1607-6729/01/0304- $25.00 © 2001

MAIK “Nauka /Interperiodica”0110

Doklady Biochemistry and Biophysics, Vol. 377, 2001, pp. 110–112. Translated from Doklady Akademii Nauk, Vol. 377, No. 4, 2001, pp. 553–555.Original Russian Text Copyright © 2001 by Isaeva, Chernyshev, Shkuratov.

Even relatively simple linear fractal systems of bothorganic and inorganic origin differ from ideal computerfractals, because their structure is reproduced incom-pletely and without sufficient precision. In fact, they arequasi-fractal rather than true fractal systems. Manymulticellular animals contain various quasi-fractalstructures. Fractal organization of epithelial structures(e.g., the bronchial tree and vascular system of mam-mals) was a subject of extensive theoretical study andsimulation [1–4]. Although the patterns of both earlybranching of drosophila tracheoli and pulmonary bron-chial tree of mammals are stereotypical, and geneti-cally controlled, the terminal branches of the patternsare variable [4]. In spite of the fact that the absence ofstrict determination at late morphogenetic stages wasnot emphasized in [4], we believe that this feature is ofcrucial importance. This raises the problem of possible(or even inevitable) emergence of structural chaos inorganismal morphology.

We found elements of chaos in the morphologicalorganization of branching canals of the gastrovascularsystem of the Scyphozoan jellyfish

Aurelia aurita

: onlytwo to tree initial stages of branching were stereotypi-cal, whereas the order-to-chaos transition scenario istypical of subsequent quasi-fractal stages of morpho-genesis of the gastrovascular canals. The disorderingwas found to increase with consecutive stages ofdichotic branching (bifurcation).

It was shown earlier that chaotic fractal dynamics istypical of many physiological processes [5–8]. Chaos(variability and unpredictability) was found to be a nor-mal functional characteristic of a healthy organism,whereas a more ordered pattern was evidence of pathol-

ogy [5, 6]. In other words, “chaos is a way to adapta-tion” [7].

Like oscillographic records of chaotic dynamics offunctional activity, morphological quasi-fractal struc-tures of an organism register the chaotic dynamics ofmorphogenetic processes during ontogeny and may beregarded as structural visualization of the organism’smorphogenesis.

The gastrovascular system of the jellyfish

A. aurita

was the object of the study. Because this system con-sists of branching canals located in one plane (the jelly-fish umbrella), it is suitable for topological analysis.The functional role of the jellyfish gastrovascular sys-tem is to transport nutrients, excretions, and sex prod-ucts. The gastrovascular canals of

A. aurita

fall intothree types: (1) eight nonbranching adradial canals, (2)four branching perradial canals, and (3) four branchinginterradial canals [9]. The jellyfish body is character-ized by radial four-ray symmetry, and the four seg-ments (antimeres) are morphologically and function-ally identical to each other. The so-called perradialcanals are the most appropriate for fractal analysis. Thecommon trunk of each perradial canal is locatedbetween gastric pockets (Fig. 1a). To obtain contrastingpatterns, the gastrovascular system was stained with aspecific dye (e.g., eosin). Then, the jellyfish wasallowed to dry on a piece of filter paper. As a result, thegastrovascular system canals were viewed as a flatimage, which could be scanned and processed with acomputer.

To provide precise quantitative assessment of bio-logical fractal patterns, their branching structures canbe represented as standard deterministic fractal trees.Such a normalized presentation of the perradial canalsof one specimen of

A. aurita

is shown in Fig. 1b. Thebranches of the tree were subjected to quantitative tax-onomic classification using the modified method ofTokunaga [10] with subsequent construction of the cor-responding matrices. This allowed us to quantitativelyestimate the degree of pattern ordering or disordering(chaos).

Quasi-fractal Organization of the Gastrovascular Systemof the Jellyfish

Aurelia aurita

: Order and Chaos

V. V. Isaeva, A. V. Chernyshev, and D. Yu. Shkuratov

Presented by Academician V.L. Kas’yanov December 19, 2000

Received January 5, 2001

Institute of Marine Biology, Far East Division,Russian Academy of Sciences,ul. Pal’chevskogo 17, Vladivostok, 690041 RussiaFar East State University,ul. Sukhanova 8, Vladivostok, 690600 Russia

BIOCHEMISTRY, BIOPHYSICS,AND MOLECULAR BIOLOGY

DOKLADY BIOCHEMISTRY AND BIOPHYSICS

Vol. 377

2001

QUASI-FRACTAL ORGANIZATION 111

All specimens tested were characterized by signifi-cant variability of the branching pattern of the gastro-vascular system canals in an organism. None of the pat-terns of quasi-fractal branching of the gastrovascularcanals was found to be completely identical to anyother pattern studied. The total number of the branchesof the second to eighth rank (nos. 2–8) is shown in thetable. These data are shown separately for the left (l)and right (r) parts of the four perradial canals (I–IV) ofthree specimens of

A. aurita

designated as A, B, and C,respectively. The branching pattern of perradial canalsof specimen A is shown in Fig. 1.

It is obvious that the first and second dichoticbranchings (bifurcations) are completely stereotypical.Although the third bifurcation (divergence of thefourth-rank branches) is also usually stereotypical,branching errors sometimes occur. After the fourthbifurcation, the branching ordering (repeated and regu-lar branching) is lost, and the branching patternbecomes chaotic. Thus, the border between order andchaos in the structural organization of canals isobserved at the level of the four- to five-rank branchesboth in one organism (clone of cells with originallyidentical genomes) and in different jellyfish specimens.A similar pattern of transition from order to chaos wasobserved in other

A. aurita

specimens studied in thiswork.

It should be noted that this approach does not takeinto account additional difference in branching patternrelated to branch curvature, linear and angular size ofbranches, or such common topological characters asbranch anastomosis. In actual natural systems, chaoti-zation of the branching pattern is more pronouncedthan in the model considered in this work, and its onsetis observed at earlier stages of jellyfish individualdevelopment. As jellyfish grows, the difference in thepattern of branching and anastomosing becomes morepronounced, thereby disturbing the radial symmetry ofjellyfish body.

Probably, there is a universal scenario of transitionfrom order to chaos during morphogenesis of branch-ing canals of jellyfish. According to this scenario, thedegree of disordering increases with the rank of branch-ing (“downstream” the cascade of bifurcations).

Thus, only the initial stages and the most generalfeatures of fractal morphogenesis of the gastrovascularsystem of the jellyfish

A. aurita

are under strict geneticcontrol; these are the radial four-ray symmetry, forma-tion of eight nonbranching and eight branching canals,and two to three initial steps of branching. Subsequentquasi-fractal morphogenesis goes out of strict geneticcontrol and becomes chaotic and flexible.

In our opinion, lack of strict genetic determinationof the terminal stages of branching of the jellyfish gas-trovascular canals is a factor of adaptation to variableand unpredictable environment. The terminal stages offractal morphogenesis of epithelial canals are charac-terized by certain freedom not only in jellyfish but also

in higher animals [4]. This freedom can also beregarded as a factor of adaptation to variable and unpre-dictable environment (e.g., this may provide moreeffective regeneration after injuries). It was shown inthe preceding work [11] that fractal self-organization ofcells may serve as a mechanism of adaptation of cellsystems. Systems with chaotic dynamics of morpho-genesis are more flexible and resistant to environmental

12

3

4

567

1

2

34

56

7

I

IIIV

III

(a)

(b)

Fig. 1.

Perradial canal branching in the jellyfish

A. aurita

:(a) the natural pattern; (b) standard fractal trees.

112

DOKLADY BIOCHEMISTRY AND BIOPHYSICS

Vol. 377

2001

ISAEVA

et al

.

perturbations than ordered systems. Therefore, the cha-otic mode of morphogenesis provides a certain degreeof freedom, autonomy of cell and tissue systems, andbetter adaptation.

ACKNOWLEDGMENTS

We are grateful to Academician V.L. Kas’yanov forcontinuous support and valuable advice.

This study was supported by the Russian Founda-tion for Basic Research (project nos. 99-04-48843 and00-15-97938), Ministry of Education of Russian Feder-ation (project no. 97-0-10. 0-110), and State Program“Integration” (project no. 937).

REFERENCES

1. Mandelbrot, B.B.,

The Fractal Geometry of Nature

, NewYork: Freeman, 1983.

2. West, B.J. and Goldberger, A.L.,

Amer. Sci.

, 1986,vol. 75, no. 4, pp. 354–365.

3. Weibel, E.R.,

Am. J. Physiol.

, 1991, vol. 261, no. 6,pp. 361–369.

4. Metzger, R.J. and Krasnow, M.A.,

Science

, 1999,vol. 284, no. 5420, pp. 1635–1639.

5. Goldberger, A.L., Rigney, D.R., and West, B.J.,

Sci.Amer.

, 1990, vol. 262, no. 2, pp. 43–49.6. Goldberger, A.L.,

Perspect. Biol. Med.

, 1997, vol. 40,no. 4, pp. 543–561.

7. Gaponov-Grekhov, A.V. and Rabinovich, M.I.,

Vestn.Akad. Nauk

, 1997, vol. 67, no. 7, pp. 608–614.8. Ivanov, P.Ch., Amaral, L.A., Goldberger, A.L.,

et al.

,

Nature

, 1999, vol. 399, no. 6735, pp. 461–465.9. Southward, A.J.,

J. Mar. Biol. Assoc. UK

, 1955, vol. 34,no. 2, pp. 201–216.

10. Turcotte, D.L., Pelletier, J.D., and Newman, W.I.,

J. Theor. Biol.

, 1998, vol. 193, no. 4, pp. 577–592.11. Derzhavin, D.K. and Isaeva, V.V.,

Dokl. Akad. Nauk

,2000, vol. 373, no. 2, pp. 254–256.

Transition from order to chaos in the structural organization of four branching canals in three specimens of

A. aurita

(see thetext for designations and explanations)

No.

A B CStructural

organizationI II III IV I II III IV I II III IV

l r l r l r l r l r l r l r l r l r l r l r l r

1 1 1 1 1 1 1 1 1 1 1 1 1 Order

2 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1

3 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2

4 4 2 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 0

5 4 4 6 4 8 6 8 6 6 6 6 6 8 8 6 6 8 6 6 8 8 6 0 0 Chaos

6 8 4 2 2 8 2 4 6 8 6 4 4 4 4 8 8 10 4 6 2 2 4 0 0

7 4 2 0 2 0 0 0 0 6 2 6 0 6 0 2 4 4 2 2 0 0 2 0 0

8 0 0 0 0 0 0 0 0 0 0 2 0 0 2 0 2 0 0 0 0 0 0 0 0