Changes of the power coefficient in the ‘metabolism  /mass ’ relationship in the evolutionary process of animals Atanas T. Atanasov a , Borislav D. Dimitrov b,c, * a Department of Biophysics, Faculty of Medicine, Thracian University, Stara Zagora 6000, Bulgaria b Department of Social Medicine, Medical University, Plovdiv 4000, Bulgaria c Laboratory of Biostatistics, Clinical Research Center for Rare Diseases ‘Aldo e Cele Dacco’, Mario Negri Institute for Pharmacological Research, Villa Camozzi, 24020 Ranica (BG), Italy Received 27 April 2001; received in revised form 28 October 2001; accepted 29 April 2002 Abstract The power coefficient k decreases along evolution in an allometric relationship between the oxygen consumption rate and the body mass of animals. This theoretical study investigated the role of the power coefficient k and its behavior along evolution. The animals were organized in three groups according to the values of the power coefficient k as follows: (I) from unicellular Prokaryotes to Eukaryotes ; (II) from Mytilus and Annelida to Pisces; (III) from Reptilia to Mammals and Aves . At the beginning of each animal group (stage), the value of k was close to 0.9  /1.0 and at the end of the stage it was close to 0.67  /0.70. Exponential sharp increase of the power coefficient k was observed during the biological transition from Protozoa to simply organized Metazoa and in the transition from Poikylothermic to Homothermic organisms (e.g. from Pisces to Reptilia). Also, when using the periodogram regression analysis, a cyclic (periodic) pattern in this increase was observed (i.e. period T :/8  /11 units, P B/0.05). It was postulated that the power coefficient k , as with the coefficient a , might represent the increase of complexity of animal organization within each group. # 2002 Elsevier Science Ireland Ltd. All rights reserved. Keywords: Relationship ‘metabolism /mass’; Evolution of animals; Cyclicity; Transformations 1. Introduction The energetics of organisms is strongly depen- dent upon their mass (Brody, 1945; Kleiber, 1961; Prosser, 1977; Schmidt-Nielsen, 1984; Gillooly et al., 2001). The relationship between the mass W of an organism and the rate of metabolism Vo 2 (quantity of oxygen consumed per unit of time ), or the intensity of metabolism vo 2 (quantity of oxygen consumed by unit of mass per unit of time ), is represented by the allometric dependence: Vo 2 / aW k and vo 2 /aW k1 , where a and k are coefficients being characteristic for each group of animals. The coefficient a represents the rate of metabolism of a conventional organism with a mass equal to 1 kg. It is used for comparing the standard metabolism of organisms with different * Corresponding author. Tel.:  /39-035-511-111; fax:  /39- 035-514-503 E-mail addresses: dimitrov@marionegri.it, bdd11@yahoo.com (B.D. Dimitrov). BioSystems 66 (2002) 65  /71 www.elsevier.com/locate/biosystems 0303-2647/02/$ - see front matter # 2002 Elsevier Science Ireland Ltd. All rights reserved. PII:S0303-2647(02)00034-5