J. Phys. Chem. 1995, 99, 2238
2238
Reply to Comment on “Monte Carlo Simulation of a First-Order Transition for Protein Folding” Ming-Hong Hao and Harold A. Scheraga” Baker Laboratory of Chemistry, Come11 University, Ithaca, New York, 14853-1301 Received: October 5, 1994
Berg et a1.I point out that the “multicanonical Monte Carlo” (MMC)*s3 method is equivalent to that of “entropy sampling Monte Carlo” (ESMC).4 First of all, it is clear (see ref 4)that both MMC2%3 and ESMC4 were developed to carry out MC simulations with a uniform probability distribution. Therefore, it is not surprising that these two methods are mathematically equivalent. However, the formulation of ESMC differs from that of MMC in a significant way that sheds lights on the generalization of the method. This difference is the basis of our choice of the ESMC method in our work.5 In the MMC method,’-3 the weight factor for the updating of configurations is defined by two parameters (see eq 5 of ref 1): @(E), which is interpreted as an “effective” inverse temperature, and a(E),which is a free parameter. Theoretically, @(E) is defined as the derivative of the entropy (or logarithm of the density of states) with respect to the energy. In simulations, these two parameters can be obtained by iteration with eqs 10 and 6 of ref 1, respectively. The rationale for this formulation is that, as stated by Berg et ale,’ “this choice is particularly appealing when one wants to combine canonical and multicanonical simulations”. This argument is valid for cases in which the density of states of the system is known or can be obtained without much expenditure of computer time because only then can the multicanonical temperatures be readily defined, and multicanonical Monte Carlo then has a meaning. But, there are cases in which the determination of the density of states is the major problem, and the multicanonical temperature cannot be defined until the simulation has converged. For example, the determination of the density of states of a protein is a very difficult problem. In fact, the major goal of our work is to determine the density of states of proteins5 In our computations, over 80% of the simulations were carried out with a distribution of states that is very different from the correct density of states. In this situation, the calculated parameter P(E), according to the MMC method,’ has little relation to the multicanonical temperature of the protein; it is, therefore, quite difficult to relate such simulations to multicanonical Monte Carlo. A generalization of the method is needed. The ESMC method4 provides a natural way to characterize such simulations. Following the formulation of the ESMC method, the probability, P(E), of occurrence of states at energy level E can be defined as
P(E) = N(E>e-J‘Q
(1)
where N(E) is the density of states, In the simulations, the histogram H(E) is collected, and the function J(E) is updated by
Jol,(E) Jnew(E)= Jold(E)
{
+ In H(E)
if H(E)> 0 if H(E) = 0
(2)
In this formulation, the function J(E) need not be related to the properties of the system; J(E) is simply a scaling function. The key property of the above formulation that is useful for simulating proteins is that the sequence of iterations based on eqs 1 and 2 moves toward a fixed point where the probability of occurrence, P(E), becomes uniform. This can easily be demon~trated.~ When the convergent condition is attained, P(E) = constant; then, from eq 1, one obtains the desired density of states by the relation
N(E) = constant x exp[Jco,,(E)l
(3)
In essence, we have used the ESMC formulation as a thermodynamic scaling technique in our determination of the density of states of proteins; this has been pointed out in our paper? Designating such a simulation as multicanonical Monte Carlo would miss an essential point. The ESMC method provides the natural way to describe the problem of interest to us. Of course, the MMC method could be interpreted in a similar way provided that both a(E)and P(E) are interpreted as free parameters instead of as the multicanonical temperature. But, then, the choice of the two parameters, one of which involves a derivative of the scaling function, to represent the scaling function becomes obscure. The formulation of ESMC is obviously simpler than that of MMC. For these reasons, we chose the ESMC method in our work.5 Even though the ESMC method was developed4 on the basis of a similar idea as that of MMC,’-3 it is more convenient to start with the ESMC formulation in order to develop a more general algorithm. Finally, as for our “failure” to cite ref 6 in our paper: ours was submitted before ref 6 appeared in the literature; therefore, we were unable to cite it. After seeing ref 6, we did cite it in our two follow-up paper^.^.^
References and Notes (1) Berg, B. A.; Hansmann, U. H. E.; Okamoto, Y. J . Phys. Chem. 1995,99, 2236. (2) Berg, B. A,; Neuhaus, T. Phys. Rev. Left. 1992,68, 9. (3) Berg, B. A,; Celik, T. Phys. Rev. Lett. 1992,69, 2292. (4)Lee, J. Phys. Rev. Lett. 1993,71, 211;Erratum 1993,71, 2353. (5) Hao, M.-H.; Scheraga, H.A. J . Phys. Chem. 1994,98, 4940. (6) Hansmann, U.; Okamoto, Y . J . Compur. Chem. 1993,14, 1333. (7) Hao, M.-H.; Scheraga, H. A. J . Phys. Chem. 1994,98, 9882. (8) Hao, M.-H.; Scheraga, H. A. J . Chem. Phys., in press.
JP9427239
0022-3654/95/2099-2238$09.00/0 0 1995 American Chemical Society
