c) Diffusion in Drosophila

Ventilatory movements have not been recorded in these small insects either at restor during flight although C M . Williams (unpublished) saw small rhythmic pulsationsof the dorsal air sacs at a frequency unrelated to the wing movements (quoted byChadwick, 1953). Chadwick & Gilmour (1940) clearly demonstrated that the flight ofD. repleta does not involve an oxygen debt and that the performance can go on attensions corresponding to 6 % oxygen in the ambient air. It is therefore almost certainthat diffusion accounts fully for the exchange of gases in such small insects even duringflight, as already suggested by Krogh, (1920a). In order further to test this, Prof.V. B. Wigglesworth kindly lent me some of his injection preparations of D. melano-gaster. Fig. 11A shows a newly hatched specimen in which the air sacs are not yet


expanded and appear flattened while Fig. 11B is a fully developed female in which theair sacs are greatly enlarged and expanded (Wigglesworth, 1950) to an extent which,as to general organization and magnitude, is reminiscent of the intricate system foundin locusts (Weis-Fogh, 1964a). Although more exact estimates are lacking, a com-parison between Fig. 11 and Tables 2 and 3 (m = 2 ml. O2/g./min.) indicates thatwith such large primary and secondary supplies diffusion should be completelyadequate in itself, the total diffusion distance from spiracle to fibre being less than0-5 mm. throughout the thorax. As to the tracheoles, I have counted the density in adorso-ventral muscle at different levels (cf. Wigglesworth, 1950, plate II, fig. 2, 75 ftthick section) and found that the average cross-section area supplied by a tracheole is55 p? corresponding to a radius of 4-2 fi, i.e. somewhat smaller than the figure of 5-4 ftin Table 5C. Since it is unlikely that all existing tracheoles have been successfullyinjected or counted, the combined results strongly indicate that the entire gaseousexchange can take place by diffusion during rest as well as during flight.

The question remains why such small insects have air sacs. Wigglesworth hasrecently suggested (1963) that the expanding air sacs tend to reduce the volume ofblood to about one-third and therefore to reduce the work of circulation as well as thetotal amount of dissolved blood sugar without decreasing its concentration. To thisshould be added the advantage of flexible air sacs with respect to a distributed andthorough mixing of the blood within the thorax and inside the muscles, as discussedin the present paper. Although the flight movements of Diptera hardly result in anynet change of thoracic volume, they must give rise to pressure changes and some move-ment of blood and air within the thorax. While the air movements are of little functionalsignificance, the blood movements may be essential for an adequate supply of fuel. Therhythmic pulsations of the dorsal air sacs observed by Williams may then reflect thework of the aorta and the accessory pulsating organs.

[d) Other vmg muscles

Many data on wing-muscle tracheation offered by Tiegs (1955) are relevant in thepresent context.

Like dragonflies, the cockroach Periplaneta americana has radial lamellar fibres30-60 ji in diameter and with no trace of internal tracheoles, but the thickness of thecortical part containing sarcosomes and fibrils is only 10-15 /*• According to Polacek& Kubista (1960) this species can perform flight of some minutes' duration and withan oxygen consumption as high as 4 ml. Og/g. muscle/min. However, unless thelamellar arrangement offers some special advantage as to diffusion (as suggested fordragonfly muscle on p. 241), Table 5 shows that the limiting thickness is 8-4/* forAp = 0-05 arm. In other words, the system is working near to the limit. Thisaccords well with Polacek & Kubista's (i960) observation that pyruvate (and probablyglycerophosphate) accumulates in the muscles and the insect soon becomes exhaustedin contrast to locusts and other good flyers.

The cicada Cyclochila australasiae also has radial lamellar fibres 60-90 ji in dia-meter but here numerous tracheoles radiate inwards from the periphery in such a waythat no part of the interior is more than 2-3 fi away from an air tube. I have cal-culated a in Table 7 to be 1-3 x io~3 so that all fibres are amply supplied with oxygen.With respect to tissue diffusion, even the highest metabolic rates can be accounted for


{Table 5). In the less active jassid Erytkroneura ix the many internal tracheolesfollow a longitudinal path each supplying about 80 /i2 of the fibre cross section whichshould permit metabolic rates of about 2 ml. Og/g./min. (Table 5).

The most interesting group consists of the large-fibred Diptera already discussedin connexion with the diffusion of dissolved fuel (p. 243). Here the fibres are servedby a regular system of small transverse tracheae about 2 /i in diameter at the surface,tapering and giving off tracheolar side branches in the interior. At the surface thedistances between tracheae vary from 30-40 fi in Musca and CalHphora to 20 /i inTrichophthalma. They serve a depth of 100 and 50 /i respectively. Thus the supplyis of the latero-linear type with a hole fraction of about io~a which is sufficient forintensely working fibres which are two to four times thicker than the fibres foundin these insects (Table 3). The tracheoles form a space network (although actualanastomoses may not occur) with a distance between the tubes of from 2-3 fi{Calliphora, Gastrophilus, Trichophthalma) to 3-5 fi (Musca) and this will ensureadequate tissue diffusion at the highest metabolic rates encountered as yet (Table 5),the safety factor being of the order of 2-3.

(e) General

So far, all estimates are based on the assumption that diffusion in air is not in-fluenced by the proximity of solid walls (normal diffusion). However, the diameterof the smallest tracheoles, o-i /i, approaches the mean free path of most gas moleculesat atmospheric pressure, about 0-06 fi, so that it is reasonable to discuss the possible•effects of the small dimensions. I have been unable to find any literature bearing onmy particular problem since the so-called Knudsen flow is related to pore diametersapproaching those of the molecules themselves, i.e. smaller by at least two orders ofmagnitude (cf. Hewitt & Sharratt, 1963). According to discussions with Prof. J.Kofoed and Dr Sten-Knudsen it seems more likely that the diffusion is unhamperedbecause gas molecules adsorbed to the walls should behave as perfectly elastic bodiesand be replaced almost instantaneously if removed. In fact, the replacement may takeplace by lateral displacement within the adsorbed layer at speeds which in some systemsmay approach that of sound. As far as tracheoles are concerned, Prof. Kofoed sug-gested that, due to this effect, the transport may even be facilitated to some extentcompared with diffusion through wider tubes of similar cross-sectional area.

The conclusions drawn in this paper should therefore not be significantly influencedby the smallness of the tracheolar tubes.

CONCLUSIONS

Transport of respiratory gases and of fuel in insect wing muscle depends on three<lifferent mechanisms: ventilatory movements executed by thorax and abdomen,muscular pumping due to length changes, and diffusion. The first is analysed elsewhere(Weis-Fogh, 1964 a, b) while the other two are treated here.

(1) The arrangement of muscle fibres, blood spaces and air tubes was found to besufficiently regular in dragonflies (Aeshna spp.) to make possible a detailed analysis.The account is extended so as to include other tracheated systems, partly by means ofa number of calculated examples (Tables 2-7), partly from observations on Schisto-cerca and Drosophila, and partly by including data from the literature.


(2) During flight the primary tracheae supplying the wing muscles in dragonfliesand other large insects must be strongly ventilated, but in small insects like Drosophiladiffusion of respiratory gases is sufficient to account for the entire transport betweenthe spiracles and the end of the tracheoles even at the highest rates of metabolism.

(3) Beyond the primary tracheae and air sacs, air-tube diffusion is sufficient in allinsects studied except in the giant belostomid bugs where the secondary tracheae arealso ventilated. It is shown that muscular pumping plays an insignificant role in thetransport of respiratory gases.

(4) The efficiency of the tracheal system is due to the large hole fraction charac-teristic of wing muscle. It varies from io~x to io~2 as far as the secondary supply isconcerned, and from icr2 to io"3 in the tertiary and tracheolar systems. The rate ofdiffusion of O2 is then io3 to io6 times greater than in the liquid phase and for CO2 it isabout 50-5000 times greater.

(5) Diffusion in the liquid phase between the tracheoles and the sites of com-bustion is sufficient in all fibres investigated. It is shown that the main exchange mustoccur via the tracheoles since, at the high metabolic rates, the diffusion distances aremuch too long for larger tubes to be effective.

(6) In giant and medium-sized fibres with ' internal' tracheoles indenting the surfacethe safety factor is of the order of 2-3, but in fibres supplied only from the surface theoxygen-consuming zone has reached the limiting thickness of about 8-10 fi in Odonataand Blattidae. It is argued that the radial-lamellar fibres in the latter groups are anadaptation which improves oxygen diffusion inside the fibre.

(7) Glucose was used as a model substance for the diffusion of fuel into wholemuscle and into the interior of the fibres. Muscular pumping of the blood is essentialin almost all active wing muscles but, because of its tidal nature, the concentration ofdissolved fuel in the haemolymph must be of the order of 0-5-1 %. It is thereforesuggested that the high content of trehalose and of lipid materials in insect blood hasevolved as an adaptation to flapping flight.

(8) In many Hymenoptera and in Diptera with giant fibres the difference in fuelconcentration between the inside of the fibre membrane and the interior must be about0-5-1 %. In insects depending on fat for sustained flight (Orthoptera, Lepidoptera,Coleoptera), the fibres are smaller in diameter and appear to be loosely packed so thatrelatively small amounts of free fatty acids in the sarcoplasma should be sufficient(0-001-0-02 %) while muscular pumping and protein-bound lipid serve to transportfuel into the muscle.

(9) All available data taken into consideration, no other mechanisms or transportroutes than those mentioned are essential in order to account quantitatively for theexchange of gas and fuels. Since this applies to the most active systems known andsince, for instance, diffusion in the tracheal system of Bombyx larvae was shown to bemuch more effective than claimed by Nunome (1944, 1951), there is no reason tobelieve that essential factors have been overlooked.


SUMMARY

1. The tracheal system of insect wing muscle is so dense that between 1 cr1 and 1 o"3

of any cut area is occupied by air tubes. In most cases, air tube diffusion of O2 andC08 through the muscle is therefore several thousand times quicker than diffusion inthe liquid phase.

2. In large insects the primary tracheal supply must be strongly ventilated whilediffusion is sufficient in the remaining part of the air tubes, even at the highestmetabolic rates encountered in any insect.

3. The tracheoles represent the main site of exchange between the gaseous and theliquid phase while the tracheae are of little significance in this respect. The fibrescannot exceed about 20/1 in diameter unless the tracheoles indent the surface andbecome 'internal'.

4. Muscular pumping of air and blood due to shortening is of little importance forthe exchange of gases but of major importance for the supply with fuel for combustion.However, the large fibre diameters and the tidal nature of the pumping necessitates avery high concentration of fuel in the haemolymph. The high concentration oftrehalose in insect blood is considered to be an essential adaptation to flapping flight.

5. The transport by diffusion of O2 and COa was followed in detail in a number ofconcrete examples in the gaseous as well as in the liquid phase. Within a safety factorof 2-3, the rate of transport was always found to be adequate. There is no reason tosuggest other mechanisms than a simple, normal diffusion.

I am indebted to Prof .V. B. Wigglesworth for the loan of injection preparations ofDrosophila, to Dr J. Buck for a translation of Nunome'a paper, and to Prof. J. Kofoed,Dr P. L. Miller, and Dr O. Sten-Knudsen for valuable discussions.

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the oxygen consumption considered in relation to the wing-rate. Physiol. Zool. 13, 398—410.CLARK, H. W. (1940). The adult musculature of the anisopterous dragonfly thorax (Odonata, Anisoptera).

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I3I-37-

APPENDIX

Steady-state diffusion in tissue with consumption

The rate of consumption per unit volume m and the rate of production of end pro-ducts are considered constant and uniform throughout the tissue. The problem thenis in simple terms to express the variation of volume and diffusion area A with dis-


tance. The procedure is illustrated in the case of equation (3) and is similar to thetreatment offered for instance by Erlang (in Krogh, 1929) and by Crisp (Thorpe &Crisp, 1947; Appendix). Many other examples may easily be developed. Whendealing with respiratory gases, the equations are solved with respect to the differencein partial pressure Ap necessary to account for the flow. The same formulae apply tothe diffusion of solutes like glucose if the concentration difference Ac and the diffusioncoefficient D are inserted instead of Ap and the permeability constant P respectively.Tissue diffusion means that the transport is assumed to take place in an isotropic liquid(blood and cytoplasm) while air-tube diffusion means that the transport is confined tothe system of air tubes and that transport in the liquid phase is insignificant, as dis-cussed on p. 232.

Tinue

Fig. i a. Examples on diffusion systems considered in the Appendix. A-D refer to linearone-dimensional cases and E-J to radial one-dimensional cases. A heavy arrow marked' tissue' shows the direction of flow in an isotropic tissue (tissue diffusion) and the arrowsmarked 'air' show the direction when the transport occurs in the tracheal system (air-tubediffusion).

(a) Rectilinear flow

Tissue diffusion. Consider a rectangular block of homogeneous tissue, L cm. longand with end area Ao from which diffusion takes place in the direction from x = o tox = L (Fig. 12A). For any value of x, Ax = Ao and we know that all oxygen con-sumed beyond Ax must pass this surface by diffusion due to the pressure gradientdpjdx. We then have for the net flow Jx through Ax,

Jx = Ax{L-x)m = A0{L-x) = -PA0(dp/dx),from which —dp = mjP{L — x) dx.

Let/) = p0 for x = o and/) = p^ for x = L. Integration between these limits gives

PO~PL = A/) = ^ . (3)


Air-tube diffusion. Consider a similar rectangular block of tissue pierced in thedirection of x by parallel air tubes whose summed areas <x0 for x = o amount toao = aA0, where a is the hole fraction (Fig. 12, B and C). Provided that these tubes are' tapped' at a rate corresponding to the consumption in the tissue they pass either bydiffusion in the liquid phase, as in the terminal tracheoles, or by diffusion in regularlyspaced side tubes (see Figs. 4, 5), the transport can be calculated when a = a(x) isknown.

Case A. The summed cross-sectional area of the tubes is constant and independentof x, i.e. ax = aAg, as in the main branches of Cossus larvae (Krogh, 1920), Aphelo-cheirus (Thorpe & Crisp, 1947) and Rhodnius (Locke, 1958). According to Fig. 12Band the previous example, we then get

mLi

(4)

Equation (4) also applies to branching systems provided that the air tubes run approxi-mately parallel to the #-axis.

Case B. The tubes taper so that a, i.e. the area, decreases linearly with x (cf.part of Fig. 7). We have, ax = aA^i—xfL), giving

rf (5)

Case C. The tubes taper so that their diameters decrease linearly with x (cf. Figs. 6,12C), andax = aA0(i—xjL)2. For 0 ^ x < L,

. ml?, L ...Ap = —-hxY . (6)r aP L — x v '

Case D. Diffusion in the central trachea. Consider a cylindrical muscle of length Land radius R supplied from an axial primary trachea of radius r0 from which secondarytracheae branch off (Fig. 12D). When r0 ^ R/10, the metabolic rate of the muscle isapproximately mnR2L. The problem is to what extent diffusion in the primarytrachea can account for transport in the axial direction x. The flow across a crosssection of the tube is Jx = mnR\L—x) = —irr^P (dp/dx), which when integratedfrom x = o and/) = p0 to x = L and/) = pL and solved with respect to L gives

L-ro/i?(2A/)P/m)i (7)

(b) Radial flow

Only cylindrical cases are treated, the radius of the tissue cylinder being R, its lengthL and the distance from the axis r.

Tissue diffusion. The diffusion takes place either from a co-axial cylindrical tube ofradius r0 towards the periphery, or from the peripheral surface towards the cylinderaxis.

Case A. Diffusion from inner cylindrical tube towards the periphery. At distancer, the flow through the tissue surface Ar is (cf. Fig. 12 E),

Jr = nLmiRt-r2) = -z-nrLP{dp\dr).


Integrated from r = roandp = p0tor = Randp = pR, we get

(8)

Case B. When the diffusion occurs from the surface of the cylinder correspondingto r = R towards an inner cylinder surface of radius r0 (cf. Fig. 12 F), the solution is

^ (9a)

For diffusion to the centre axis where r0 ->• o, we have

A^^p- (9b)

Air-tube diffusion. The first case is directly related to the preceding one.Case A. Consider a cylindrical piece of tissue supplied from the outside by tubes

radiating towards the axis and where the summed cross-sectional area of the tubes a isproportional to the distance from the axis (Fig. 12G). We have then aB = aznRL.If r0 -> o, the solution is

Case B. The tissue cylinder, is supplied from a central co-axial tube of radius r0

which is 'tapped' by means of radially arranged secondary tubes whose summed areaat the origin is aTo = azTrrJL. Let us first assume ar = aro = constant (Fig. 12H).Jr = nmL(R2-r2) = -2nr0LdP{dpjdr). Integrated from r = r0 and p = p0 tor = R and p = pR, we get

Le t / = rojR be inserted in equation (11) and solved for/,

Case C. As above, but with linearly decreasing area (Fig. 121), so that

ar =

. m

For any value of r, we have

Af = jjfeR2^—ro) + i?(r2 — rl)). (^^O

If, again, / = ro/R, we have

D. As the two preceding examples, but with linearly increasing area so thataT = a27rr0Lr. m I R

17 Exp. Biol. 41, 2


Case E. A tissue cylinder is supplied from thin tubes radiating out from the centralaxis (Fig. 12 J; cf. outer half cylinder of the lobe in Fig. 5). The hole fraction is heredefined as a = ct^lzRL. When ar = a^ = constant, we have

nmR2

Case F. Same as (E) but with linearly decreasing area, ar = 2aL(R — r). The solu-

tion i s

diffusion in insect wjng muscle, the most ...intricate system of unidirectional valves in the...

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