Modelling universality and scaling

Abstract

Of the equations that have been used to describe ontogenetic growth in terms of the rate of increase in mass, m, as a function of time, t, most are merely statistical descriptions with no mechanistic basis. Our model¹ is derived from fundamental biological and physical principles and relates growth to metabolic power at the cellular level. It is based on the allocation of resources to the maintenance and replacement of existing tissue and the production of new tissue, with the whole-body metabolic rate B = N_cB_c + E_c(dN_c/dt), where B_c is the cellular metabolic rate, E_c is the energy needed to create a cell, and N_c is the total number of cells. As m = N_cm_c, where m_c is the average cell mass, this gives

where a = B₀m_c/E_c, b = B_c/E_c, β = 1 and α is the allometric exponent for B (≡B₀m^α), taken to be 3/4 in accordance with a large body of data and with theoretical arguments^2,3. The asymptotic mass, for which dm/dt = 0, is predicted to be M = (B₀m_c/B_c)⁴.