Sticking coefficient-orchestrated selection in a kinetic model for the first origin

The mathematical formalism of the steady-state Poisson equation is applied to a variant of Freeman Dyson's "Toy Model" for a first origin. Our kinetic approach allows for an examination of the requisite conditions under which metabolism is quantized into discrete eigenstates (e.g. Dyson's disordered, saddle point, and metabolically active toy cell states). The surface reaction machinery additionally allows for more realistic modeling, whence the crucial role of sticking coefficients (catalyst precursors) as prebiotic selectors emerges. In our interior source model, a steady influx of vent nutrients fuels the intracellular synthesis of (impermeable) monomers within a rock-encradled cavity. Random adsorptions and desorptions occur at inactive "cell" wall sites (where the inert monomers remain impermeable until their eventual return to the intracellular metabolite pool). Occasionally, metabolizing reactions also occur due to endogeneous source monomers adsorbing at their "active" sites. Dyson's mean field approach is used to simplify the species-specific sticking coefficients at empty active (substrate) sites to functions of the fraction x of sites occupied by (catalytically) active monomers. In short, our work suggests that disorder-order transition models based on random drift between discrete metabolic eigenstates (Dyson's Toy Model) do not, in general, extend to more realistic metabolisms. From a perspective based on quasi-random feedback kinetics, the contraindication for discretization (spontaneous generation) in non-autocatalytic metabolisms is consistent with the emergence of ordered metabolism under hydrothermal driving forces, a provisio the occurrence of each period of vent dormancy coincides with a discrete zero-source (dormant) metabolic state. Cell drift to higher order is induced by the random reactions which happen to enhance the substrate specificity (chemical selectivity) of the sticking coefficients for active monomers. The result is stronger sink effects for metabolizing species, whence active adsorptions are promoted in favor of inactive adsorptions at substrate sites. Positive feedback plays a crucial role in preserving ("propagating") order in the cell wall reaction kinetics and is held in check by negative feedback inhibition of excessive cell growth. Finally, the eventual desorption of randomly growing dysfunctional proteins is postulated as a deterrent to deterioration catastrophes.