The Lagrangian Structure of Long-Time Torsional Dynamics Leading to RNA Folding

Abstract

Within the context of biopolymer renaturation in vitro, a principle of maximization in the economy of the folding process has been previously formulated as the principle of sequential or stepwise minimization of conformational entropy loss (SMEL). When specialized to the RNA folding context, this principle leads to a predictive folding algorithm under the assumption that an “adiabatic approximation” is valid. This approximation requires that conformational microstates be lumped up into base-pairing patterns (BPPs) which are treated as quasiequilibrium states, while folding pathways are coarsely represented as sequences of BPP transitions. In this work, we develop a semiempirical microscopic treatment aimed at validating the adiabatic approximation and its underlying SMEL principle. We start by coarse-graining the conformation torsional space X = 3N-torus, with N = length of the chain, representing it as the lattice (Z ₂)^3N, where Z ₂ = integers modulo 2. This is done so that each point in the lattice represents a complete set of local torsional isomeric states coarsely specifying the chain conformation. Then, a coarse Lagrangian governing the long-time dynamics of chain torsions is identified as the variational counterpart of the SMEL principle. To prove this statement, the Lagrangian computation of the coarse Shannon information entropy σ associated to the specific partition of X into BPPs is performed at different times and contrasted with the adiabatic computation, revealing (a) the subordination of torsional microstate dynamics to BPP transitions within time scales relevant to folding and (b) the coincidence of both plots in the range of folding time scales.