Effizientes Abtasten der Gleichgewichtsensembles gelöster Polypeptide: Neue Methoden und ihre Anwendungen

Replica exchange (RE) molecular dynamics (MD) simulations are frequently applied to sample the folding–unfolding equilibria of b-hairpin peptides in solution, because efficiency gains are expected from this technique. Using a three-state Markov model featuring key aspects of b-hairpin folding we show that RE simulations can be less efficient than conventional techniques. Furthermore we demonstrate that one is easily seduced to erroneously assign convergence to the RE sampling, because RE ensembles can rapidly reach long-lived stationary states. We conclude that typical REMD simulations covering a few tens of nanoseconds are by far too short for sufficient sampling of b-hairpin folding–unfolding equilibria. 2008 Elsevier B.V. All rights reserved. Replica exchange molecular dynamics (REMD) is considered to be a method for the efficient canonical sampling of biomolecular properties. Therefore, the REMD simulation technique has been frequently applied to generate equilibrium ensembles of biomolecules in solution [1–12]. Here, systems of particular interest were peptides folding into a b-hairpin. Reported simulation times covered the range from a few nanoseconds up to a few tens of nanoseconds [9–12]. As an important result, free energy differences DF between various conformational states were given [9–11]. A reliable calculation of these values requires that the REMD simulations generate approximately the associated conformational equilibrium ensembles. It is not easy to judge whether a necessarily finite simulation time was actually long enough to sample the most important regions of conformational space. As an indication of convergence one may take the observation that the calculated DF values do not change too much upon a substantial elongation of the simulation time. Particularly for a b-hairpin peptide, which folds on time scales of ls at ambient temperatures [13,14], one may ask whether the equilibrium folding–unfolding ensemble can be sampled by REMD within simulation times spanning a few tens of nanoseconds. In this Letter, we will address this question by constructing a Markovian three-state model for b-hairpin folding–unfolding and by executing extended replica exchange Monte Carlo (REMC) simulations for this model. In addition, we check to what extent replica exchange can help to speed up the sampling in the given case. Our three-state model system consists of a folded state f, a transition state t, and an unfolded state u. For a b-hairpin peptide in solution, f will cover a small and compact region in the peptide’s configurational space. Correspondingly, f is characterized by a small entropy Sf and a low energy Uf, which accounts for the hydrogen bonds linking the b-strands. We assume that the configurational space covered by the transition state t is not much larger. Thus, St is also small. However, the energy Ut must be higher because some inter-strand hydrogen bonds will be broken. The unfolded state u is extended and highly flexible with a large entropy Su and a large energy Uu because of the broken hydrogen bonds. The concepts sketched above can be immediately transferred into the set-up of a three-state Markov model for b-hairpin folding–unfolding. Table 1 lists our choice for the model parameters Si and Ui, i 2 ff; t; ug f1;2;3g. According to this choice, the free energies F f and Fu are equal at the temperature T 1⁄4 T0. Note that the equilibrium probability PiðTÞ of a state i at temperature T is expð F i=kBTÞ=1⁄2 P k expð Fk=kBTÞ , where kB is the Boltzmann constant. Consequently, PfðT0Þ equals PuðT0Þ. Fig. 1 specifies the structure of the transition matrix b T 1⁄4 ðtijÞ defining the Markov model and illustrates the energetics of the model at a temperature T > T0. Direct transitions from f to u and vice versa are forbidden. Therefore, t13 and t31 vanish. Furthermore, we have chosen t22 1⁄4 0 implying that the transition state t cannot be occupied for more than one Markov step. We have chosen t12 1⁄4 t32 1⁄4 1=2, which is plausible for all temperatures T > ð2=3ÞT0 because at these temperatures F f and Fu are smaller than Ft (cf. Table 1). We assume detailed balance implying that t21Pf 1⁄4 t12Pt and t23Pu 1⁄4 t32Pt, where Pf, Pt, and Pu are the equilibrium probabilities of the states i. One gets t21 1⁄4 exp1⁄2ðF f FtÞ= kBT =2 and t23 1⁄4 exp1⁄2ðFu FtÞ=kBT =2. b T becomes a stochastic matrix for t11 1⁄4 1 t21 and t33 1⁄4 1 t23. The associated stationary distribution is then ðPf; Pt; PuÞ T at all T > 2T0=3. 0009-2614/$ see front matter 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2008.04.114 * Corresponding author. Fax: +49 89 2180 9220. E-mail address: [email protected] (P. Tavan). Chemical Physics Letters 458 (2008) 244–248