Solitons and Collapse in the lambda-repressor protein

TL;DR

This paper analyzes the structure, stability, and folding pathways of the lambda-repressor protein, revealing novel soliton-based modeling of its loops and demonstrating the importance of soliton formation order in proper folding.

Contribution

It introduces a soliton Ansatz for modeling protein loops and explores the folding dynamics, highlighting the sequence of soliton formation as crucial for correct protein structure.

Findings

Soliton modeling accurately describes protein loops beyond experimental fluctuation limits.

Folding pathways are dominated by the order of soliton formation, especially the third soliton.

The folded configuration is stable under heating and cooling simulations.

Abstract

The enterobacteria lambda phage is a paradigm temperate bacteriophage. Its lysogenic and lytic life cycles echo competition between the DNA binding $\lambda$-repressor (CI) and CRO proteins. Here we scrutinize the structure, stability and folding pathways of the $\lambda$-repressor protein, that controls the transition from the lysogenic to the lytic state. We first investigate the super-secondary helix-loop-helix composition of its backbone. We use a discrete Frenet framing to resolve the backbone spectrum in terms of bond and torsion angles. Instead of four, there appears to be seven individual loops. We model the putative loops using an explicit soliton Ansatz. It is based on the standard soliton profile of the continuum nonlinear Schr\"odinger equation. The accuracy of the Ansatz far exceeds the B-factor fluctuation distance accuracy of the experimentally determined protein configuration. We then investigate the folding pathways and dynamics of the $\lambda$-repressor protein. We introduce a coarse-grained energy function to model the backbone in terms of the C$_\alpha$ atoms and the side-chains in terms of the relative orientation of the C$_\beta$ atoms. We describe the folding dynamics in terms of relaxation dynamics, and find that the folded configuration can be reached from a very generic initial configuration. We conclude that folding is dominated by the temporal ordering of soliton formation. In particular, the third soliton should appear before the first and second. Otherwise, the DNA binding turn does not acquire its correct structure. We confirm the stability of the folded configuration by repeated heating and cooling simulations.