Hierarchical Autocatalytic Systems as a Bridge between Maximum Entropy Production and Bayesian Posterior Contraction: A Numerical Study with Stochastic-Thermodynamic Bounds

Abstract

We construct a three-layer reaction-diffusion model of an autocatalytic chemical system in which raw molecules ($a_i$), catalytic proteins ($p_l$) and large RNA/protein ``genes'' ($W_p^{(k)}$) interact through a mass-action stoichiometry tensor ${\mathrm{Coef}}_{ijk}$ whose magnitude is modulated by the fold-stable activity of the largest polymers.Mass-action is broken by an $\epsilon$-noise term so that the system is nonequilibrium. We compute the total entropy production $\sigma (t)$, the genetic Shannon entropy $S_{\mathrm{gene}}$ and the thermodynamic uncertainty relation (TUR) and thermodynamic speed limit (TSL) bounds on growth and evolution rates. The hierarchical model exhibits the expected co-occurrence of ${\sigma }_{\mathrm{env}}\uparrow$ and $S_{\mathrm{gene}}\downarrow$ predicted by Schrodinger's negentropy argument and reformulated as maximum-entropy-production-principle (MEPP)-driven adaptation. In contrast to a single kinetic-proofreading-like cycle, whose TUR products of $\sim 5$, matching the experimentally reported regime of the ribosome.The hierarchical model's TUR product sits $10^4$-$10^5$ above the universal bound of 2, and the TSL ratio sits $10^6$-$10^8$ above its bound of 1. And scaling number of molucules leaves the looseness intact for the hierarchical model but tightens it monotonically with particle number for the minimal model. We close by drawing an explicit correspondence between the autocatalytic system and diffusion-model training: $a_{\mathrm{ext}}\to a$ flux $\iff$ data-information flow, $\tanh (\beta Wp-\theta )$ $\iff$ score network, replication noise $\iff$ forward-diffusion noise, $S_{\mathrm{gene}}\searrow$ $\iff$ $H[q(\theta |\mathcal{D})]\searrow$. All code and figures are available https://github.com/xiangze/DiverseCells/Hier_Autocatalysis