^{**}=7/3ψ

# The cell, initial metabolism, and minimum mass

Small self-replicating cells are selected by a mass dependent metabolism

Let the origin of life be replicating molecules with no internal metabolism, practically no mass, and a replication that is driven directly by an extrinsic source of energy. This replicator may evolve into a small self-replicating cell by the evolution of an internal metabolism that is able to generate the essential net energy that is required for self-replication.

To show this we note that the initial metabolism of the self-replicator is dependent upon the mass of the molecules in the metabolic pathways, and on the mass of the heritable code that codes for the self-replicator and its metabolic pathways. An organised metabolism is also dependent on a compartment - like a cell – where the metabolic molecules can concentrate. Hence, when comparing across variants with similar mass specific metabolism (w_{|β}) we find that the quality-quantity trade-off

∂ r / ∂ ln w_{|β} = - 1

is selecting a partial attractor (w_{β}^{*}) that has a minimum cell with the minimum mass that is required to sustain the given mass specific metabolism (β).

When looking along these partial attractors with minimum masses we find a rate of mass specific metabolism that increases with an increase in minimum mass

∂ ln β / ∂ ln w_{β}^{*} = β_{β}^{•}

with the degree of increase in metabolism (β_{β}^{•}) declining from an initial value (β_{β,0}^{•}) towards zero as larger cells have more fully developed metabolic pathways.

Now with fitness given as r = ln [p α β τ / w], and ∂ r / ∂ ln β = 1, we may combine the dependence of metabolism on minimum mass with the selection of the quality-quantity trade-off

∂ r / ∂ ln w_{β}^{*} = [ ∂ r / ∂ ln β ] [ ∂ ln β / ∂ ln w_{β}^{*} ] + ∂ r / ∂ ln w_{β}^{*}

and obtain an overall selection of mass

∂ r / ∂ ln w_{β}^{*} = β_{β}^{•} - 1

Consider now a line of replicating molecules where the initial dependence of mass specific metabolism on mass is stronger than linear [ β_{β,0}^{•} > 1 ]. The selection will then proceed towards an equilibrium [Fig. 1, ∂ r / ∂ ln w_{β}^{*} = 0 ], where the dependence of mass specific metabolism on mass has declined to a linear dependence [ β_{β}^{•} = 1 ]. These selected self-replicating cells are so small that they will have an underdeveloped metabolism in the sense that it is biochemically possible to increase the metabolic efficiency per unit mass with an increase in mass.

Evolution will take a different direction in a line of replicating molecules that have a biochemical constellation where the initial mass specific metabolism is increasing sub-linearly with mass [ β_{β,0}^{•} < 1 ]. There will then be no selection of a metabolism, mass, and cell; but only a sustained selection for the evolutionary maintenance of replicating molecules with no internal metabolism and practically no mass.

### References

- Witting, L. 2016. The natural selection of metabolism and mass selects lifeforms from viruses to multicellular animals. Preprint at bioRxiv http://dx.doi.org/10.1101/087650.