Computer simulation of the free energy of peptides with the local states method: Analogues of gonadotropin releasing hormone in the random coil and stable states

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Abstract
The Helmholtz free energy F (rather than the energy) is the correct criterion for stability; therefore, calculation of F is important for peptides and proteins that can populate a large number of metastable states. The local states (LS) method proposed by H. Meirovitch [(1977) Chemical Physics Letters, Vol. 45, p. 389] enables one to obtain upper and lower bounds of the conformational free energy, FB (b, l) and FA (b, l), respectively, from molecular dynamics (MD) or Monte Carlo samples. The correlation parameter b is the number of consecutive dihedral or valence angles along the chain that are taken into account explicitly. The continuum angles are approximated by a discretization parameter l; the larger are b and l, the better the approximations; while FA can be estimated efficiently, it is more difficult to estimate FB. The method is further developed here by applying it to MD trajectories of a relatively large molecule (188 atoms), the potent “Asp4‐Dpr10” antagonist [cyclo(4/10)‐(Ac‐Δ3Pro1‐D‐pFPhe2‐D‐Trp3‐Asp4‐Tyr‐5‐D‐Nal6‐Leu7‐Arg8‐Pro9‐Dpr10‐NH2)] of gonadotropin releasing hormone (GnRH). The molecule was simulated in vacuo at T = 300 K in two conformational states, previously investigated [J. Rizo et al. Journal of the American Chemical Society, (1992) Vol. 114, p. 2860], which differ by the orientation of the N‐terminal tail, above (tail up, TU) and below (tail down. TD) the cyclic heptapeptide ring. As in previous applications of the LS method, we have found the following: (1) While FA is a crude approximation for the correct F, results for the difference, ΔFA = FA(TD) – FA(TU) converge rapidly to 5.6(1) kcal/mole as the approximation is improved (i.e., as b and l are increased), which suggests that this is the correct value for ΔF; therefore TD is more stable than TU. (The corresponding difference in entrophy. TΔSA = 1.3(2) kcal/mole, is equal to the value obtained by the harmonic approximation.) (2) The lowest approximation, which has the minimal number of local states, i.e., based on b = 0 (no correlations) and l = 1 (the angle values are distributed homogeneously), also leads to the correct value of ΔF, within the error bars. This is important since the lowest approximation can be applied even to large proteins. (3) The method enables one to define the entropy of a part of the molecule and thus to measure the flexibility of this part. We have verified that the results for T[SA(TU) – SA(TD)] of the tail alone converged to 2.4(1) kcal/mole, which demonstrates the relatively high flexibility of the tail in the TU state. In order to study the random coil state, the Asp4‐Dpr10 analogue and its linear version were simulated by MU at 1000 K. We have been able to calculate a lower bound, ∼ 25 kcal/mole for T[S(linear) – S(cyclic)], which is the reduction in the conformational entropy caused by the ring closure. © 1994 John Wiley & Sons, Inc.