Hitting Times of Some Critical Events in RNA Origins of Life

Can a replicase be found in the vast sequence space by random drift? We partially answer this question through a proof-of-concept study of the times of occurrence (hitting times) of some critical events in the origins of life for low-dimensional RNA sequences using a mathematical model and stochastic simulation studies from Python software. We parameterize fitness and similarity landscapes for polymerases and study a replicating population of sequences (randomly) participating in template-directed polymerization. Under the ansatz of localization where sequence proximity correlates with spatial proximity of sequences, we find that, for a replicating population of sequences, the hitting and establishment of a high-fidelity replicator depends critically on the polymerase fitness and sequence (spatial) similarity landscapes and on sequence dimension. Probability of hitting is dominated by landscape curvature, whereas hitting time is dominated by sequence dimension. Surface chemistries, compartmentalization, and decay increase hitting times. Compartmentalization by vesicles reveals a trade-off between vesicle formation rate and replicative mass, suggesting that compartmentalization is necessary to ensure sufficient concentration of precursors. Metabolism is thought to be necessary to replication by supplying precursors of nucleobase synthesis. We suggest that the dynamics of the search for a high-fidelity replicase evolved mostly during the final period and, upon hitting, would have been followed by genomic adaptation of genes and to compartmentalization and metabolism, effecting degree-of-freedom gains of replication channel control over domain and state to ensure the fidelity and safe operations of the primordial genetic communication system of life.

Asymptotic distribution of hitting times for critical maps of the circle

It is well known that the renormalization group transformation $\mathcal{R}$ has a unique fixed point $f_{cr}$ in the space of critical $C^{3}$-circle homeomorphisms with one cubic critical point $x_{cr}$ and the golden mean rotation number $\overline{\rho}:=\frac{\sqrt{5}-1}{2}.$ Denote by $Cr(\overline{\rho})$ the set of all critical circle maps $C^{1}$-conjugated to $f_{cr}.$ Let $f\in Cr(\overline{\rho})$ and let $\mu:=\mu_{f}$ be the unique probability invariant measure of $f.$ Fix $\theta \in(0,1).$ For each $n\geq1$ define $c_{n}:=c_{n}(\theta)$ such that $\mu([x_{cr},c_{n}])=\theta\cdot\mu([x_{cr},f^{q_{n}}(x_{cr})]),$ where $q_{n}$ is the first return time of the linear rotation $f_{\overline{\rho}}.$ We study convergence in law of rescaled point process of time hitting. We show that the limit distribution is singular w.r.t. the Lebesgue measure.