Unfolding Hidden Barriers by Active Enhanced Sampling

Collective variable (CV) or order parameter based enhanced sampling algorithms have achieved great success due to their ability to efficiently explore the rough potential energy landscapes of complex systems. However, the degeneracy of microscopic configurations, originating from the orthogonal space perpendicular to the CVs, is likely to shadow “hidden barriers” and greatly reduce the efficiency of CV-based sampling. Here we demonstrate that systematic machine learning CV, through enhanced sampling, can iteratively lift such degeneracies on the fly. We introduce an active learning scheme that consists of a parametric CV learner based on deep neural network and a CV-based enhanced sampler. Our active enhanced sampling algorithm is capable of identifying the least informative regions based on a historical sample, forming a positive feedback loop between the CV learner and sampler. This approach is able to globally preserve kinetic characteristics by incrementally enhancing both sample completeness and CV quality.

Figure 1

FESs of alanine dipeptide (b) in gas phase with respect to (a) $\mathrm{\Phi }$ and $\mathrm{\Psi }$, (b) RG and NH are shown. Each FES was constructed from configurations sampled by 100 ns WTMs with individual set of CVs. AES was applied on this system with $\mathrm{\Phi }$, $\mathrm{\Psi }$, ${\omega }_1$, and ${\omega }_2$ [inset of (b)] as CVs. Unbiased configurations from all AES iterations are mapped onto the final StKE CVs to generate the FES shown in (c). Configurations from $C{7}_{eq}$, $C5$, and $C{7}_{ax}$ are scattered as cyan, yellow, and magenta dots, respectively. Different $C(t)$ are calculated from WTM trajectories using RG and NH, StKE CVs trained in each AES iteration, and $\mathrm{\Phi }$ and $\mathrm{\Psi }$. Each WTM was 100 ns and samples for calculating $C(t)$ were collected after 50 ns such that biasing potentials were well converged. (f) FES convergence was calculated by unbiased samples from AES, WTM with $\mathrm{\Phi }$ and $\mathrm{\Psi }$, and WTM with RG and NH. The benchmark FES is in (a).

Figure 2

Configurations sampled in AES first and second iterations were mapped onto StKE CVs from iteration 1, generating FES shown in (a). Fixing the StKE CVs at one minimum, the FES along RG is also generated with samples from iteration 1 (blue line) and with samples from both iterations (pink line), as shown in (c). Samples in this minimum with RG values within the green block (“Min 1”) and pink block (“Min 2”) are mapped out in (a) as megenta dots and black dots. It is clear that “Min 2” consists of new states sampled in iteration 2. After training StKE CVs with samples from both iterations, the structures with respect to “Min 2” were separated out, forming different minima on the FES with updated StKE CVs (b).

Figure 3

The number of ${\tilde{\mathbf{s}}}^h$ versus total simulation time is recorded for both AES and WTM in (a). This number approximates the size of sampled conformational space. Panel (b) records the average time in which one structure changes to another with $\mathrm{RMSD}=1\AA$. The average was taken from individual 400 ns WTM with CVs used in each AES iteration. Only the last 200 ns trajectory was used to calculate the average, after biasing potential has converged.