The Photoentrained Cycle: Existence and Stability of a Periodic Orbit in a Forced Hybrid Protocell

Abstract

The corpus supplies an enclosure, a strand anchored within it, a clock and a readable state. It does not say whether these compose. A cycle is not a set of steps but a closed orbit, and closure is a property of the composition. The question is posed here as the existence of a fixed point of a return map on a periodically forced hybrid system with an impulsive reset at division, and it receives two answers.

The flow is assembled from four parts. The clock is monotone at constant illumination, and the mechanochemical loop that couples it to the membrane is shown to have small gain and a sign that sharpens rather than destabilises, so the system does not oscillate autonomously; what the architecture has is entrainment, and since the cooperative coupling it needs for commitment makes the period exponentially sensitive to that coupling, a free-running clock would be unusable. Separation is a driven front in a quenched sequence landscape: processivity is topological rather than regulatory, the linking number is taken up by lateral sliding of the anchors at negligible cost and only in a fluid regime, and the origin is the low-force end of a compositional gradient rather than a marked site. Polymerisation is entropically catalysed by the scaffold itself, and the requirement that the encoding survive forces re-templating rather than polymerisation from free monomer. Division requires a reduced volume below Q(α)^(-3/2), of which symmetric fission is the most expensive case.

The first answer is unconditional and is the paper's principal result. Every process of the architecture leaves the number of impermeant internal particles unchanged or lowers it, so that number is conserved across a lineage. Each division restores the reduced volume exactly — shape is homeostatic with no feedback and no sensor, the restoration being a property of the reset — while area and particle number both fall. A founder therefore buds some forty times before reaching a size floor, and across a population total membrane grows only as the cube root of cell number. The architecture reproduces and does not grow.

The second answer is conditional. A steady lineage requires the cell to generate impermeant particles at the rate a bud removes them, about six per ten seconds at a bud fraction of a tenth. Under that condition the return map has a fixed point; under supply-limited growth it is a contraction of multiplier one half, so that size is stable and the contraction is supplied by the halving at division rather than by any measurement, and under surface-catalysed growth there is no stable size and the division interval is quantised to an integer multiple of the forcing period. The two growth laws are distinguishable by observation of a population.

Three caps on the length of the sequence carried — an information threshold, a geometric span and a diurnal clock — admit a common band of one hundred to four hundred base pairs, and the only free parameter among them is measurable. What the account does not supply is named rather than assumed: the missing particle source is the first requirement in this arc that is metabolic in character, and it is located not at the genome, the membrane or the clock but at the osmolyte.