Year I: Validation of new method

In order to compare the new HMC algorithm with other methods, it is essential to define a measure of the efficiency and to determine whether a given sampling method is ergodic [102]. If the true normalized probability distribution, $\rho_{exact}(\Gamma) = Z^{-1} exp(-\beta V(\Gamma))$, of the conformational states is known, one can monitor the quantity  [18]

$$\chi^{2}(t)=\int_{-\infty}^{+\infty} d\Gamma[\rho(\Gamma,t)-\rho _{exact}(\Gamma)]^{2},$$

as a function of run length. Here, $\Gamma$ denotes the conformational space. The root mean square deviation should decay to zero if the sampling method is ergodic and if the run length exceeds the mixing time of the sampling algorithm. Then the rate of decay of $\chi(t)$ is a measure of how efficient a simulation method is. In one dimension, a Fourier sum of sine waves can be used to generate a rugged potential surface, so that $\chi(t)$ offers a simple measure of sampling efficiency.

Thirumalai et al.  [89] have introduced an approximate method for determining whether or not a sampling algorithm is ergodic, which also produces a measure of the mixing time. Several ergodic measures, such as a force measure (based on vector force of an atom), and an energy measure (based on the non-bonded energy of an atom) and pair distance measure (based on the pair distance between a certain pair of sites in a protein), can be used. In general, for an interesting quantity of g of the jth atom or jth pair, the average over time for a particular trajectory can be calculated. The mean square difference of these quantities between two separate trajectories can be used as the metric to measure the ergodicity of the sampling method.

We have chosen the following test systems because various sampling methods, including conventional HMC, have been tested on them. Enkephalins are secreted in the brain, mainly from the hypothalamus. Enkephalins are an endogenous opiate. They behave in a similar way to opium in that they have an analgesic effect, thus the similar name. There are opiate receptors throughout the body, in the brain and peripheral tissues. Although it is a short peptide (five amino acids) there are numerous functions: a)acts as a ligand, altering signal transmission at the synapse. b)enhances natural killer cells in an immune response. c)reduces growth and metastases of tumor cells.

The amino acid sequence of Met-enkepalin is: Tyr-Gly-Gly-Phe-Met-OH. It has large torsion angles. Melittin  [88] is a toxin extracted from honey bee venom (PDB ID: 2mlt) consisting of 54 residues. This method will also be applied to the ionic liquid systems that are described in the collaborative applications.