^{**}=7/3ψ

# Protozoa - the interacting self-replicating cell

Protozoa are selected from prokaryotes by incomplete population dynamic feed-back selection

With selection for interactive behaviour, the cost gradient (ψ) of interactive competition is increasing. This is generating a bias in the distribution of net energy [ ε (w_{i}/w)^{ψι} with ψι > 0 ] that reflects that the larger (w_{i}) than average (w) individuals monopolise (ψ) the resource to some degree. This emergence of a resource bias selects for larger self-replicating cells with higher rates of mass specific metabolism; as defined by the β-dependent minimum mass that is required to sustain a pre-mass mass specific metabolism that increases sub-linearly with mass [ β_{β}^{•} = 1 - ψι^{*} ]. This selection can continue with a maximal resource bias exponent that increases towards unity [ ψι^{*} → 1 ], causing the evolution of a self-replicating cell with fully developed metabolic pathways and a β_{β}^{•} exponent of zero.

This partial unfolding of population dynamic feed-back selection is illustrated in Fig. 1, right. With a maximal resource bias exponent below unity, the interactive competition is not yet so strong that it can select mass independently of the dependence of mass specific metabolism on mass. The result is a mass that is still being selected as a β-dependent minimum. This implies maximal interference and an incomplete unidirectional feed-back; where interference is influencing the selection of mass, but where the level of interference is unaffected by the selection of mass.

_{β}), generates maximal population growth (r), and a density regulated (γ) abundance (n^{*}) with a maximal resource bias below unity (0 < ψι^{**} < 1). This maximum bias assists in the selection of mass, but it is too weak to generate a complete feed-back, where it is the interactive selection of mass that is determining the level of interference in the population.

_{β}^{•}) for mass specific metabolism that evolves from unity to zero, the predicted allometric exponent for mass specific metabolism b = β_{β}^{•} + β_{w}^{•} is declining from 5/6 (≈0.83) over zero to -1/6 (≈-0.17), because β_{w}^{•}=-1/6 for three dimensional interactions (Witting, 1995, 2016a). This coincides with an empirically estimated exponent that declines from 0.61 over zero to -0.20 across protozoa with an increasing mass (Fig. 1, left).

### References

- DeLong, J.P., J.G. Okie, M.E. Moses, R.M. Sibly and J.H. Brown 2010. Shifts in metabolic scaling, production, and efficiency across major evolutionary transitions of life. Proceedings of the National Academy of Sciences 107:12941--12945.
- Witting, L. 1995. The body mass allometries as evolutionarily determined by the foraging of mobile organisms. Journal of Theoretical Biology 177:129--137.
- Witting, L. 1997. A general theory of evolution. By means of selection by density dependent competitive interactions. Peregrine Publisher, Århus, 330 pp, URL http://mrLife.org.
- Witting, L. 2002. From asexual to eusocial reproduction by multilevel selection by density dependent competitive interactions. Theoretical Population Biology 61:171--195.
- Witting, L. 2016a. The natural selection of metabolism and mass selects allometric transitions from prokaryotes to mammals. bioRxiv http://dx.doi.org/10.1101/084624.
- Witting, L. 2016b. The natural selection of metabolism and mass selects lifeforms from viruses to multicellular animals. bioRxiv http://dx.doi.org/10.1101/087650.