Empirical lognormality of biological variation: implications for the 'zero-force evolutionary law'

Abstract

The zero-force evolutionary law (ZFEL) of McShea et al. states that independently evolving entities, with no forces or constraints acting on them, will tend to accumulate differences and therefore diverge from each other. McShea et al. quantified the law by assuming normality on an additive arithmetic scale and reflecting negative differences as absolute values, systematically augmenting perceived divergence. The appropriate analytical framework is not additive but proportional, where logarithmic transformation is required to achieve normality. Logarithms and logarithmic differences can be negative but the proportions they represent cannot be negative. Reformulation of ZFEL in a proportional or geometric reference frame indicates that when entities evolve randomly and independently, differences smaller than any initial difference are balanced by differences larger than the initial difference. Total variance increases with each step of a random walk, but there is no statistical expectation of divergence between random-walk lineages.