Entropy and Free Energy of a Mobile Protein Loop in Explicit Water

Abstract

Estimation of the energy from a given Boltzmann sample is straightforward since one just has to average the contribution of the individual configurations. On the other hand, calculation of the absolute entropy, S (hence the absolute free energy F) is difficult because it depends on the entire (unknown) ensemble. We have developed a new method called, "the hypothetical scanning molecular dynamics" (HSMD) for calculating the absolute S from a given sample (generated by any simulation technique). In other words, S (like the energy) is "written" on the sample configurations, where HSMD provides a prescription of how to "read" it. In practice, each sample conformation, i is reconstructed with transition probabilities and their product leads to the probability of i, hence to the entropy. HSMD is an exact method where all interactions are considered and the only approximation is due to insufficient sampling. In previous studies HSMD (and HS Monte Carlo -HSMC) has been extended systematically to systems of increasing complexity, where the most recent is the 7-residue mobile loop, 304-310 (Gly-His-Gly-Ala-Gly-Gly-Ser) of the enzyme porcine pancreatic α-amylase modeled by the AMBER force field and AMBER with the implicit solvation GB/SA (paper I). In the present paper we make a step further and extend HSMD to the same loop capped with TIP3P explicit water at 300 K. As in paper I, we are mainly interested in entropy and free energy differences between the free and bound microstates of the loop, which are obtained from two separate MD samples of these microstates. The contribution of the loop to S and F is calculated by HSMD and that of water by a particular thermodynamic integration procedure. As expected, the free microstate is more stable than the bound microstate by a total free energy difference, F free − F bound = −4.8 ± 1, as compared to −25.5 kcal/mol obtained with GB/SA. We find that relatively large systematic errors in the loop entropies, S free (loop) and S bound (loop) are cancelled in their difference which is thus obtained efficiently and with high accuracy, i.e. with a statistical error of 0.1 kcal/mol. This cancellation, which has been observed in previous HSMD studies, is in accord with theoretical arguments given in paper I.

Relative stability of the open and closed conformations of the active site loop of streptavidin

“…(11)] will be too large leading to a too low T S A loop ; as n f increases the entropy increases (due to the decrease in n init ), approaching its correct value (from below) for a large n f where the effect of the initial conditions is gradually eliminated. Note that the required equilibration time is system-dependent, where the present time (16 ps) is significantly larger than the 2.5 ps used in our previous studies, 13,14,26 probably due to the larger loop studied here and its relatively high stability as discussed in Sec. III A.…”

“…. , K /2) can leave the microstate during long MD simulations; such an "overflow" is more likely to happen for small residues such as Gly and for small k.] To control an overflow, we have suggested carrying out the reconstruction in j shorter repetitive "units," 13,14,25 each based on n f < n f conformations where n f = jn f . Using units of increasing length (n f ) and larger values of n f (i.e., larger j) enables one to gain control on the extent of coverage of a microstate by the future simulations (again, very small n f values will lead to undercoverage of the microstate while large n f might lead to an overflow).…”

“…, α 1 )] which indeed has been found in several previous studies. 13,14,23,25 In the second scenario, the future conformation is located within m but n f is too small for adequately calculating a TP [Eq. (11)], i.e., the ratio n visit /n f as yet has not been stabilized.…”

“…Notice that the present trajectories of 2 ns are significantly larger than the ∼0.5 ns trajectories studied in our previous papers. 13,14,[23][24][25][26]54 First, we calculated for each run (trajectory) the RMSD of the loop's conformations from its initial one, and the results are presented in Fig. 4 as a function a α k (min), α k (max), and α k are defined in Eq.…”

“…13 In a subsequent paper, the loop was capped with 70 TIP3P water molecules and an HSMD-TI procedure was developed in which the contribution of water to F is calculated by a TI process, which is more efficient than HSMD. 14 Subsequently, HSMD-TI was applied to the loop 287-290 with the bulky residues, Ile, Phe, Arg, and Phe of acetylcholinesterase (AChE) from Torpedo California; 26 reaction of AChE with the inhibitor diisopropylphosphorofluoridate (DFP) leads to a displacement of the loop's backbone by roughly 4 Å, and experimental evidence suggests that the free-energy penalty for the loop displacement is on the order of 4 kcal/mol (i.e., F free -F bound ∼ −4 kcal/mol). Therefore, AChE has been an ideal system for checking the performance of HSMD-TI, and in particular for examining the minimal number of water molecules needed to cap the loop.…”