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The Wang-Landau method estimates the relative density of states (DOS) by performing random walk in energy space. However, estimation of the DOS near the ground state minimum is highly challenging because of the dearth of states in the low-energy region compared to that at the high-energy region. Ideally the derivative of the logarithm of the DOS with respect to energy, which is proportional to the inverse of temperature, should become steeper with decrease in energy. However, in actual estimation of the DOS for molecular systems, it is nontrivial to achieve this. In the current work, the accuracy of the Wang-Landau method in estimating the DOS near the ground state minimum is investigated for two peptides, Met-enkephalin and (Alanine)_{5}. It has been found that the steepness of the DOS can be achieved if the correct ground state energy is found, the bin used to discretize the energy space is extremely small (0.1 kcal/mol was used in the current case) and the energy range used to estimate the DOS is small. The findings of this work can help in devising new protocols for calculating the DOS with high accuracy near the ground state minimum for molecular systems.

The density of states (DOS) is a fundamental quantity to understand the thermodynamic properties of a system. For any molecular system except for the smallest ones, calculation of the DOS is an extremely difficult problem because of the high dimensionality of the system. The Wang-Landau (WL) technique is one popular method to estimate the DOS [

In the current work, we have thoroughly investigated a continuous system, Met-enkephalin, which is a peptide containing five amino acids with the sequence Tyr-Gly-Gly-Phe-Met. Met-enkephalin is often used as a model system for testing performance of new simulation techniques [_{5} by gradually increasing the lowest energy range as done for Met-enkephalin. From the results of this work, we have found that unless the system ground state or a structure very close to the ground state is found it is difficult to get a steep derivative of the log of the DOS. Even for an error of 1 kcal/mol in finding the ground state, the slope of the derivative deviates significantly. The energy range used, that is, the lid value used to contain the DOS, is also important to get proper sampling of low energy states. A very fine bin size (of the order of 0.1 kcal/mol) is required to get high accuracy estimation of the DOS near the ground state energy.

This paper is arranged in the following way. Next section discusses the WL method. Section three describes the details of simulation. Results and discussions are given in section four. The paper ends with a conclusion.

The essence of the WL method is given briefly in this section. For details, the readers are referred to the original literature [^{−1}

Both the peptides Met-enkephalin and (Alanine)_{5} were represented with the ECEEP/3 force field, which has electrostatics, Van der Waals, hydrogen bonding, and torsional terms [_{5} was determined by doing initial set of WL calculation, where lowest energy range was decreased continuously to search for the lowest energy structure. The error bars in the DOS were taken as the standard deviations of the DOS calculated over multiple trajectories.

At first we investigated the difference between the two flavours of the WL method, the original WL implementation and the ^{−1}^{−1}^{−1}^{−1}^{−8} as the convergence threshold in a Xeon@2.7 GHz machine. We have found that the DOS calculated by the original WL method converges smoothly with the lowering of the convergence parameter ^{−1}^{−1}^{−1}^{−7} to 1 * 10^{−7}. For the ^{−8} it decreased even more but increased for the ^{−8}^{−1}^{−1}^{−1}

(a) The variation of the DOS for Met-enkephalin for different ^{−1}

Next we investigated the dependence of the DOS on the bin size. Figure

The DOS calculated for Met-enkephalin from its ground state energy with three bin sizes.

One question is how the steepness of the derivative of the log(DOS) with respect to energy changes as we move the lowest energy range gradually from the ground state minimum to higher energy value. This might be important in general investigation of protein systems, where knowledge of global minimum is not always present. Figure _{e}(DOS) changes as a function of shifting the lowest energy state. All the calculations were done for 1 kcal/mol range with 0.1 kcal/mol as the bin width. Five different trajectories were run to calculate the error bars for each energy range. As we shift the lowest energy range from the known lowest energy state of −12.4, the steepness monotonically decreases. We also found that the DOS has maximum increase for the first energy bin of 0.1 kcal/mol from the ground state. The rate of change of the DOS with energy decreases as we sample higher energy bins. The findings from this plot indicate that if the ground-state energy is not known for the system, it will be difficult to get steep derivative of the entropy with respect to the energy.

The DOS calculated for Met-enkephalin by shifting the lowest energy range of energy. All the calculations are done with 0.1 kcal/mol bin size.

Finally we have focused on how the log_{e}(DOS) changes as we shift the maximum of the energy range (or the lid value in the language of reference [_{e}(DOS) for first 0.9 kcal/mol energy to a straight line for both the plots. It was found from the fitted straight lines that calculations done with the smaller energy range gave a slightly higher gradient than that came from the larger energy range.

(a) The DOS calculated for Met-enkephalin for 0.9 kcal/mol energy range with 0.1 kcal/mol bin size. (b) The DOS calculated for Met-enkephalin for 4.9 kcal/mol energy range with 0.1 kcal/mol bin size.

To reconfirm our findings for Met-Enkephalin, we have performed the estimation of the DOS at low energy for another system, (Alanine)_{5}. The steepness of the DOS for 1 kcal/mol range by shifting the energy range is shown in Figure

The DOS calculated for (Alanine)_{5} by shifting the lowest energy range of energy. All the calculations are done with 0.1 kcal/mol bin size.

From the results of this work, we can qualitative say that to get high-accuracy DOS near the ground-state energy of small peptides, the knowledge of ground state minimum or a structure very close to the ground state is necessary. Moreover, the calculation should be done with very fine bin size. The energy range has an effect on the steepness of the derivative but the difference is not significant for Met-enkephalin. The conclusions that came from this work should be applicable to other molecular systems as well. However, the results of this work should be taken as preliminary, as investigations on more systems are required to get a quantitative understanding.

Poor sampling of the DOS near the ground state of a molecular system by the WL method is well known. In the current work, we have investigated the accuracy of the WL method to get highly accurate DOS for a peptide, Met-Enkephalin, near its ground-state minimum. The findings for Met-enkephalin was reconfirmed by calculating the DOS for (Alanine)_{5}. Our results indicate that it is necessary to know the ground-state minimum or a structure very close to the ground-state minimum of the system and only stringent convergence criteria can increase the sampling efficiency of the DOS at low-energy region. The dependence on bin size, energy range to use for the DOS estimation has also been investigated in this work. However, it is to be mentioned that in general for larger peptides and proteins, knowledge of ground-state energy minimum is rarely known. The steepness of the DOS can be used to check if the system is close to the ground-state minimum albeit with increasing computational cost.