Extended Hamiltonian approach to continuous tempering

We introduce an enhanced sampling simulation technique based on continuous tempering, i.e., on continuously varying the temperature of the system under investigation. Our approach is mathematically straightforward, being based on an extended Hamiltonian formulation in which an auxiliary degree of freedom, determining the effective temperature, is coupled to the physical system. The physical system and its temperature evolve continuously in time according to the equations of motion derived from the extended Hamiltonian. Due to the Hamiltonian structure, it is easy to show that a particular subset of the configurations of the extended system is distributed according to the canonical ensemble for the physical system at the correct physical temperature.

Figure 1

Shown on the left is the system under consideration in its typical double tetrahedron configuration. On the right is the interparticle distance distribution. The reference distribution, computed as described in the text, is compared to the distribution of distances between single pairs obtained using our method. The inset shows, in black, the distribution of the additional degree of freedom $\xi$. The function $f\left(\xi \right)$ is shown in blue. The dashed red lines delimit the range of $\xi$ for which $f=0$. Configurations of the physical system corresponding to these values are canonically distributed at the physical temperature.

Figure 2

Shown on top is the free energy of alanine dipeptide as a function of the $\phi$ and $\psi$ dihedral angles obtained with our tempering scheme. The bottom is the same quantity obtained from four independent molecular dynamics simulations, in total $5.64\mu \mathrm{s}$ long, using the integrator and the Nosé-Hoover thermostat natively implemented in NAMD-Lite. Both plots were obtained by dividing the $(\varphi ,\psi )$ space into $70\times 70$ bins and then applying a smoothing with a width of three bins.

Figure 3

Shown on top is the autocorrelation function of $\xi$ and of the end-to-end distance (black and blue lines, respectively). The estimated autocorrelation time is less than $100\mathrm{ps}$ in both cases. The lower inset shows, in black, the evolution of the end-to-end distance during the first $400\mathrm{ns}$ of simulation and in red the same quantity for the subset of configurations subject to the condition $\left|\xi \right|<\mathrm{\Delta }$. Only the latter were used to compute the free-energy profile shown in the top inset. The bottom shows the deca-alanine molecule in its folded helical state. A dashed gray line connects the two extremal carbonyl carbon atoms used to compute the end-to-end distance.