A self-adaptive tube restraint for free-energy calculations along path collective variables

Path collective variables reduce a high-dimensional transition to a progress coordinate, s, and an orthogonal distance, z. Computing the free energy along s often requires restraining z, so that sampling embraces a tube centered on the reference path. However, a conventional fixed half-harmonic wall demands an a priori wall position that is both system dependent and s-dependent. Too tight a tube biases the path-projected free energy. Conversely, too loose a tube induces numerical instability and inter-channel leakage. We present an adaptive tube restraint, the wall position of which evolves on the fly to track a fixed contour ΔF∗ of the orthogonal free energy F(z∣s), widening automatically in flat basins and narrowing at pinched saddles, using a user-selected transverse energy scale. We prove that, for a locally harmonic perpendicular well, a contour-following tube captures an s-independent fraction of the orthogonal partition function, so that in the hard-wall limit, its bias cancels out. In stark contrast, the bias of a fixed-width tube varies with s and distorts the free-energy landscape. We probe the algorithm on several model potentials, on N-acetyl-N'-methylalanylamide isomerization, on the folding of the mini-protein chignolin, and on the folding-upon-binding of the ribose-binding protein. The method is implemented in the open-source Colvars library and is, therefore, usable within popular MD engines, such as NAMD, LAMMPS, GROMACS, and Tinker-HP.