Microscopic dynamics from a coarsely defined solution to the protein folding problem

In this work we introduce a least-action formulation of the protein folding problem casted within a coarse description of the soft mode dynamics of the peptide chain. Ultimately, we show that this coarse variational approach can be lifted to yield the microscopic long-time torsional dynamics responsible for the actual folding process. As a first step, a binary coding of local topological constraints associated to each structural motif is introduced to coarsely mimic the long-time dynamics. Folding pathways are initially resolved as transitions between patterns of locally encoded structural signals. Our variational approach is aimed at identifying the most economic pathway with respect to the stepwise cost in conformational freedom. Our treatment allows us to account for the expediency of the process in proteins effectively capable of in vitro renaturation. We identify the dominant pathway by introducing a coarse version of Lagrangian microscopic dynamics. The coarse folding pathways are generated by a parallel search for structural patterns in a matrix of local topological constraints (LTM) of the chain. Each local topological constraint represents a coarse description of a local torsional state and each pattern is evaluated, translated, and finally recorded as a contact matrix (CM), an operation that is subject to a renormalization feedback loop. The renormalization operation periodically introduces long-range correlations on the LTM according to the latest CM generated by translation. Local topological constraints may form consensus regions in portions of the chain that translate as secondary structure motifs or tertiary interactions. Nucleation steps and cooperative effects are accounted for by means of the renormalization operation, which warrants the persistence of seeding patterns upon successive LTM evaluations. Relevant folding time scales beyond the realm of molecular dynamics simulations become accessible through the coarsely codified representation of local torsional constraints. The validity of our approach is tested vis à vis experimentally probed folding pathways generating tertiary interactions in proteins that may recover their active structure under in vitro renaturation conditions. We focus on determining significant folding intermediates and the late kinetic bottlenecks that occur within the first $\frac{1}{100}\mathrm{s}$ of the renaturation process. After the computational accessibility of this coarse solution of the folding problem becomes apparent, we show how to lift our variational problem to microscopic dynamics of the peptide chain. The consistency of our approach is revealed by an actual generation of Newton’s laws at the microscopic level through an inverse projection of the coarse dynamics originally generated through the pattern recognition computation.