Proc. Natl. Acad. Sci. USAVol. 88, pp. 10357-10361, November 1991Physiology

The concept of symmorphosis: A testable hypothesis ofstructure-function relationship

(respiratory system/mitochondria/capillaries/maximl oxygen consumption)

EWALD R. WEIBEL*, C. RICHARD TAYLOR, AND HANS HOPPELERDepartment of Anatomy, University of Bern, Buhlstrasse 26, CH 3000 Bern 9, Switzerland; and Museum of Comparative Zoology, Harvard University,Cambridge, MA 02138

Contributed by Ewald R. Weibel, August 12, 1991

ABSTRACT The hypothesis that, in biological organisms,structural design is matched to functional demand is difficult totest because it is largely based on anecdotal evidence suggestingeconomic design. The hypothesis of symmorphosis postulates aquantitative match of design and function parameters within adefined functional system; because of its stringency it is refut-able and can, therefore, be subjected to empirical test, forexample, by assessing whether the structures that support thepathway for oxygen from the lung to the consumer in musclecells are quantitatively adjusted to the limit of functionalperformance of the respiratory system. The study of allometricand adaptive variation leads to the conclusion that the hypoth-esis of symmorphosis is acceptable for all internal compart-ments of the respiratory system (blood, heart, muscle capil-laries, and mitochondria), whereas it must be refuted for thelung that forms the interface to the environment.

The concept that animals, and humans as well, should bedesigned economically (i.e., that structural design should bematched to functional demand) follows from common sense,but it is also supported by many observations. Thus bloodvessel architecture is such that blood flow distribution occurswith minimal energy loss; bone structure is patterned ac-cording to stress distribution and also quantitatively adaptedto total stress; with training, athletes can specifically adjustthe structure of their muscles and of their cardiovascularsystem to higher functional demands, and these modifica-tions are soon reversed when training is stopped. Theseexamples suggest optimization of structural design based onthe cost-to-benefit relationship: the larger an organ the moreit will cost in terms of construction, maintenance, and load onthe body. Most of this evidence is anecdotal, however, andone can come up with similar anecdotal evidence that ques-tions this conclusion.To test the validity of the concept of economic design as an

important principle of structure-function relationship in bi-ology requires a hypothesis that postulates a quantitativematch between parameters of design and function. Thisappears fulfilled by the hypothesis of symmorphosis, definedas the state of structural design resulting from morphogenesisregulated to match functional demand (1). From this hypoth-esis we would predict that, if the functional requirements onan organ system vary, structural design should vary inparallel.Symmorphosis has recently been discussed from various

aspects (2-4). We wish to summarize the insight we havegained in studying the mammalian respiratory system and themorphometric variations that accompany varying functionaldemands.

The Respiratory System and Symmorphosis

The respiratory system ofmammals is a good test case for thehypothesis of symmorphosis for a number of reasons.

(i) It serves one dominant overall function, the supply of02from environmental air to the mitochondria in the cells insupport of oxidative phosphorylation.

(ii) This overall function has a measurable limit, the aerobiccapacity or maximal rate of 02 consumption, VO2max-

(iii) This functional limit varies between individuals andspecies.

(iv) The respiratory system involves a sequence of struc-tures: the lung, the blood, the heart, microcirculation, and themitochondria in (muscle) cells, which all need coadjustmentwhen aerobic capacity is varied.

(v) At each step a number of functional parameters areinvolved that can be rapidly adjusted, whereas the structuralparameters are "fixed" quantities that are very slow inadapting (days to months) and thus potentially set the ca-pacity of the system for instantaneous functional perfor-mance.The model ofthe respiratory system shown in Fig. 1 breaks

the pathway for 02 into four steps (5-7): two steps of diffusivegas exchange in the lung and in the peripheral microvascu-lature are connected by the convective transport of 02 bycirculation of the blood; the final step represents the con-sumption of02 in the process of oxidative phosphorylation inthe mitochondria. Under steady-state conditions, the flow of02 through each ofthese steps is equal to Vo2 measured at themouth.The functional and the structural parameters that affect the

transfer functions at every step are identified in Fig. 1 wherethe functional factors are shown to the left and the structuralfactors are to the right of the central multiplication dot. Theproduct of the factors shown in boldface type determine Vo2directly; the parameters shown in italic type within braces aredeterminants of these factors. These equations will be dis-cussed step by step below.

Testing the Hypothesis of Symmorphosis by ComparativePhysiology

The test of the hypothesis is to set the morphometric mea-surements into relation to the physiological variables perti-nent for each transfer step and, particularly, to VO2max as theoverall estimate ofaerobic capacity. The simplest question toask is how the variables vary with changes in Vo2max.Alternatively, the hypothesis of symmorphosis predicts thatthe design parameters form an invariant ratio with Vo2maunder all modes of variation, ifthe morphometric features arelinked with aerobic capacity.The test we propose is based on the study of variations of

VO2max between animals or species. The approach we have

*To whom reprint requests should be addressed.

10357

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Proc. Natl. Acad. Sci. USA 88 (1991)

FUNCTION -DESIGN

V02 - (PAO2 - PbO2)jtif; DLO2 S A 1.S(.'. V

Vo2 -(7a7 Pao2 - -v Pvo2)JfH Vs V . Vv(ec)

Vo= (Pbo2 -PCO2)!;.*DTO2

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parameters that affect 02 diffusion, capillary volume V(c) andsurface S(c), and the characteristic diffusion distance to thevicinity of the mitochondria 8(c,m) (Fig. 1, line 3). Thearchitectural situation is not simple, and a model does notexist for estimating DTO2. However, the basic design patternof muscle microvasculature is sufficiently similar in differentspecies and muscles that a comparative study of the relationbetween capillaries and mitochondria can be based on simpleestimators of design. Thus capillary volume and surface aswell as the radial diffusion distance (radius of Krogh cylinder)are all related to the length density of capillaries in muscle,a quantity that can be estimated reliably (26, 27).

In first approximation, we predict from the hypothesis ofsymmorphosis that the volume of capillary blood in themuscles is proportional to the muscles' capacity to consume02: in individual muscles capillary volume should be propor-tional to mitochondrial volume, and total muscle capillaryvolume V(c) should vary directly with Vo2ma.

In allometric variation we found that V(c)/Mb varies indirect proportion with Vo2max/Mb and with weight-specificmitochondrial volume (26). In adaptive variation (22), it wasfound that athletic species have, paradoxically, a relativelysmaller capillary volume than sedentary animals of the samesize (Table 1). However, the athletic species were found tohave a hematocrit, Vv(ec), about 1.5 times higher than theother species with the result that their blood has a larger 02capacity. The product of capillary volume and hematocritestimates capillary erythrocyte volume; Table 1 shows thatthe ratio of total capillary erythrocyte volume to VO2max isinvariant between the species pairs of adaptive variation. Inallometric variation there is no systematic trend for differ-ences in hematocrit or hemoglobin concentration.We therefore conclude that, in both instances, the invariant

ratio of capillary parameters to aerobic capacity is

[V(c)'Vv(ec)]/V02max = [1.8 ml(ec)]/(ml of 02 per min)=invariant.

We note that in this instance two morphometric parametersare involved: the quantity of capillaries and the concentrationOf 02 carrying erythrocytes in capillary blood. Thus themicrovasculature and the blood contribute about equally toestablishing conditions for 02 supply that are proportional toneeds.

Convective 02 Transport by the Circulation of Blood

The transport of 02 by the circulation depends on theproperties of the heart as the pump and on those of the bloodas 02 carrier as expressed by the Fick equation (Fig. 1, line2): the arteriovenous 02 concentration difference is affectedby the Po2 in arterial and venous blood, by the hematocrit,and by the 02 capacitance of the blood, which we haveexpressed as a "hematocrit-specific" variable 0'02 to sepa-rate functional and structural variables. Cardiac output Qis the product of a functional parameter, heart frequency fH,and stroke volume V., a structural parameter that is propor-tional to heart weight (unpublished observations).On the basis of the hypothesis of symmorphosis, we would

predict that the two morphometric parameters, ventricular orstroke volume and hematocrit, should be matched to Vo2maxas the global measure of 02 transport capacity. Table 1 showsthat in adaptive variation the stroke volume of athleticspecies is about 1.6 times higher than in sedentary species;when considering the differences in hematocrit, we find thatthe product of erythrocyte volume with stroke volume, whatone could call the "erythrocyte stroke volume", is increasedin proportion to Vo2max in the athletic species. It is notewor-thy that the maximal heart frequencies are pairwise identical

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Proc. Natl. Acad. Sci. USA 88 (1991)

in the size classes (Table 1). Accordingly, the higher 02transport capacity of athletic species is achieved purely bystructural adaptation (23).

In allometric variation, heart frequency at rest variesdirectly with 02 consumption (5, 9, 10); maximal heartfrequency is not sufficiently well documented but, from thedata in Table 1, it appears that it should follow about the sameallometric regression as resting heart rate. On the other hand,heart weight makes up the same fraction of body mass overthe size range of mammals from mouse to cow, namely,0.58% (28), so that the ratio VS/Mb should be invariant withbody size because we have found Vs to be linearly related toheart mass (unpublished data). Likewise, hemoglobin con-centration in blood and hematocrit do not vary systematicallywith body size (29). Although allometric measurements ofcirculatory parameters under conditions of VO2max arescarce, the available evidence suggests that the size-dependent change in Vo2max/Mb is matched by a change inthe functional parameter heart frequency whereas the struc-tural parameters Vj/Mb and hematocrit are invariant.We therefore conclude that the prediction derived from the

hypothesis of symmorphosis, namely, that stroke volume andhematocrit should match Vo2n., is fulfilled in adaptive variation;in allometric variation, heart frequency comes into play, which,like metabolic rate, is a dominant functional parameter whenbody size varies. When testing the hypothesis of symmorphosisby seeking an invariant ratio of structural-to-functional parame-ters, this must be taken into account. We find that

[VsVv(ec)]/(Vo2max/fHmax) = 2.8[ml(ec)]/(ml of 02)= invariant

holds for both adaptive and allometric variation.

Uptake of 02 in the Lung

The transfer rate of 02 from the air into the blood (Fig. 1, line1) is driven by the partial pressure difference from alveolar airto capillary blood, (PAO2 - Pb02), a functional variable thatdepends on the ventilation of alveoli and the perfusion ofcapillaries. The conductance or diffusing capacity DLO2, incontrast, is largely determined by structural parameters (7,30) and has two main components: the membrane diffusingcapacity DMO2 depends on the alveolar and capillary surfaceareas S(A) and S(c) and on the harmonic mean thickness ofthe tissue and plasma barriers Tht and Thp, whereas thecapillary blood volume V(c) and the hematocrit, together withthe 02 binding rate coefficient 602, determine the 02 uptakecapacity of the blood.The hypothesis of symmorphosis predicts that the com-

pound parameter of structural design DLO2 should be propor-tional to Vo2max. This is not the case. In allometric variationwe find that DLO2 has a different slope from Vo2m. whenplotted against body mass (Fig. 2) with the result that the ratioDLO2/VO2max increases with Mg 2 (20). Thus a 300-kg cow hassix times as much diffusing capacity as a 30-g mouse toaccomplish 02 uptake at Vo2max, so that the driving force for02 uptake is smaller in the cow than in the mouse.

In adaptive variation we have found that athletic specieshave a larger gas exchange surface area and a higher hemat-ocrit than nonathletic animals, which results in a largerDLO2/Mb in the athletic species (Table 1), but this increaseis not proportional to the differences in V02max/Mb (15,24). When combining physiological measurements with themorphometric estimate of DLO, we found that capillarytransit time t- was invariant ana that all lungs used only afraction of tc for equilibration of capillary to alveolar P02; butthis fraction was greater in athletic species, namely, 80%versus 50% in the sedentary species (31).

In conclusion, we find that the ratio

DLO2/V02max #A invariant,

neither in allometric nor in adaptive variation. As a result thefunctional parameter (PAO2 - PB02) = APo2 must vary as wellbecause DLO2/(VO2max/APO2) must be invariant by definition(Fig. 1, line 1). This result leads to the conclusion that thehypothesis of symmorphosis does not apply at the level ofthepulmonary gas exchanger.

Discussion

The hypothesis of symmorphosis predicts that structuralparameters are matched to functional capacity. We havebased the test of this hypothesis on a comparative study ofthemammalian respiratory system. We have asked whether thedesign parameters of the structures that support 02 flow arematched to the aerobic capacity of the organism estimated bythe maximal rate of 02 consumption, Vo2m.a, because this isthe only condition where we can expect all of the availablestructure to be utilized. We compared the effects oftwo typesof natural variation in VO2max, allometric versus adaptivevariation, and accepted the test of the hypothesis if the twoconditions gave identical results.

This study reveals that it does not make sense to test thehypothesis of symmorphosis on single structural parametersbut that the entire structural system must be considered incontext. The simplest structure-function relationship wasfound at the level of the mitochondria where we found thetotal muscle mitochondrial volume to vary in strict propor-tion to aerobic capacity; it appears that the molecular con-stitution of muscle mitochondria is invariant so that a highercapacity for oxidative phosphorylation can only be achievedby building more of the same mitochondria into the musclecells. At the other levels of the respiratory system, we foundthat more than one structural parameter could be varied andthat this was utilized in varying capacity. Thus, the capacityof the microcirculation to deliver 02 to the cells can bemodified either by increasing the density of the capillaries orby changing the blood composition. Likewise, 02 transportby the circulation can be increased either by pumping moreblood with a bigger heart or by packing more erythrocytesinto the blood. Which option is used may depend on eco-nomic considerations: increasing the number of erythrocytesper unit blood volume will reduce the amount of blood theheart must pump, but it will, conversely, increase pumpingcost due to higher blood viscosity, so that there will be anoptimal range of erythrocyte concentration (between 30 and50%) where pumping cost is minimal (32, 33). In the per-spective of symmorphosis it, therefore, appears economic topartition the adaptive effort between the two structuralsystems, and we found, indeed, that it was the products ofthemorphometric parameters of vessels and of blood that wereproportional to the functional parameter: hematocrit timescapillary or heart volume, respectively. That heart frequencymust be involved as an additional functional factor in allo-metric variation Of V02ma was discussed and justified above:time is a dominant factor when body size changes (10).When considering all the morphometric parameters of the

pulmonary diffusing capacity, we did not find them to beproportional to Vo2max, neither singly nor in combination. Itrather appeared that the lung maintains a considerable excessdiffusing capacity, up to a factor of 2 in sedentary species,that can be partly exploited in athletic species to achievehigher uptake rates; this also allows sedentary animals tomaintain their aerobic capacity in a hypoxic environment(31). We conclude that with the currently available evidencethe hypothesis of symmorphosis must be refuted for thepulmonary gas exchanger.

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In general conclusion, it appears from this analysis that theprinciple of economic design, as reflected in the hypothesisof symmorphosis, is upheld for all the internal compartmentsof the respiratory system, but it does not appear to apply tothe lung. Future work will have to address mainly twoquestions: (i) to examine whether symmorphosis as evi-denced in this study is indeed related to an economic designprinciple ofminimizing cost and (ii) to ask why the pulmonarygas exchanger is different from the other parts of the sys-tem-whether it is, for example, because the lung forms theinterface to the environment and is, therefore, exposed toadditional stresses. The most important question will be toelucidate in other systems whether symmorphosis reflects abasic principle, to which there may be some exceptions, orwhether it must be refuted in the end. For the time being thishypothesis still has considerable heuristic value.

This work was supported by grants from the Swiss NationalScience Foundation to E.R.W. and H.H. (31.9439.88) and from theU.S. National Science Foundation to C.R.T. (DCB 8918371). Addi-tional support was received from The M. E. Muller Foundation,Bern, Switzerland.

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