Figure 2.

Results from simulations of dialanine peptide in SPC/E water used for the validation of the sampling along $N_R$ multiple biasing increments $\nabla H{\left(B\right)}_{ab}$ ($d{L}_{a{b}_{ik}}\left(t\right)$, adaptive bias MD) and $\nabla H{\left(B\right)}_{\sigma }$ ($d{L}_{{\sigma }_{ik}}\left(L_{ik}\left(t\right)\right)$, path-sampling) to accelerate the sampling, which results in a biased action integral $L_s$ and a modification of the un-biased Hamiltonian $H\left(A\right)$, which results in a hybrid Hamiltonian $H\left(C\right)$. (a) Free energy landscapes as function of backbone dihedral angles $\mathrm{\Phi }$ and $\mathrm{\Psi }$ from simulations at the temperatures 300 K, 350 K, 400 K, and 450 K using multipe path sampling in combination with optimization techniques (using principal components of the biasing Hamiltonian $H\left(B\right)$), a number of $N_R=10$ biases, ${\beta }_{md}^{\prime }={\beta }^{\prime }={10}^{-3}$, ${\tau }_1$ = 0.5 ps, and ${\tau }_2$ = 0.1 ps. Energy values on the color bar are in units of $k_{B}T$. (b) Comparison of the logarithm of the transition frequency of the dihedral angle $\mathrm{\Phi }$ as function of the inverse temperature (300–450 K) in the biased (red and green curve) and 1 $\mathsf{\mu }$s MD-trajectories (black curve) (blue curve, fit of the biased data to the MD-data for the determination of the linear scaling factor ln($\rho$)). In this Kinetic analysis, we determined an acceleration factor $\rho$ equal to 10.68 by which the algorithm accelerates the sampling of dialanine peptide. (c) Static permittivities $ϵ\left(0\right)$ as function of MD-simulation time using different coupling ${\alpha }^{\prime }$ and ${\alpha }_{md}$-parameters in the individual simulations of SPC/E and TIP3P water compared with conventional MD simulations.